(*,s)-Dualities

Journal of Mathematical Sciences, Vol. 115, No. 4, 2003

(∗, s)-DUALITIES J. Get´ an, J.-E. Mart´ınez-Legaz, and I. Singer

UDC 517.977

We introduce and study (∗, s)-dualities ∆ : AX → AW , where A = (A, ) is a complete lattice, ∗ : A × A → A is a binary operation satisfying 
inf ai ∗ a = inf (ai ∗ a) i∈I

i∈I

for all {ai }i∈I ⊆ A and a ∈ A, s is an antiautomorphism of A, and X and W are two sets. This notion provides us with a general theory which encompasses, as particular cases, our earlier theories of ∗-dualities, where ∗ is a group operation and s(a) = a−1 , ∨-dualities, where A = R, ∗ = ∨, and s(a) = −a, and ⊥-dualities, where A = R, ∗ = ⊥ = the residual of the mapping x → a ∧ x, and s(a) = −a. 1. Introduction It is well known that various concepts of conjugation have important applications, for example, to duality in optimization theory. We recall that if X is a (real) locally convex space with conjugate space X ∗ , then the (usual) Fenchel conjugate of any function f : X → R = [−∞, +∞] is the function f ∗ : X ∗ → R defined by f ∗(Φ) = sup {Φ(x) − f (x)},

Φ ∈ X ∗.

(1.1)

x∈X

This concept has been extended by Moreau (see, e.g., [14] and the references therein) to the case where X, X ∗ , and the bilinear function (x, Φ) → Φ(x) occurring in (1.1) are replaced respectively by two arbitrary sets X and W and an arbitrary function ψ : (x, w) ∈ X × W → ψ(x, w) ∈ R, called a “coupling function,” and the operation − in (1.1) is extended to R in a suitable way. Namely, we recall (see, e.g., [14]) that . the usual addition + on R = (−∞, +∞) admits two extensions to R = [−∞, +∞], + and + . , called upper and lower addition, respectively, defined by

.

a+b=a+ . b = a + b if either R ∩ {a, b} = ∅ or a = b = ±∞,

.

a + b = +∞, a + . b = −∞ if a = −b = ±∞.

(1.2) (1.3)

Then the so-called Fenchel–Moreau conjugate of a function f : X → R associated to the coupling function ψ : X × W → R is the function f ∆ψ (·) : W → R defined by f ∆ψ (w) = sup {ψ(x, w) + . −f (x)}, x∈X

w ∈ W.

(1.4)

An axiomatic approach to this concept has been started in [18] by showing that for a mapping X W X ∆ : f ∈ R → f ∆ ∈ R , where R denotes the set of all functions f : X → R, there exists a uniquely X determined coupling function ψ : X × W → R such that f ∆ = f ∆ψ (of (1.4)) for all f ∈ R if and only Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 92, Optimization and Related Topics–1, 2001.

2506

c 2003 Plenum Publishing Corporation 1072–3374/03/1154–2506 $ 25.00 

if ∆ satisfies the following two conditions: for any index set I (including the empty set ∅, with the usual conventions inf ∅ = +∞ and sup ∅ = −∞), ∆
X = sup fi∆ , {fi }i∈I ⊆ R , (1.5) inf fi i∈I

i∈I

.

∆

(a + f ) = f

∆

X

+ . −a,

f ∈ R , a ∈ R;

(1.6)

in this case, ∆ has been called in [18] a conjugation. X W The mappings ∆ : R → R satisfying condition (1.5), called dualities, were systematically studied X W in [7, 19]. In [8], there have been introduced and studied mappings ∆ : R → R satisfying (1.5) and another “second condition,” namely, X

(f ∨ a)∆ = f ∆ ∧ −a,

f ∈ R , a ∈ R;

(1.7)

such mappings were called in [8] “∨-dualities.” Among other results, it was shown in [8] that a mapping X W ∆ : R → R is a ∨-duality if and only if it admits a representation in the form f ∆ (w) = sup {ψ(x, w) ∨ −f (x)}, x∈X

X

f ∈ R , w ∈ W,

(1.8)

with a uniquely determined coupling function ψ : X × W → R. W X X W The problem of determining the “dual” ∆ : R → R of a duality ∆ : R → R led to the introduction, in [8], of a different class of dualities, called “⊥-dualities,” defined by another “second condition” that involves two further (noncommutative and nonassociative) binary operations on R denoted by ⊥ and  respectively. While these operations will be recalled in Sec. 3 below, let us mention here that X W the “second condition” defining ⊥-dualities ∆ : R → R is X

(f ⊥a)∆ = f ∆  − a, X

→R

and that a mapping ∆ : R

W

f ∈ R , a ∈ R,

(1.9)

is a ⊥-duality if and only if it admits a representation of the form X

f ∆ (w) = sup {−f (x) − ψ(x, w)},

f ∈ R , w ∈ W,

x∈X

(1.10)

with a uniquely determined coupling function ψ : X × W → R. X W A unified approach to conjugations, ∨-dualities, and ⊥-dualities ∆ : R → R , where X and W are two sets, was developed in [10, 13], namely, that of dualities associated to a binary operation ∗ on R, with the property that for any index set I, we have 
(1.11) inf ai ∗ a = inf (ai ∗ a), {ai }i∈I ⊆ R, a ∈ R; i∈I

X

these are the dualities ∆ : R

i∈I

→R

W

satisfying the “second condition”

(f ∗ a)∆ = f ∆ ∗a,

X

f ∈ R , a ∈ R,

(1.12)

where ∗ is the binary operation on R defined by a∗b = −(−a ∗ b),

a, b ∈ R;

(1.13)

they are called [10] “∗-dualities.” Under certain assumptions, ∗-dualities admit a representation of the form f ∆ (w) = sup {ψ(x, w)∗f (x)}, x∈X

X

f ∈ R , w ∈ W,

(1.14)

with a uniquely determined coupling function ψ : X ×W → R, and conversely, under certain assumptions, X W every mapping ∆ : R → R of form (1.14) is a ∗-duality [10]. 2507

This general concept encompasses, as particular cases, the conjugations, ∨-dualities, and ⊥-dualities . X W ∆ : R → R . Indeed, for ∗ = +, one has a∗b = a + . −b,

a, b ∈ R,

(1.15)

and one obtains again the conjugations of (1.4), (1.5), and (1.6); for ∗ = ∨, one has a∗b = a ∧ −b,

a, b ∈ R,

(1.16)

and one obtains again the ∨-dualities of (1.8), (1.5), and (1.7); for ∗ = ⊥, one has a∗b = a − b,

a, b ∈ R,

(1.17)

and one obtains again the ⊥-dualities of (1.10), (1.5), and (1.9). Furthermore, using that (R, ≤), where ≤ is the usual total order on R, is a complete lattice and that + is a group operation on R, in [9] the axiomatic approach (1.5), (1.6) to the Fenchel–Moreau conjugations (1.4) was extended to introducing and studying conjugations for functions with values in “the canonical . enlargement” G = (G, , ⊗, ⊗ . ) of a complete totally ordered group G = (G, , ⊗) (this concept will be recalled in Sec. 3 below), and in [12] the total order  has been replaced by a partial order  and further results on “conjugations” have been obtained in this framework, with some new applications. The main interest in this extension lies in the fact that it contains also the important particular case where G = (G, , ⊗) = (R++ , ≤, ×), where R++ = {d ∈ R | d > 0}, and ≤ and × are the usual total order and the usual multiplication, respectively. In this case,

.

.

G = (G, , ⊗, ⊗ . ) = (R+ , ≤, ×, × . ),

.

where × and × . are, respectively, the “upper” and “lower” Moreau multiplication on R+ = [0, +∞] (see, X

e.g., [14]). In this approach, the “⊗-conjugations” ∆ψ : G

→G

−1 f ∆ψ (w) = sup G {ψ(x, w) ⊗ . f (x) }, x∈X

W

are X

f ∈ G , w ∈ W,

and conditions (1.5) and (1.6) are extended, respectively, to conditions (1.5) (with {fi }i∈I ⊆ G inf and sup taken in the respective complete lattices) and

.

−1 (f ⊗ a)∆ = f ∆ ⊗ . a ,

X

f ∈ G , a ∈ G.

(1.18) X

and with (1.19)

At this point, it was raised in [10] the problem whether there exists a general theory which encom. X W passes, as particular cases, all the above, i.e., ⊗-conjugations ∆ψ : G → G , where G = (G, , ⊗, ⊗ .) is the canonical enlargement of a complete lattice ordered group G = (G, , ⊗), and ∨- and ⊥-dualities X W ∆ : R → R . The main difficulty is that ∨ and ⊥ are not group operations on R ∪ {−∞}; for example, although the neutral element with respect to ∨ is e = −∞ (i.e., a ∨ −∞ = −∞ ∨ a = a for all a ∈ R ∪ {−∞}), no element a ∈ R admits an “inverse” a−1 for ∨ (i.e., such that a ∨ a−1 = a−1 ∨ a = −∞). In particular, the elements −a, −b, and −f (x) occurring in the above formulas involving ∨ or ⊥ are not “inverses” of a, b, f (x) ∈ R with respect to ∨ or ⊥, respectively. Nevertheless, in [10, Sec. 5], it was observed that the answer to the above question is affirmative. Namely, the key observation is that (R, ≤) is a complete lattice and the mapping s : R → R defined by s(a) = −a, 2508

a ∈ R,

(1.20)

is an antiautomorphism of the complete lattice (R, ≤). Therefore, the following general framework has been introduced in [10, Sec. 5]: Let A = (A, ) be a complete lattice, ∗ : A × A → A be a binary operation satisfying
 inf ai ∗ a = inf (ai ∗ a), {ai }i∈I ⊆ A, a ∈ A, (1.21) i∈I

where inf ai = inf i∈I

i∈I

Aa i

i∈I

and s : A → A is a bijection such that 
s inf ai = sup s(ai ) i∈I

i∈I

for every index set I and every family {ai }i∈I ⊆ A (or, equivalently, s is an antiautomorphism of A). Then, one defines in [10] a new binary operation ∗s : A × A → A by a ∗s b = s(s−1 (a) ∗ b),

a, b ∈ A.

(1.22)

Next, given two sets X and W , a mapping ∆ : AX → AW is called a “(∗, s)-duality” [10] if it satisfies (1.5) (with R, sup, and inf replaced by A, supA , and inf A respectively) and ∆(a ∗ f ) = ∆(f ) ∗s a,

f ∈ AX , w ∈ W ;

(1.23)

also, it is natural to expect that, under certain assumptions, the general form of a (∗, s)-duality ∆ : AX → AW will be ∆(f )(w) = ∆ψ (f )(w) = sup A {ψ(x, w) ∗s f (x)}, x∈X

f ∈ AX , w ∈ W,

(1.24)

with a coupling function ψ : X × W → A uniquely determined by ∆. . In particular, for A = G = (G, , ⊗, ⊗ . ), the canonical enlargement of a complete lattice ordered

.

group G = (G, , ⊗), ∗ = ⊗, and s : G → G defined by s(a) = a−1 ,

a ∈ G,

(1.25)

.

the binary operation ∗ = ⊗ satisfies condition (1.21), where inf is taken in the complete lattice G, and (1.22) yields −1 a ∗s b = (a−1 ⊗ . b) ,

a, b ∈ G;

(1.26)

.

s thus, for G = (R, ≤, +, + . ) (hence s(a) = −a, a ∈ R), the binary operation ∗ of (1.26) reduces to ∗

of (1.13). Hence, by the above, the framework of (∗, s)-dualities encompasses, as particular cases, ⊗X W X W conjugations ∆ψ : G → G and ∨-dualities, ⊥-dualities ∆ : R → R , and even the new cases of ∨-dualities and ⊥-dualities ∆ : AX → AW , where A = (A, ) is a complete lattice in which ∗ = ∨ and ∗ = ⊥ satisfy condition (1.21). The importance of these concepts lies in the fact that even when ∗ is not a group operation on A, one can still prove many results using an antiautomorphism s : A → A and the binary operation ∗s of (1.22), which “replaces” the inversion of elements of a group. The fact that s is an antiautomorphism of A captures one of the basic properties of s(a) := a−1 and allows one to develop a rich theory of generalized “conjugation operators.” In [10, Sec. 5], it was stated, without any details, that “some of the results” of [10, Secs. 1–4] on ∗-dualities for (R, ≤, ∗) and ∗ of (1.13) “can be extended to results on (∗, s)-dualities” for (A, , ∗) and ∗s of (1.22). The aim of the present paper is to carry out this program, i.e., to develop a general theory of (∗, s)-dualities. The outline of the paper is as follows. In Sec. 2, we introduce the concepts of upper and lower inverses and s-conjugate of a binary operation ∗ on a complete lattice, where s is a bijection of the complete lattice onto itself. In Sec. 3, we give three main examples of those notions. The first example 2509

refers to the canonical enlargement of a boundedly complete lattice ordered group, while the second and third examples correpond to the cases where the binary operation is the join ∨ and the lower inverse ⊥ of the meet ∧, respectively. In Sec. 4, we introduce and study (∗, s)-dualities and their duals. In particular, we show that, under suitable assumptions on ∗, any (∗, s)-duality can be represented with the aid of coupling functions. We also study a special class of dualities associated to a subset of the Cartesian product of two sets, which are characterized as those dualities that are (∗, s)-dualities for any ∗ satisfying a mild condition. Finally, in Sec. 5, we study subdifferentials with respect to (∗, s)-dualities. 2. Upper and Lower Inverses and s-Conjugates of Binary Operations on a Complete Lattice First, we recall (see, e.g., [3, 5]) that a partially ordered set (poset) A = (A, ) is a set A endowed with a (partial) order , i.e., a reflexive, antisymmetric, transitive binary relation . As usual, by the notation a ≺ b we mean that a  b and a = b. The order  is called a total order and A = (A, ) is called a chain if for each pair a, b ∈ A, we have either a  b or b  a. A poset A is called a lattice if for each pair a, b ∈ A, there exists the least upper bound, denoted sup(a, b), or supA (a, b), or a ∨ b, or a ∨A b, and the greatest lower bound, denoted inf(a, b), or inf A (a, b), or a ∧ b, or a ∧A b. A lattice A is called a complete lattice if for each nonempty subset M of A, there exists the least upper bound, denoted sup M or supA M , and the greatest lower bound, denoted inf M or inf A M . We denote by +∞ or (+∞)A the greatest element of A and by −∞ or (−∞)A the least element of a complete lattice A. In any complete lattice A, we adopt the usual conventions: inf ∅ = +∞,

sup ∅ = −∞.

(2.1)

The following concepts play a fundamental role in the sequel. Definition 2.1. Let A = (A, ) be a complete lattice endowed with a binary operation ∗ : (a, b) ∈ A × A → a ∗ b ∈ A (i.e., an internal composition law), called “multiplication.” We say that ∗ satisfies (a) condition (inf-d) if for any index set I (including the empty set ∅), we have 
inf ai ∗ a = inf (ai ∗ a), {ai }i∈I ⊆ A, a ∈ A; (2.2) i∈I

i∈I

(b) condition (sup-d) if for any index set I (including the empty set ∅), we have
 sup ai ∗ a = sup(ai ∗ a), {ai }i∈I ⊆ A, a ∈ A. i∈I

(2.3)

i∈I

Remark 2.1. (a) In Definition 2.1, no additional assumption is made on the binary operation ∗. (b) By (2.1), condition (2.2) for I = ∅ means that +∞ ∗ a = +∞,

a ∈ A,

(2.4)

a ∈ A.

(2.5)

and condition (2.3) for I = ∅ means that −∞ ∗ a = −∞,

(c) If ∗ satisfies condition (inf-d), then the postmultiplication ∗ by any c ∈ A is isotone, i.e., we have the implication b1 , b2 ∈ A, b1  b2

⇒

b1 ∗ c  b2 ∗ c.

(2.6)

Indeed, if b1  b2 , then, by (2.2), b1 ∗ c = (inf{b1 , b2 }) ∗ c = inf{b1 ∗ c, b2 ∗ c}  b2 ∗ c. Dually, if ∗ satisfies condition (sup-d), then the premultiplication ∗ by any c ∈ A is isotone, i.e., we have the implication b1 , b2 ∈ A, b1  b2 2510

⇒

c ∗ b1  c ∗ b2 .

(2.7)

Indeed, if b1  b2 , then, by (2.3), c ∗ b1  sup{c ∗ b1 , c ∗ b2 } = c ∗ (sup{b1 , b2 }) = c ∗ b2 . (d) In [10], we used for (2.2) and (2.3) the terms “condition (α)” and “condition (β),” respectively. Instead, here we use for them the terms “condition (inf-d)” and “condition (sup-d),” respectively, to indicate that they mean infinitary distributivity, from the left, of inf (respectively, sup) over the binary operation ∗. We denote by min (respectively, max) an infimum (respectively, supremum) which is attained. Let us recall some well-known easily shown facts from classical residuation theory (see, e.g., [1–4]). Proposition 2.1. Let A = (A, ) be a complete lattice. (a) A binary operation ∗ : A × A → A satisfies condition (inf-d) if and only if for each pair of elements a, c ∈ A, there exists the element a ∗−1 u c := min{x ∈ A | a  x ∗ c}.

(2.8)

(b) A binary operation ∗ : A × A → A satisfies condition (sup-d) if and only if for each pair of elements a, c ∈ A, there exists the element a ∗−1 l c := max{x ∈ A | x ∗ c  a}.

(2.9)

−1 Remark 2.2. (a) For some pairs a, c ∈ A, the elements a ∗−1 u c in (2.8) or a ∗l c in (2.9) may exist even without assuming that they exist for all pairs a, c ∈ A, i.e., without assuming condition (inf-d) or (sup-d) (see, e.g., Remark 3.3 below). (b) The above notation is justified as follows. Since for a, c ∈ A, the equation x ∗ c = a need not have a solution in A, it is natural to consider its “upper solutions” and “lower solutions” (called also “supersolutions” and “subsolutions,” see, e.g., [4]), i.e., the elements x ∈ A such that a  x ∗ c −1 (respectively, x ∗ c  a). Then a ∗−1 u c and a ∗l c are the greatest upper solution and the least lower solution; therefore, in this notation, the subscripts u and l stand for emphasizing “upper” and “lower.” Furthermore, it is well known and easy to see that we have the equivalences

ab∗c

⇔

a ∗−1 u c  b,

b∗ca⇔b

a ∗−1 l

c,

(2.10) (2.11)

∗−1 l

which show that the binary operations ∗−1 are some kind of “inverses,” for inequalities, of the u and binary operation ∗; therefore, in the above notation, the supersript −1 stands for emphasizing this fact. −1 the lower inverse of ∗. We call ∗−1 u the upper inverse of ∗ and ∗l −1 (c) In [10], we used the notation ∗l (respectively, ∗u ) instead of ∗−1 u (respectively, ∗l ), and in [10, Remarks 2.8(a) and 2.12(b)], these binary operations were called the “(left) epi-hypo-inverse” (respectively, the “(left) hypo-epi-inverse”) of ∗. In the literature on residuation theory, various other terms and notation for these operations are also used, but we do not mention them here. The next result is also known in classical residuation theory. Proposition 2.2.

(a) If ∗ satisfies condition (inf-d), then ∗−1 u satisfies condition (sup-d) and −1 (∗−1 u )l = ∗;

(2.12)

(b) if ∗ satisfies condition (sup-d), then ∗−1 satisfies condition (inf-d) and l −1 (∗−1 l )u = ∗.

(2.13)

The following concept introduced in [10] will be fundamental in the sequel. Definition 2.2. Let A = (A, ) be a complete lattice endowed with a binary operatio ∗ : A × A → A, and let s : A → A be a bijection (i.e., a one-to-one mapping of A onto A). Then 2511

(a) the binary operation ∗s : (a, b) ∈ A × A → a ∗s b ∈ A defined by a ∗s b := s(s−1 (a) ∗ b),

a, b ∈ A,

(2.14)

is called the s-conjugate of ∗; (b) applying Definition 2.2(a) to the bijection s−1 , we obtain the s−1 -conjugate of ∗, namely, a ∗s

−1

b := s−1 (s(a) ∗ b),

a, b ∈ A;

(2.15)

(c) applying Definition 2.2(a) to the bijection s−1 and the binary operation ∗s , we obtain the binary −1 −1 −1 operation ∗ss : A × A → A defined by ∗ss := (∗s )s , i.e., a ∗ss

−1

b := s−1 (s(a) ∗s b),

a, b ∈ A,

(2.16)

which we call the bi-s-conjugate of ∗. −1

One can recuperate ∗ from ∗s and ∗s . Indeed, we have the following assertion. Lemma 2.1. Let A = (A, ), ∗ : A × A → A, and s : A → A be as in Definition 2.2. Then ∗ss

−1

= ∗.

(2.17)

Proof. By (2.16) and (2.14), we have, for any a, b ∈ A, a ∗ss

−1

b = s−1 (s(a) ∗s b) = s−1 (s(s−1 (s(a)) ∗ b)) = a ∗ b.

In Definition 2.2 and Lemma 2.1, no additional assumption was made about the bijection s : A → A. Now we consider a special class of bijections. Definition 2.3 (see [10, 20]). Let A and B be two complete lattices. A mapping s : A → B is called (a) a duality if for any index set I, we have 
s inf ai = sup s(ai ), i∈I

i∈I

{ai }i∈I ⊆ A,

(2.18)

where inf ai = inf A ai , i∈I

i∈I

sup s(ai ) = sup B s(ai ); i∈I

i∈I

(b) a bijective duality if, in addition, s is a bijection. Remark 2.3. (a) Taking a two-element index set I = {1, 2}, we see that every duality s : A → B is antitone, i.e., for any a1 , a2 ∈ A, a1  a2

⇒

s(a2 )  s(a1 ).

(2.19)

(b) By conventions (2.1), condition (2.18) for I = ∅ means that s(+∞) = −∞,

(2.20)

where +∞ = +∞A and −∞ = −∞B . Lemma 2.2. Let A and B be two complete lattices. A mapping s : A → B is a bijective duality if and only if it is an antiisomorphism of A onto B. 2512

Proof. Clearly, each antiisomorphism of A onto B is a bijective duality. Conversely, assume that s : A → B is a bijective duality and let {ai }i∈I ⊆ A. Then 
s sup ai = s (inf{a ∈ A | ai  a, i ∈ I}) i∈I

= sup{s(a) | ai  a, i ∈ I} = sup{b ∈ B | ai  s−1 (b), i ∈ I}

(2.21)

= sup{b ∈ B | b  s(ai ), i ∈ I)} = inf s(ai ), i∈I

which, together with (2.18), proves that s is an antiisomorphism of A onto B. In the sequel, we consider the case where A = B and s : A → A is an antiautomorphism. Proposition 2.3. Let A = (A, ) be a complete lattice endowed with a binary operation ∗ : A × A → A and s : A → A be an antiautomorphism. (a) ∗ satisfies condition (inf-d) if and only if ∗s satisfies condition (sup-d). Moreover, in this case, we have s s −1 (∗−1 u ) = (∗ )l ,

(2.22)

s −1 ((∗−1 u ) )u

(2.23)

=∗ . s

(b) Dually, ∗ satisfies condition (sup-d) if and only if ∗s satisfies condition (inf-d). Moreover, in this case, we have s s −1 (∗−1 l ) = (∗ )u ,

(2.24)

s −1 ((∗−1 l ) )l

(2.25)

=∗ . s

Proof. (a) Since s : A → A is an antiautomorphism, s−1 is also an antiautomorphism, whence, for any {ai }i∈I ⊆ A,
 −1 −1 sup ai . (2.26) inf s (ai ) = s i∈I

i∈I

Let us observe that for any {ai }i∈I ⊆ A and a ∈ A, by (2.14) and (2.26) we have 
 



  inf s−1 (ai ) ∗ a , sup ai ∗s a = s s−1 sup ai ∗ a = s i∈I

i∈I

i∈I

(2.27)

and, by (2.14) and (2.18), sup(ai ∗ a) = sup s(s s

i∈I

i∈I

−1


 −1 (ai ) ∗ a) = s inf (s (ai ) ∗ a) . i∈I

Now assume that ∗ satisfies condition (inf-d). Then, by (2.2), we have

 
 −1 −1 s inf s (ai ) ∗ a = s inf (s (ai ) ∗ a) , {ai }i∈I ⊆ A, a ∈ A, i∈I

i∈I

(2.28)

(2.29)

whence, using (2.27) and (2.28), we obtain (2.3) for ∗s . This proves that ∗s satisfies condition (sup-d). Conversely, assume that ∗s satisfies condition (sup-d). Then, by (2.3) for ∗s , (2.27), and (2.28), we have (2.29), whence, since s is a bijection, we obtain (2.2). This proves that ∗ satisfies condition (inf-d). 2513

s −1 Next, observe that, by the first part of the above proof, ∗−1 u and (∗ )l are well defined. Let a, b ∈ A. Then, by (2.14), (2.8), (2.18), (2.19), and (2.9), we obtain s −1 −1  −1  b(∗−1 u ) a = s(s (b) ∗u a) = s(min{a ∈ A | s (b)  a ∗ a})

= max{s(a ) ∈ A | a ∈ A, s−1(b)  a ∗ a} = max{b ∈ A | s−1 (b)  s−1 (b ) ∗ a} = max{b ∈ A | s(s−1 (b ) ∗ a)  b} = max{b ∈ A | b ∗s a  b} = b(∗s )−1 l a. This proves (2.22). Furthermore, by (2.22) and (2.13) (applied to ∗s satisfying condition (sup-d), by the first part of the above proof), we have s −1 s −1 −1 s ((∗−1 u ) )u = ((∗ )l )u = ∗ . This proves (2.23). (b) For the case where s : A → A satisfies condition (sup-d), the proofs of the dual statements are similar. Corollary 2.1. Let A = (A, , ∗) be as in Definition 2.2 and s : A → A be an antiautomorphism. s (a) If ∗ satisfies condition (inf-d), then (∗−1 u ) also satisfies this condition. −1 s (b) If ∗ satisfies condition (sup-d), then (∗l ) also satisfies this condition. Proof. (a) Assume that ∗ satisfies condition (inf-d). Then, by Proposition 2.2(a), ∗−1 u satisfies condition s satisfies condition (inf-d). ) (sup-d). Hence, by Proposition 2.3, (∗−1 u The proof of part (b) is similar. Proposition 2.4. (a) Under the assumptions of Proposition 2.3, if ∗ satisfies condition (inf-d), then for any a, b, c ∈ A, we have the equivalence b ∗s a  c

−1 s−1 (c) ∗−1 u a  s (b).

⇔

(2.30)

(b) If, in addition, ∗ is commutative, then for any a, b, c ∈ A, b ∗s a  c

−1 s−1 (c) ∗−1 u s (b)  a.

⇔

(2.31)

Proof. (a) By (2.14), (2.19), and (2.10), b ∗s a  c

⇔

s(s−1 (b) ∗ a)  c

⇔

s−1 (c)  s−1 (b) ∗ a

⇔

−1 s−1 (c) ∗−1 u a  s (b).

(b) If, in addition, ∗ is commutative, then, by the above equivalences and (2.10), b ∗s a  c

⇔

s(s−1 (b) ∗ a)  c

⇔

−1 s−1 (c) ∗−1 u s (b)  a.

⇔

s−1 (c)  s−1 (b) ∗ a = a ∗ s−1 (b)

Proposition 2.5. Under the assumptions of Proposition 2.4, ∗ is commutative if and only if b ∗s a = s(a) ∗s s−1 (b),

a, b ∈ A.

Proof. Let a, b ∈ A. If ∗ is commutative, then, by (2.14) and a = s−1 (s(a)), we have   b ∗s a = s(s−1 (b) ∗ a) = s(a ∗ s−1 (b)) = s s−1 (s(a)) ∗ s−1 (b) = s(a) ∗s s−1 (b). Conversely, if (2.32) holds, then, by (2.14), (2.32), and the relation s−1 (s(a)) = a,   s(s−1 (b) ∗ a) = b ∗s a = s(a) ∗s s−1 (b) = s s−1 (s(a)) ∗ s−1 (b) = s(a ∗ s−1 (b)), whence, since s is bijective, we obtain s−1 (b) ∗ a = a ∗ s−1 (b). 2514

(2.32)

3. Some Examples First,we recall the concept of canonical enlargement of a boundedly complete lattice ordered group, which will be the framework for our examples. A lattice ordered group G = (G, , ⊗) is a lattice (G, ) endowed with a group operation ⊗, called “product” or “multiplication”, such that ⊗ is isotone, i.e., a, b, c ∈ G, a  b

⇒

c ⊗ a  c ⊗ b, a ⊗ c  b ⊗ c.

(3.1)

A lattice ordered group G = (G, , ⊗) is said to be a boundedly complete lattice ordered group (some authors use, instead of “boundedly complete,” the term “conditionally complete” [5] or “complete” [9]), if G = (G, ) is a conditionally complete lattice, i.e., a lattice in which every nonempty (order-) bounded subset admits a supremum and an infimum. By a classical result of Iwasawa (see, e.g., [3, Chap. 13, Sec. 15, Theorem 28]), every boundedly complete lattice ordered group G is commutative. In the sequel, we assume, without any special mention, that G is not a singleton. It is well known (see, e.g., [3, Chap. 13, Sec. 1, Corollary of Lemma 1]) that a lattice ordered group G (which is not a singleton) has no least element and no greatest element. Therefore, it is convenient to adjoin to any lattice ordered group G a least element −∞ and a greatest element +∞, i.e., to consider the set G := G ∪ {−∞} ∪ {+∞},

(3.2)

−∞  a  +∞,

(3.3)

where  is extended to G by a ∈ G,

.

and with the product ⊗ extended to two different operations ⊗ and ⊗ . on G defined by

.

a⊗b =a⊗ . b = a ⊗ b,

.

a, b ∈ G,

.

+∞ ⊗ a = a ⊗ +∞ = +∞,

(3.4)

a ∈ G,

(3.5)

−∞ ⊗ a = a ⊗ −∞ = −∞,

a ∈ G ∪ {−∞},

(3.6)

+∞ ⊗ . a =a⊗ . +∞ = +∞,

a ∈ G ∪ {+∞},

(3.7)

.

.

−∞ ⊗ . a=a⊗ . −∞ = −∞,

a ∈ G.

(3.8)

.

Then, following [9, Definition 1.1], G = (G, , ⊗, ⊗ . ) is called the canonical enlargement of G = (G, , ⊗).

.

Remark 3.1. Since ⊗ is associative and commutative on G, its extensions ⊗ and ⊗ . to G are also associative and commutative on G. Also, by (3.4)–(3.8), isotony (3.1) extends to G. Furthermore, if e is the unit element of the group (G, ⊗), then, by (3.7) and (3.8), e also acts as a unit element in the . −1 semigroups (G, ⊗) and (G, ⊗ . ). We extend the notation a from G to G by the conventions (+∞)−1 = −∞,

(−∞)−1 = +∞.

(3.9)

Finally, we note that by [9, Lemma 1.4] and [12, Lemma 1.1], for any index set I, we have the distributivity properties G sup G (a ⊗ . ai ) = a ⊗ . sup ai , i∈I

.

.

inf G (a ⊗ ai ) = a ⊗ inf G ai , i∈I

{ai }i∈I ⊆ G, a ∈ G,

(3.10)

{ai }i∈I ⊆ G, a ∈ G;

(3.11)

i∈I

i∈I

2515

in particular, for I = ∅, this means (3.8) and (3.5). In the subsequent examples, whenever the complete lattice is A = G, the canonical enlargement of a boundedly complete lattice ordered group G = (G, , ⊗), we endow A with various binary operations ∗ and consider the mapping s(a) = a−1 ,

a ∈ G,

(3.12)

which is an antiautomorphism (by [9, Lemma 1.1]).

.

Example 3.1. Let A = G = (G, , ⊗, ⊗ . ) be the canonical enlargement of a boundedly complete lattice ordered group G = (G, , ⊗), and let

.

∗ = ⊗.

(3.13)

Then, by (3.11), ∗ satisfies condition (inf-d). Furthermore, we have

. −1

−1 a ⊗u c = a ⊗ .c ,

a, c ∈ A.

(3.14)

Indeed, by [9, Lemma 1.5], for any a, b, c ∈ A = G, the equivalence

.

ab⊗c

−1 a⊗ . c b

⇔

.

(3.15)

.

holds, i.e., these ∗ = ⊗ and ∗−1  of (3.12), we have, by definition (2.14) u satisfy (2.10). Also, for ∗ = ⊗ and s of ∗s and [9, Lemma 1.3],

. s


.

−1 a ⊗ c = (c−1 ⊗ a)−1 = a ⊗ .c ,

a, c ∈ A.

(3.16)

.

Finally, note that, by (3.12), (2.14), (3.14) (for ∗ = ⊗), and [9, Lemma 1.3],

. −1

. −1

.

−1 −1 a(⊗u )s
c = s( s−1 (a) ⊗u c) = (a−1 ⊗ . c ) = a ⊗ c,

a, c ∈ A.

(3.17)

Let us mention two important particular cases. Example 3.1(a). Let G = (R, ≤, +), where ≤ is the usual total order on R = (−∞, +∞), and + is the . usual addition on R. Then G = R = [−∞, +∞], and the binary operations ⊗ and ⊗ . defined by (3.4)–(3.8)

.

(for ⊗ = +) are the “upper” and “lower” additions + and + . on R, of (1.2), (1.3). Furthermore, in this case, (3.12) is the antiautomorphism s(a) = −a,

a ∈ R.

(3.18)

Example 3.1(b). Let G = (R+ \{0}, ≤, ×), where ≤ is the usual total order and × is the usual multi. plication on R+ \{0}. Then G = R+ = [0, +∞] and the binary operations ⊗ and ⊗ . defined by (3.4)–(3.8)

.

(for ⊗ = ×) are the “upper” and “lower” multiplications × and × . (see, e.g., [14]) on R+ . Furthermore, in this case, (3.12) is the antiautomorphism 1 s(a) = , a ∈ R+ (3.19) a with the conventions 1/0 = +∞ and 1/ + ∞ = 0. The groups G of Examples 3.1(a) and 3.1(b), and hence also their canonical enlargements G, are isomorphic by the logarithm function (see also Remark 3.2(a) below), but both are important for applications. 2516

Example 3.2. Let A = (A, ) be a complete lattice, s : A → A be an antiautomorphism, and ∗ = ∨.

(3.20)

Then, by (2.18) and (2.21) applied to a two-element index set I = {1, 2}, we have s(a ∧ b) = s(a) ∨ s(b), s(a ∨ b) = s(a) ∧ s(b),

a, b ∈ A.

(3.21)

Hence, by (2.14), a ∨s b = s(s−1 (a) ∨ b) = a ∧ s(b),

a, b ∈ A.

(3.22)

Recall (see, e.g., [3, Chap. 2, Sec. 11, and Chap. 5, Theorem 24]) that a complete lattice A = (A, ) is called Brauerian if ∗ = ∧ satisfies condition (sup-d) of (2.3), i.e., 
sup ai ∧ a = sup(ai ∧ a), {ai }i∈I ⊆ A, a ∈ A, (3.23) i∈I

i∈I

or, equivalently, if the element   a⊥c := a ∧−1 l c = max{b ∈ A | b ∧ c  a}

(3.24)

(of (2.9), for ∗ = ∧) exists for all a, c ∈ A. We say that a complete lattice A = (A, ) is dual Brauerian if ∗ = ∨ satisfies condition (inf-d) of (2.2), i.e., 
inf ai ∨ a = inf (ai ∨ a), {ai }i∈I ⊆ A, a ∈ A, (3.25) i∈I

i∈I

or, equivalently, if the element   ac := a ∨−1 u c = min{b ∈ A | a  b ∨ c}

(3.26)

(of (2.8), for ∗ = ∨) exists for all a, c ∈ A. It is well known (see, e.g., [3, Chap. 13, Sec. 14, Theorem 25]) that every complete lattice ordered group is both Brauerian and dual Brauerian. If A = (A, ) is a dual Brauerian complete lattice and s : A → A is an antiautomorphism, then s a(∨−1 u ) b = a⊥s(b),

a, b ∈ A.

(3.27)

Indeed, by (2.14) (for ∗ = ∨−1 u ), (3.26), and (3.34) below, s −1 −1 −1 a(∨−1 u ) b = s(s (a) ∨u b) = s(s (a)b) = a⊥s(b),

a, b ∈ A.

Now assume, in particular, that A = G is the canonical enlargement of a boundedly complete lattice ordered group G and consider the antiautomorphism s : A → A defined by (3.12). Then, by (3.22) and (3.12), a ∨s
b = a ∧ b−1 ,

a, b ∈ A.

(3.28)

Also, by (3.27) applied to s = s of (3.12), s
−1 a(∨−1 u ) b = a⊥b ,

a, b ∈ A.

(3.29)

Remark 3.2. (a) If A = G is the canonical enlargement of a boundedly complete totally ordered group G, then (see, e.g., [6, p. 110, Proposition 5]) the group (G, , ⊗) is isomorphic to the complete lattice ordered group (R, ≤, +), where ≤ and + are the usual total order and addition on R = (−∞, +∞). . Hence, all results of [8] on the canonical enlargement R = (R, ≤, +, + . ) of the group (R, ≤, +) carry over by this isomorphism to the general case of the canonical enlargement A = G of a boundedly complete totally ordered group G. 2517

(b) As was observed in [9], the binary operations  and ⊥ are noncommutative and nonassociative. Indeed, for example, in A = (R, ≤), we have (00)0 = −∞0 = −∞,

(3.30)

0(00) = 0 − ∞ = 0.

(3.31)

(c) There also exists other notation for ac of (3.24), for example, a  c [1, 4], a : c [3, 21], etc., but we follow here the notation of [8], since it emphasizes the symmetry with the properties of ⊥ of (3.26). Note that in the literature on dioids (see, e.g., [1, 4]) the symbols  and ⊥ are used to denote the “top element” and the “bottom element” of a complete lattice; this leads to no confusion, since in the present paper we denote those elements by +∞ and −∞, respectively. Example 3.3. Let A = (A, ) be a Brauerian complete lattice, s : A → A an antiautomorphism, and ∗ = ⊥.

(3.32)

Then, by (3.24), (2.21), (3.21), and (3.26), we have s(a⊥b) = s(max{a ∈ A | a ∧ b  a}) = min{s(a ) ∈ A | s(a)  s(a ∧ b)} = min{s(a ) ∈ A | s(a)  s(a ) ∨ s(b)} = s(a)s(b),

a, b ∈ A,

(3.33)

and, similarly, a, b ∈ A.

s(ab) = s(a)⊥s(b),

(3.34)

Hence, by (2.14) and (3.34), we obtain a⊥s b = s(s−1 (a)⊥b) = as(b),

a, b ∈ A.

(3.35)

Furthermore, since by our assumption that A is Brauerian, ∧ satisfies condition (sup-d), it follows from Proposition 2.2(b) that ⊥ = ∧−1 satisfies condition (inf-d) and l −1 −1 a⊥−1 u b = a(∧l )u b = a ∧ b,

a, b ∈ A.

(3.36)

Also, we have s a(⊥−1 u ) b = a ∨ s(b),

a, b ∈ A.

(3.37)

Indeed, by (2.14), (3.36), and (3.21), for any a, b ∈ A, we have s −1 −1 −1 a(⊥−1 u ) b = s(s (a)⊥u b) = s(s (a) ∧ b) = a ∨ s(b).

(3.38)

Now assume, in particular, that A = G is the canonical enlargement of a boundedly complete lattice ordered group G and consider the antiautomorphism s : A → A defined by (3.12). Then, by (3.35) and (3.12), a⊥s
b = ab−1 ,

a, b ∈ A.

(3.39)

Also, by (3.37) applied to s = s of (3.12), we have s
−1 a(⊥−1 u ) b=a∨b ,

a, b ∈ A.

(3.40)

Some other remarks on the values of a⊥c and ac are collected in the following remark. Remark 3.3. (a) If A = (A, ) is a complete lattice and a, c ∈ A and c  a, then the element a⊥c of (3.24) exists and a⊥c = +∞,

a, c ∈ A, c  a.

(3.41)

Indeed, this follows from (3.24) and the fact that b ∧ c  c  a for all b ∈ A. Similarly, for a, c ∈ A with a  c, the element ac of (3.26) exists and ac = −∞, 2518

a, c ∈ A, a  c.

(3.42)

(b) If A = (A, ) is a complete lattice, then the elements a⊥ ± ∞ and a ± ∞ exist for all a ∈ A and we have a⊥ − ∞ = +∞,

a ∈ A,

(3.43)

a⊥ + ∞ = a,

a ∈ A,

(3.44)

a − ∞ = a,

a ∈ A,

(3.45)

a + ∞ = −∞,

a ∈ A.

(3.46)

Indeed, since b ∧ (−∞) = (−∞) ∧ b = −∞  a for all b , a ∈ A, from (3.24) for c = −∞ we obtain a⊥ − ∞ = +∞; the proofs of (3.44)–(3.46) are similar. (c) If, in addition,  is a total order on A, then it can be easily verified (and, in the particular case where A = (R, ≤), it is well known in classical residuation theory) that a⊥c and ac can be calculated and admit simple expressions for all pairs a, c ∈ A; namely, (3.24) and (3.26) yield, respectively,  a, if a ≺ c, −1 (3.47) a⊥c = a ∧l c = +∞, if c  a,  a, if c ≺ a, ac := a ∨−1 (3.48) u c= −∞, if a  c. 4. Generalized Conjugation 4.1. Preliminaries. Recall that if A = (A, ) is a complete lattice and X is a set, then the set AX of all functions f : X → A endowed with the partial order  defined as the pointwise extension of the partial order of A, i.e., by f g

⇔

f (x)  g(x),

x ∈ X,

is a complete lattice, in which sup and inf are the pointwise extensions of those in A, i.e.,
 
sup fi (x) = sup fi (x), x ∈ X. inf fi (x) = inf fi (x), i∈I

i∈I

i∈I

(4.1)

(4.2)

i∈I

We identify each element a ∈ A with the constant function fa ∈ AX defined by fa (x) = a, x ∈ X. Then AX becomes a complete lattice, whose greatest and least elements are the functions identically equal to +∞ and −∞, respectively. Furthermore, if ∗ : A × A → A is a binary operation on A, then we extend it pointwise to a binary operation ∗ : AX × AX → AX , i.e., by (f ∗ g)(x) := f (x) ∗ g(x),

x ∈ X.

(4.3)

Actually, in the sequel, we consider only the particular case where one of the functions f or g is constant. For f ∈ AX and a mapping ∆ : AX → AW , where X and W are two sets, we write f ∆ instead of ∆(f ). Recall that an element e ∈ A is called (a) a left neutral element for ∗, if e ∗ a = a,

a ∈ A;

(4.4)

a ∗ e = a,

a ∈ A;

(4.5)

(b) a right neutral element for ∗, if (c) a neutral element for ∗, if it is both a left and a right neutral element for ∗. Note that a neutral element for ∗ is necessarily unique. 2519

Definition 4.1. Let A = (A, ) be a complete lattice endowed with a binary operation ∗ : A × A → A admitting a (not necessarily unique) left neutral element e, and fix a left neutral element e. For any subset G of a set X, the indicator function of G (with respect to e) is the function χG = χG,e : X → {e, +∞} defined by  e, if y ∈ G, (4.6) χG (y) = χG,e (y) := +∞, if y ∈ X\G. Lemma 4.1. Let X be a set and A = (A, ) be a complete lattice endowed with a binary operation ∗ : A × A → A satisfying (2.4) and admitting a left neutral element e. Then f = inf {χ{x} ∗ f (x)}, x∈X

f ∈ AX .

(4.7)

Proof. By (4.6), (4.4), and (2.4), for any f ∈ AX and x, y ∈ X, we have  e ∗ f (y) = f (y), if x = y, χ{x} (y) ∗ f (x) = +∞ ∗ f (x) = +∞, if x = y, whence inf {χ{x} (y) ∗ f (x)} = min{f (y), +∞} = f (y).

x∈X

In the sequel, the following concept plays an important role. Definition 4.2. If X and W are two sets, then any function ϕ : X ×W → A is called a coupling function. 4.2. (∗, s)-Dualities. Definition 4.3. Let A be a complete lattice, ∗ : A × A → A be a binary operation on A, s : A → A be a bijection, and X and W be two sets. We say that a mapping ∆ : AX → AW is a (∗, s)-duality if for any index set I, we have (at each w ∈ W ) ∆
= sup fi∆ , {fi }i∈I ⊆ AX (4.8) inf fi i∈I

i∈I

(i.e., ∆ is a duality in the sense of Definition 2.3) and (f ∗ c)∆ = f ∆ ∗s c,

f ∈ AX , c ∈ A.

(4.9)

By the conventions (2.1), condition (4.8) for I = ∅ means that (+∞)∆ = −∞.

(4.10)

Proposition 4.1. Let A = (A, ) be a complete lattice, ∗ : A × A → A be a binary operation satisfying (2.4), and s : A → A be a bijection. The following statements are equivalent: (1) ∆ is a (∗, s)-duality; (2) ∆ satisfies (4.8) and (f ∗ c)∆ = f ∆ ∗s c,

f ∈ AX , c ∈ A\{+∞}.

(4.11)

If, in addition, ∗ is commutative and satisfies condition (inf-d), s is a (bijective) duality, and inf c∈A\{−∞,+∞}

c = −∞,

(4.12)

then these statements are equivalent to the following statement: (3) ∆ satisfies (4.8) and (f ∗ c)∆ = f ∆ ∗s c, 2520

f ∈ AX , c ∈ A\{−∞, +∞}.

(4.13)

Proof. The implications (1) ⇒ (2) and (1) ⇒ (3) are obvious. Conversely, assume that (2) holds and let us show that then (4.9) holds also for c = +∞. Indeed, by the commutativity of ∗, (2.4), (4.10), (2.14), and (2.20), we have (f ∗ (+∞))∆ = ((+∞) ∗ f )∆ = (+∞)∆ = −∞,     f ∆ ∗s (+∞) = s (s−1 ◦ f ∆ ) ∗ (+∞) = s (+∞) ∗ (s−1 ◦ f ∆ ) = s(+∞) = −∞, whence (4.9) for c = +∞. Now assume that (3) holds and let us show that then (4.9) also holds for c = −∞. Indeed, by the commutativity of ∗, (4.12), condition (inf-d), (4.8), (4.9), and (2.14), we obtain

 ∆
∆ inf c ∗f = inf (c ∗ f ) (f ∗ (−∞))∆ = ((−∞) ∗ f )∆ = c∈A\{−∞,+∞}

=

(c ∗ f )∆ =

sup

c∈A\{−∞,+∞}

=

c∈A\{−∞,+∞}

inf

s ◦ ((s−1 ◦ f ∆ ) ∗ c))

sup c∈A\{−∞,+∞}

((s

−1

c∈A\{−∞,+∞}

= f ∆ ∗s

(f ∗ c)∆

c∈A\{−∞,+∞}

(f ∆ ∗s c) =

sup

=s◦

sup

c∈A\{−∞,+∞}



−1 ∆ ◦ f ) ∗ c)) = s ◦ (s ◦ f ) ∗ ∆

inf

 c

c∈A\{−∞,+∞}

inf c∈A\{−∞,+∞}

c = f ∆ ∗s (−∞),

whence (4.9) for c = −∞. Remark 4.1. Assumption (4.12) is satisfied, for example, if the cardinality of the set of all atoms of A c, then b must be an atom (since, otherwise, is =1. Indeed, observe first that if −∞ < b := inf c∈A\{−∞,+∞}

there exists c ∈ A such that −∞ < c < b, which contradicts the definition of b). Hence, if A has no atoms, then (4.12) holds. On the other hand, if A has two atoms, say a, b, then inf c∈A\{−∞,+∞}

c ≤ inf{a, b} = −∞.

2 \{(x , 0) | 0 < However, one can have (4.12) also when A has exactly one atom. For example, let A = R+ 1 x1 < 1}, with the usual (componentwise) partial order . Then the only atom of A is (1, 0), but c = (0, 0) = −∞. inf c∈A\{−∞,+∞}

Our next aim is to give representations of (∗, s)-dualities with the aid of coupling functions. Theorem 4.1. Let X and W be two sets, A = (A, ) be a complete lattice endowed with a binary operation ∗ : A × A → A, and s : A → A be a bijection. (a) If ∗ satisfies (2.4) and admits a left neutral element e, then for each (∗, s)-duality ∆ : AX → AW , there exists a coupling function ϕ : X × W → A such that f ∆ (w) = sup {ϕ(x, w) ∗s f (x)}, x∈X

f ∈ AX , w ∈ W ;

(4.14)

for example, one can take ϕ(x, w) = χ∆ {x} (w),

x ∈ X, w ∈ W.

(4.15)

(b) Assuming that (4.14) holds, if ∗ is commutative, satisfies condition (inf-d), and admits a neutral element e and if s is an antiautomorphism, then ϕ of (4.15) is the unique coupling function for which we have (4.14). (c) Assuming that (4.14) holds, if ∗ is commutative and satisfies condition (inf-d) and if s is an antiautomorphism, then ∆ is a duality, i.e., it satisfies (4.8). 2521

(d) Assuming that (4.14) holds, if ∗ is associative and satisfies condition (inf-d), and if s is an antiautomorphism, then ∆ satisfies (4.9). Proof. (a) By (2.4) and (4.7)–(4.9), for any f ∈ AX , we have
f

∆

=

∆ s inf {χ{x} ∗ f (x)} = sup {χ{x} ∗ f (x)}∆ = sup {χ∆ {x} ∗ f (x)},

x∈X

x∈X

x∈X

which, for ϕ of (4.15), yields (4.14). (b) Assume that (4.14) holds. If x ∈ X, then, applying (4.14) to f = χ{x} and using (2.14), we obtain, for any w ∈ W ,  −1  s χ∆ {x} (w) = sup {ϕ(y, w) ∗ χ{x} (y)} = sup s s (ϕ(y, w)) ∗ χ{x} (y) . y∈X

(4.16)

y∈X

But, by our assumptions on ∗, e, and s, we have 

   s s−1 (ϕ(x, w)) ∗ e , if y = x,   −1 s s (ϕ(x, w)) ∗ (+∞) , if y = x,  ϕ(x, w), if y = x, = s(+∞) = −∞, if y = x,

 s s−1 (ϕ(y, w)) ∗ χ{x} (y) =

whence, by (4.16), we obtain χ∆ {x} (w) = ϕ(x, w), x ∈ X, w ∈ W ; this proves the uniqueness of ϕ. (c) Using (4.14), (2.14), condition (inf-d), the commutativity of ∗, and the fact that s is an antiautomorphism, we obtain, for any {fi }i∈I ⊆ AX and w ∈ W ,
∆ 
  s inf fi (w) = sup ϕ(x, w) ∗ inf fi (x) i∈I i∈I x∈X

 
−1 ϕ(x, w) ∗ inf fi (x) = sup s s i∈I x∈X 

 −1 inf fi (x) ∗ s (ϕ(x, w) = sup s i∈I x∈X
 −1 = sup s inf {fi (x) ∗ s (ϕ(x, w))} i∈I x∈X   = sup sup s fi (x) ∗ s−1 (ϕ(x, w)) x∈X i∈I   = sup sup s s−1(ϕ(x, w)) ∗ fi (x) i∈I x∈X

= sup sup {ϕ(x, w) ∗s fi (x)} = sup fi∆ (w). i∈I x∈X

2522

i∈I

(d) Using (4.14), (2.14), condition (inf-d), the associativity of ∗, and the fact that s is an antiautomorphism, we obtain, for any f ∈ AX , a ∈ A, and w ∈ W ,   (f ∗ a)∆ (w) = sup {ϕ(x, w) ∗s (f (x) ∗ a)} = sup s s−1 (ϕ(x, w)) ∗ (f (x) ∗ a) x∈X x∈X  −1  = sup s s (ϕ(x, w)) ∗ f (x) ∗ a x∈X 
 −1  = s inf s (ϕ(x, w)) ∗ f (x) ∗ a x∈X

 
−1 −1 s inf [s (ϕ(x, w)) ∗ f (x)] ∗ a =s s x∈X

   −1  −1 sup s s (ϕ(x, w)) ∗ f (x) ∗ a =s s x∈X
 
−1 s sup ϕ(x, w) ∗ f (x) ∗ a = s(s−1 (f ∆ (w)) ∗ a) =s s x∈X s

= f (w) ∗ a. ∆

Remark 4.2. (a) One can also give another proof of Theorem 4.1(a), by deducing it from a theorem of representation of arbitrary dualities ∆ : AX → B W (see [7, Theorem 3.1]) for two complete lattices (A, ≤) and (B, ≤) ⊆ (R, ≤), where ≤ is the usual total order on R and extended to arbitrary complete lattices (A, 1 ) and (B, 2 ) in [20, p. 419]. Since we use only the particular case where (A, 1 ) = (B, 2 ), we recall the result only for this case. Let X and W be two sets and A = (A, ) be a complete lattice. For a mapping ∆ : AX → AW , the following statements are equivalent: (1) ∆ is a duality. (2) There exists a function Γ = Γ∆ : X × W × A → A satisfying, for any index set I,
 (4.17) Γ∆ x, w, inf ai = sup Γ∆ (x, w, ai ), x ∈ X, w ∈ W, {ai } ⊆ A, i∈I

x∈X

and such that f ∆ (w) = sup Γ∆ (x, w, f (x)), x∈X

f ∈ AX , w ∈ W ;

(4.18)

moreover, in this case, Γ∆ is uniquely determined by ∆, namely, ∆ (w), Γ∆ (x, w, a) = ψx,a

x ∈ X, w ∈ W, a ∈ A,

(4.19)

where ψx,a : X → A is the mapping defined by

 a, if x = y, ψx,a (y) := +∞, if x = y.

(4.20)

Let us deduce Theorem 4.1(a) from this result, by calculating Γ = Γ∆ for a (∗, s)-duality ∆ : AX → AW . By (4.6), (4.4), and (2.4), for any a ∈ A and x, y ∈ X, we have  e ∗ a = a, if x = y, χ{x} (y) ∗ a = +∞ ∗ a = +∞, if x = y, and hence, by (4.20), ψx,a = χ{x} ∗ a,

x ∈ X, a ∈ A.

(4.21) 2523

If ∆ : AX → AW is a (∗, s)-duality, then, by (4.19), (4.21), and (4.11), we have s Γ∆ (x, w, a) = (χ{x} ∗ a)∆ (w) = χ∆ {x} (w) ∗ a,

x ∈ X, w ∈ W, a ∈ A,

(4.22)

which, together with (4.18), yields s f ∆ (w) = sup {χ∆ {x} (w) ∗ f (x)}, x∈X

f ∈ AX , w ∈ W.

Consequently, for the coupling function (4.15), we obtain (4.14). (b) If ∗ admits a neutral element e, then, by (4.22) and (4.15), we have Γ∆ (x, w, a) = ϕ(x, w) ∗s a,

x ∈ X, w ∈ W, a ∈ A,

(4.23)

whence, in particular (for a = e), Γ∆ (x, w, e) = ϕ(x, w) ∗s e = s(s−1 (ϕ(x, w) ∗ e)) = ϕ(x, w),

x ∈ X, w ∈ W.

(4.24)

Definition 4.4. (a) For any coupling function ϕ : X × W → A such that ∆ = ∆(ϕ) : AX → AW of (4.14) is a (∗, s)-duality, we call ∆(ϕ) the (∗, s)-duality associated to ϕ. (b) If e is a fixed left neutral element for ∗, then, for any (∗, s)-duality ∆ : AX → AW , we call ϕ∆ of (4.15) the coupling function associated to ∆. Corollary 4.1. Let X and W be two sets, A = (A, ) be a complete lattice endowed with a commutative and associative binary operation ∗ : A × A → A admitting a neutral element and satisfying condition (inf-d), and let s : A → A be an antiautomorphism. Then: (a) For each (∗, s)-duality ∆ : AX → AW , there exists a coupling function ϕ = ϕ∆ : X × W → A such that we have (4.14). Moreover, ϕ is uniquely determined by ∆, namely, it is given by (4.15). (b) Conversely, for each coupling function ϕ : X × W → A, the mapping ∆ = ∆(ϕ) : AX → AW defined by (4.14) is a (∗, s)-duality. Proof. This follows from Theorem 4.1 and Remark 2.1(b). Remark 4.3. As was shown above, under the assumptions of Corollary 4.1, there is a one-to-one correspondence between (∗, s)-dualities ∆ : AX → AW and coupling functions ϕ : X × W → A. Corollary 4.2. Under the assumptions of Theorem 4.1(a), for any f ∈ AX , x ∈ X, and w ∈ W , we have the following inequalities: ϕ(x, w) ∗s f (x)  f ∆ (w),

(4.25)

−1 (s−1 ◦ f ∆ (w)) ∗−1 u f (x)  s (ϕ(x, w)).

(4.26)

If, in addition, ∗ is commutative, then we also have −1 (s−1 ◦ f ∆ (w)) ∗−1 u s (ϕ(x, w))  f (x).

(4.27)

Proof. Let f ∈ AX , x ∈ X, and w ∈ W . Inequality (4.25) follows from (4.14). Furthermore, (4.25) means, by (2.14), that s(s−1 (ϕ(x, w)) ∗ f (x))  f ∆ (w), which, by (2.19) (applied to s−1 ), is equivalent to (s−1 ◦ f ∆ (w))  s−1 (ϕ(x, w)) ∗ f (x).

(4.28)

But, by (2.10), the latter inequality is equivalent to (4.26). Finally, if ∗ is commutative, we have, by (4.28), (s−1 ◦ f ∆ (w))  f (x) ∗ s−1 (ϕ(x, w)), which, by (2.10), is equivalent to (4.27). Remark 4.4. Extending the terminology of [11], we call (4.25) the “Fenchel–Young inequality.” Assuming condition (inf-d), one can also give another expression for f ∆ (w) of (4.14), using ∗−1 u instead of ∗s , as follows. 2524

Proposition 4.2. Let X and W be two sets, A = (A, ) be a complete lattice, ϕ : X × W → A be a coupling function, ∗ : A × A → A be a binary operation satisfying condition (inf-d), s : A → A be an antiautomorphism, and ∆ : AX → AW be the mapping defined by (4.14). Then −1 f ∆ (w) = min{b ∈ A | s−1 (b) ∗−1 u s (ϕ(·, w))  f },

f ∈ AX , w ∈ W.

(4.29)

Proof. By (4.14), (2.14), (2.19), and (2.10) we obtain, for any f ∈ AX and w ∈ W , f ∆ (w) = min{b ∈ A | f ∆ (w)  b} = min{b ∈ A | ϕ(·, w) ∗s f  b} = min{b ∈ A | s(s−1 (ϕ(·, w)) ∗ f  b} = min{b ∈ A | s−1 (b)  s−1(ϕ(·, w)) ∗ f } −1 = min{b ∈ A | s−1 (b) ∗−1 u s (ϕ(·, w))  f }.

4.3. Some examples.

.

.

Example 4.1 ((⊗, s)-dualities). As in Example 3.1, let A = G = (G, , ⊗, ⊗ . ) be the canonical enlarge-

.

ment of a boundedly complete lattice ordered group G = (G, , ⊗), and let ∗ = ⊗, s(a) = a−1 , a ∈ A. . Furthermore, let X and W be two sets. Then, by Definition 4.3, a mapping ∆ : AX → AW is a (⊗, s)duality, if we have (4.8) and

.

−1 (f ⊗ c)∆ = f ∆ ⊗ .c ,

f ∈ AX , c ∈ A, w ∈ W.

(4.30)

Moreover, since condition (4.12) holds (see [9, (1.21)]), by Proposition 4.1 it suffices to require, instead of (4.30), that (f ⊗ c)∆ = f ∆ ⊗ c−1 ,

f ∈ AX , c ∈ G, w ∈ W.

(4.31)

.

Now observe that ∗ = ⊗ is commutative, associative, admits a neutral element (namely, the neutral element e of the group G), and satisfies condition (inf-d) (by (3.11)). Hence, by Corollary 4.1, for each . (⊗, s)-duality, there exists a coupling function ϕ : X × W → A such that −1 f ∆ (w) = sup {ϕ(x, w) ⊗ . f (x) }, x∈X

f ∈ AX , w ∈ W.

(4.32)

Moreover, ϕ is uniquely determined by ∆, namely, it is given by (4.15) with χ{x} defined by (4.6). Conversely, for each coupling function ϕ : X × W → A, the mapping ∆ : AX → AW defined by (4.32) is . a (⊗, s)-duality. The above results specialized to this case yield some known results on conjugations for functions with values in extensions of boundedly complete lattice ordered groups (see, e.g., [9, 12, 20] and the references therein). Note that, in the particular case of Example 3.1(a), χ{x} of (4.6) is given by  0, if w = x, (4.33) χ{x} (w) = χ{x},0 (w) = +∞, if w = x

.

(since e = 0 for ∗ = +), while in the case of Example 3.1(b), it is  1, if w = x, χ{x} (w) = χ{x},1 (w) = +∞, if w = x

(4.34)

.

(since e = 1 for ∗ = ×). 2525

Example 4.2 ((∨, s)-dualities). Let A = (A, ) be a complete lattice, ∗ = ∨, s : A → A be an antiautomorphism, and X and W be two sets. Then, by Definition 4.3 and (3.22), a mapping ∆ : AX → AW is a (∗, s)-duality if and only if we have (4.8) and (f ∨ c)∆ = f ∆ ∧ s(c),

f ∈ AX , c ∈ A.

(4.35)

Furthermore, ∗ = ∨ is commutative, associative, and admits a neutral element (namely, e(∨) = −∞, the least element of A). Now assume, in addition, that (A, ) is dual Brauerian, i.e., ∗ = ∨ satisfies condition (inf-d) (see Example 3.2). Then, by Corollary 4.1 and (3.22), for each (∨, s)-duality ∆ : AX → AW , there exists a coupling function ϕ : X × W → A such that f ∆ (w) = sup {ϕ(x, w) ∧ (s ◦ f )(x)}, x∈X

f ∈ AX , w ∈ W.

(4.36)

Moreover, ϕ is uniquely determined by ∆, namely, it is given by (4.15) with χ{x} defined by (4.6), where (since e = −∞ for ∗ = ∨)  −∞, if w = x, χ{x} (w) = χ{x},−∞ (w) = (4.37) +∞, if w = x. Conversely, for each coupling function ϕ : X × W → A, the mapping ∆ : AX → AW defined by (4.36) is a (∨, s)-duality. The above results, specialized to this case, yield some new results for (∨, s)-dualities; also, for A = (R, ≤), they encompass some results of [8] on “∨-dualities.” Example 4.3 ((⊥, s)-dualities). Let A = (A, ) be a Brauerian complete lattice, ∗ = ⊥ (of (3.24)), s : A → A be an antiautomorphism, and X and W be two sets. Then, by Definition 4.3 and (3.35), a mapping ∆ : AX → AW is a (⊥, s)-duality if and only if we have (4.8) and (f ⊥c)∆ = f ∆ s(c),

f ∈ AX , c ∈ A.

(4.38)

However, the problem of a representation of (⊥, s)-dualities, corresponding to (4.32) and (4.36), is more delicate. Indeed, as was observed in Remark 3.2(b), ∗ = ⊥ is neither commutative nor associative. Moreover, by (3.43) or (3.44), there exists no left neutral element for ∗ = ⊥ (but, by (3.44), e = +∞ is a right neutral element for ⊥). Thus, the assumptions of Corollary 4.1 are not satisfied. In order to obtain a representation of (⊥, s)-dualities, we use the technique of duals of (∗, s)-dualities, which will be developed below. 4.4. The dual of a (∗, s)-duality. Recall that if X and W are two sets and A = (A, ) a complete lattice, then the dual of any mapping ∆ : AX → AW is the mapping ∆ : AW → AX defined by


g∆ := inf{f ∈ AX | f ∆  g},

g ∈ AW ,

(4.39)

g ∈ AW , x ∈ X.

(4.40)

i.e.,


g∆ (x) = inf{f (x) | f ∈ AX , f ∆  g}, We have the equivalence f∆  g

⇔




g∆  f,

f ∈ AX , g ∈ AW .

(4.41)

Also, it is well known that the dual ∆ : AW → AX of any mapping ∆ : AX → AW is a duality, and that for any duality ∆ : AX → AW , we have ∆ := (∆ ) = ∆.

(4.42)

Proposition 4.3. Let X and W be two sets, A = (A, ) be a complete lattice endowed with a binary operation ∗ : A × A → A satisfying condition (inf-d), s : A → A be an antiautomorphism, and ∆ : AX → AW be a mapping. s −1 (a) If ∆ is a (∗, s)-duality, then its dual ∆ : AW → AX is a ((∗−1 u ) , s )-duality. 2526

s −1  W → AX is a (∗, s)-duality. (b) If ∆ is a ((∗−1 u ) , s )-duality, then its dual ∆ : A

Proof. (a) Assume that ∆ : AX → AW is a (∗, s)-duality. Clearly, the mapping ∆ defined by (4.39) is a duality, i.e., it satisfies (4.8), mutatis mutandis. Furthermore, it also satisfies (4.9), mutatis mutandis, since by (4.39), (2.14), (2.10), (2.19), (4.9), (4.41), and (2.17) (applied to ∗−1 u ), we have


s ∆ s (g(∗−1 = inf{f ∈ AX | f ∆  g(∗−1 u ) c) u ) c}

= inf{f ∈ AX | f ∆  s ◦ ((s−1 ◦ g) ∗−1 u c)} −1 = inf{f ∈ AX | (s−1 ◦ g) ∗−1 ◦ f ∆} u cs

= inf{f ∈ AX | s−1 ◦ g  (s−1 ◦ f ∆ ) ∗ c} = inf{f ∈ AX | s ◦ ((s−1 ◦ f ∆ ) ∗ c)  g} = inf{f ∈ AX | f ∆ ∗s c  g} = inf{f ∈ AX | (f ∗ c)∆  g}





= inf{f ∈ AX | g∆  f ∗ c} = inf{f ∈ AX | g∆ ∗−1 u c  f}





−1

∆ −1 ss = g∆ ∗−1 c, u c = g (∗u )

f ∈ AX , c ∈ A.

X W −1 s −1  (b) Assume−1that ∆ : A → A is a ((∗u ) , s )-duality. Then, by item (a), we obtain that ∆ is s −1 s , (s−1 )−1 -duality. But by (2.23), we have ((∗−1 )s )−1 = ∗s and, clearly, (s−1 )−1 = s; a (((∗−1 u ) )u ) u u −1

therefore, ∆ is a (∗ss , s)-duality. Hence, by (2.17), ∆ is a (∗, s)-duality. Corollary 4.3. Under the assumptions of Proposition 4.3, we have: s −1 (a) Every ((∗−1 u ) , s )-duality is the dual of a (∗, s)-duality. s −1 (b) Every (∗, s)-duality is the dual of a ((∗−1 u ) , s )-duality. Proof. This follows from Proposition 4.3 and (4.42). The following proposition gives an expression of the dual ∆ of a mapping ∆ : AX → AW defined by (4.14). Proposition 4.4. Let X and W be two sets, A = (A, ) be a complete lattice, ϕ : X × W → A be a coupling function, ∗ : A × A → A be a commutative binary operation, s : A → A be an antiautomorphism, and ∆ : AX → AW be the mapping defined by (4.14). Then


−1 g∆ (x) = sup {(s−1 ◦ g)(w) ∗−1 u s (ϕ(x, w))}, w∈W

g ∈ AW , x ∈ X.

(4.43)

Proof. By (4.39), (4.14), (2.19) (for s−1 ), the commutativity of ∗, and (2.10), we obtain


g∆ = inf{f ∈ AX | f ∆  g} = inf{f ∈ AX | ϕ(·, ·) ∗s f  g} = inf{f ∈ AX | s(s−1 (ϕ(·, ·)) ∗ f )  g} = inf{f ∈ AX | s−1 ◦ g  s−1 (ϕ(·, ·) ∗ f )}

(4.44)

= inf{f ∈ AX | s−1 ◦ g  f ∗ s−1 (ϕ(·, ·))} −1 = inf{f ∈ AX | (s−1 ◦ g) ∗−1 u s (ϕ(·, ·))  f },

g ∈ AW ,

whence


−1 ∆ (s−1 ◦ g) ∗−1 u s (ϕ(·, ·))  g ,

g ∈ AW .

(4.45)

On the other hand, for each g ∈ AW , the function hg : X → A defined by −1 hg (x) := sup {(s−1 ◦ g) ∗−1 u s (ϕ(·, ·))},

x ∈ X,

(4.46)

w∈W

2527

belongs to the set

−1 {h ∈ AX | (s−1 ◦ g) ∗−1 u s (ϕ(·, ·))  h},

whence, by (4.44),


g∆  hg .

(4.47)

Consequently, by (4.45)–(4.47), we obtain (4.43). Using a more general result on duals of arbitrary dualities ∆ : AX → B W , one can obtain the conclusion (4.43) of Proposition 4.4, under different assumptions. Proposition 4.5. Let X and W be two sets, A = (A, ) be a complete lattice endowed with a binary operation ∗ : A × A → A admitting a left neutral element, s : A → A be an antiautomorphism, ∆ : AX → AW be a (∗, s)-duality, and ϕ : X × W → A be a coupling function satisfying (4.14) (for example, by Theorem 4.1(a), one can take ϕ of (4.15)). Then we have (4.43). Proof. By [7, Theorem 3.5], for any duality ∆ : AX → B W , we have


g∆ (x) = sup Γ∆
(w, x, g(w)), w∈W

g ∈ AW , x ∈ X,

(4.48)

where Γ∆
(w, x, b) := min{a ∈ A | Γ∆ (x, w, a)  b},

w ∈ W, x ∈ X, b ∈ A,

(4.49)

with Γ∆ of Remark 4.2. Then, by (4.49), (4.23), (2.14), (2.19) (applied to s−1 ), and (2.10), we obtain Γ∆
(w, x, b) = min{a ∈ A | ϕ(x, w) ∗s a  b} = min{a ∈ A | s(s−1 (ϕ(x, w)) ∗ a)  b} = min{a ∈ A | s−1 (b)  s−1 (ϕ(x, w)) ∗ a}

(4.50)

−1 = min{a ∈ A | s−1 (b) ∗−1 u s (ϕ(x, w))  a} −1 = s−1 (b) ∗−1 u s (ϕ(x, w)),

w ∈ W, x ∈ X, b ∈ B,

which (applied to b = g(w)), together with (4.48), yields (4.43). Corollary 4.4. Let A = (A, ), ∗ : A × A → A, s : A → A, ∆ : AX → AW , and ϕ : X × W → A be as in Proposition 4.4 or Proposition 4.5. Then for any f ∈ AX and g ∈ AW , the following statements are equivalent:
(1) g∆  f ; −1 (2) (s−1 ◦ g)(w) ∗−1 u f (x)  s (ϕ(x, w)), x ∈ X, w ∈ W ; ∆ (3) f  g. Proof. The equivalence (1) ⇔ (3) is nothing else than (4.41). (2) ⇔ (3) For any x ∈ X and w ∈ W , we have, by (2.10) and (2.19), −1 (s−1 ◦ g)(w) ∗−1 u f (x)  s (ϕ(x, w))

⇔ (s−1 ◦ g)(w)  s−1 (ϕ(x, w)) ∗ f (x)   ⇔ s s−1 (ϕ(x, w)) ∗ f (x)  g(w), whence, by (4.14) and (2.14), −1 (s−1 ◦ g)(w) ∗−1 u f (x)  s (ϕ(x, w))

  ⇔ f ∆ (w) = sup s s−1 (ϕ(x, w)) ∗ f (x)  g(w). x∈X

2528

We have the following result. Proposition 4.6. Let X be a set, A = (A, ) be a complete lattice, ∗ : A × A → A be a commutative binary operation satisfying condition (inf-d), s : A → A be an antiautomorphism, and ∆ : AX → AW be a mapping for which there exist both a unique coupling function ϕ∆,∗ : X × W → A such that f ∆ (w) = sup {ϕ∆,∗ (x, w) ∗s f (x)}, x∈X

f ∈ AX , w ∈ W,

(4.51)

and a coupling function ϕ : X × W → A such that we have (4.43). Then ϕ = ϕ∆,∗ .

(4.52)

Proof. By the relation ∆ = (∆ ) , (4.39) (applied to ∆ ), (4.43), and Proposition 2.4(b) (applied to a = f (x), b = s−1 (ϕ(x, w)), c = (s−1 ◦ g)(w), where x ∈ X and w ∈ W ), we have


f ∆ (w) = inf{g(w) | g ∈ AW , g∆  f }   W −1 −1 −1 = inf g(w) | g ∈ A , sup {(s ◦ g)(w) ∗u s (ϕ(·, w))  f w∈W   W s = inf g(w) | g ∈ A , sup {ϕ(x, w) ∗ f }  g(w) x∈X

= sup {ϕ(x, w) ∗ f }, s

x∈X

f ∈ AX , w ∈ W.

Hence, by the assumption of uniqueness of the coupling function ϕ∆,∗ satisfying (4.51), we obtain (4.52). Corollary 4.5. Let X and W be two sets, A = (A, ) be a complete lattice endowed with a commutative and associative binary operation ∗ : A × A → A admitting a neutral element and satisfying condition (inf-d), s : A → A be an antiautomorphism, and ∆ : AX → AW be a (∗, s)-duality. Then there exists a uniquely determined coupling function ϕ : X × W → A such that we have (4.14) and (4.43), namely, function (4.15). Proof. This follows from Corollary 4.1(a) and Propositions 4.4 and 4.6. Proposition 4.7. Let X and W be two sets, A = (A, ) be a complete lattice endowed with a binary operation ∗ : A × A → A satisfying condition (inf-d) and admitting a neutral element e, s : A → A be an antiautomorphism, and ∆ : AX → AW be a (∗, s)-duality with associated coupling function ϕ : X × W → s A. If (∗−1 u ) is commutative and associative, then there exists a uniquely determined coupling function  s −1  W → AX admits the representation ϕ : W × X → A such that the ((∗−1 u ) , s )-duality ∆ : A


g∆ (x) = sup {ϕ (w, x) ∗−1 u g(w)}, w∈W

g ∈ AW , x ∈ X,

(4.53)

namely, ϕ (w, x) = ϕ(x, w),

w ∈ W, x ∈ X.

(4.54)

s −1 Proof. First, we recall that, by Proposition 4.3, ∆ is a ((∗−1 u ) , s )-duality. Note also that, by (2.22), (2.9), and (2.14) we have, for any a ∈ A, s s −1   s a(∗−1 u ) e = a(∗ )l e = max{a ∈ A | a ∗ e  a}

= max{a ∈ A | s(s−1 (a ) ∗ e)  a} = max{a ∈ A | a  a} = a, s −1 s and hence, since (∗−1 u ) is assumed commutative, it follows that e is a neutral element for (∗u ) . Consequently, by Theorem 4.1(a), (b) (mutatis mutandis), there exists a uniquely determined coupling function

2529

ϕ : W × X → A such that we have, for any g ∈ AW and x ∈ X,   −1
s s g∆ (x) = sup {ϕ (w, x) (∗−1 g(w)} = sup {ϕ (w, x) ∗−1 u ) u g(w)}, w∈W

w∈W

where the last equality holds by (2.17) (applied to ∗ replaced by ∗−1 u ). This proves (4.53). Furthermore, by Remark 4.2(a) applied to ∆ replaced by ∆ , we have (4.48), where


w ∈ W, x ∈ X, b ∈ B,

∆ Γ∆
(w, x, b) = ψw,b (x),

with ψw,b defined, for each w ∈ W , b ∈ B, and w ∈ W , by  b, if w = w ,  ψw,b(w ) = +∞, if w = w . Defining ϕ : W × X → A by




ϕ (w, x) := χ∆ {w} (x),

from (4.22) (with ∆, ∗, and s replaced by w ∈ W , x ∈ X, and b ∈ B,

∆ ,

∗−1 u ,

w ∈ W, x ∈ X,

and s−1 respectively) and (2.17), we obtain, for any

  −1 s s Γ∆
(w, x, b) = ϕ (w, x) (∗−1 b = ϕ (w, x) ∗−1 u ) u b.

Finally, by (4.24), ∆ = any x ∈ X and w ∈ W ,

∆ ,

(4.49) (applied to ∆ replaced by

∆ ),

(4.55)

(4.55), and (2.10), we obtain, for

ϕ(x, w) = Γ∆ (x, w, e) = Γ∆

(x, w, e) = min{b ∈ A | Γ∆
(w, x, b)  e} = min{b ∈ A | ϕ (w, x) ∗−1 u b  e} = min{b ∈ A | ϕ (w, x)  b ∗ e = b} = ϕ (w, x).


For a (∗, s)-duality ∆ : AX → AW , the representation of g∆ yields, in particular, the following

representation of the “bidual” f ∆∆ := (f ∆ )∆ ∈ AX of a function f ∈ AX . Proposition 4.8. Under the assumptions of Proposition 4.5, we have, for any f ∈ AX and x ∈ X,


−1 f ∆∆ (x) = sup {(s−1 ◦ f ∆ )(w) ∗−1 u s (ϕ(·, w))} w∈W

= sup min{a ∈ A | ϕ(x, w) ∗s a  f ∆ (w)} w∈W   −1 s (ϕ(x, w)) = sup inf {s−1 (ϕ(y, w)) ∗ f (y)} ∗−1 u w∈W

=

(4.56)

y∈X

sup w∈W, b∈A −1 (ϕ(x,w))f s−1 (b) ∗−1 u s

−1 {s−1 (b) ∗−1 u s (ϕ(x, w))}.

Proof. Let f ∈ AX and x ∈ X. The first equality in (4.56) is obtained by applying (4.43) to g = f ∆ . Furthermore, by (2.10), the commutativity of ∗, (2.19), and (2.14), we obtain, for any w ∈ W , −1 −1 ◦ f ∆ )(w)  a ∗ s−1 (ϕ(., w))} (s−1 ◦ f ∆ )(w) ∗−1 u s (ϕ(·, w)) = min{a ∈ A | (s

= min{a ∈ A | (s−1 ◦ f ∆ )(w)  s−1 (ϕ(·, w)) ∗ a}    = min a ∈ A | s s−1 (ϕ(·, w)) ∗ a  f ∆ (w) = min{a ∈ A | ϕ(x, w) ∗s a  f ∆ (w)}, which, together with the first equality in (4.56), yields the second equality of (4.56). 2530

Furthermore, by the first equality of (4.56), (4.14), (2.14), and (2.21) (applied to s−1 ), we have


−1 f ∆∆ (x) = sup {(s−1 ◦ f ∆ )(w) ∗−1 u s (ϕ(·, w))} w∈W 



s−1

= sup w∈W

w∈W

= sup w∈W

s

= sup

−1 ∗−1 u s (ϕ(·, w))

sup {ϕ(y, w)) ∗s f (y)} y∈X

 −1

sup s{s

 

−1

(ϕ(y, w)) ∗ f (y)}

y∈X



inf {s

y∈X

−1

(ϕ(y, w)) ∗ f (y)}

∗−1 u

s

−1

 ∗−1 u

s

−1

(ϕ(·, w))

 (ϕ(x, w)) .

This proves the second equality of (4.56). Finally, recall that, by [7, Theorem 3.6], for any duality ∆ : AX → AW , we have


f ∆∆ (x) = sup {Γ∆
(w, x, b) | Γ∆
(w, ·, b)  f }, w∈W b∈B

x ∈ X,

(4.57)

with Γ∆ of (4.49). But, by (4.49), (4.23), (2.14), (2.19) (for s−1 ), and (2.10), we obtain Γ∆
(w, x, b) = min{a ∈ A | Γ∆ (x, w, a)  b} = min{a ∈ A | ϕ(x, w) ∗s a  b} = min{a ∈ A | s(s−1 (ϕ(x, w)) ∗ a)  b} = min{a ∈ A | s−1 (b)  s−1 (ϕ(x, w)) ∗ a}

(4.58)

−1 = min{a ∈ A | s−1 (b) ∗−1 u s (ϕ(x, w))  a} −1 = s−1 (b) ∗−1 u s (ϕ(x, w)),

w ∈ W, x ∈ X, b ∈ A.

Substituting (4.58) in (4.57), we obtain the last equality of (4.56). s The assumption of commutativity of (∗−1 u ) in Proposition 4.7 is rather restrictive, since the commus −1 s tativity of ∗ does not imply that of (∗−1 u ) . In order to relax the assumption of commutativity of (∗u ) , let us introduce the following definition.

Definition 4.5. Let A = (A, ) be a complete lattice, ∗ : A × A → A be a binary operation satisfying condition (inf-d), and s : A → A be an antiautomorphism. We say that an element c ∈ A is (a) a singular value of ∗ if there exist a, b ∈ A such that s −1 s a(∗−1 u ) b ≺ c  b(∗u ) a;

(4.59)

(b) a regular value of ∗ if for any a, b ∈ A there holds the equivalence s c  a(∗−1 u ) b

⇔

s c  b(∗−1 u ) a.

(4.60)

s Remark 4.5. (a) If (∗−1 u ) is commutative, then ∗ has no singular value. (b) c = −∞ is always a regular value of ∗.

Proposition 4.9. Let X and W be two sets, A = (A, ) be a complete lattice endowed with a commutative binary operation ∗ : A × A → A, which admits a neutral element and satisfies condition (inf-d), s : A → A be an antiautomorphism, and ∆ : AX → AW be a (∗, s)-duality with an associated coupling function ϕ : X × W → A. The following statements are equivalent: (1) ∆ : AW → AX is a (∗, s)-duality with an associated coupling function ϕ : W × X → A satisfying (4.54); (2) ϕ takes only regular values of ∗. 2531

Proof. (1) ⇒ (2) Assume that (1) holds and let x ∈ X, w ∈ W , and a0 , b0 ∈ A be such that s ϕ(x, w)  b0 (∗−1 u ) a0 .

(4.61)

Then, by (4.52), (4.23), (4.49), (2.14), (2.19), (2.10), and (4.61), we have ϕ(x, w) ∗s b0 = ϕ (w, x) ∗s b0 = Γ∆
(w, x, b0 ) = min{a ∈ A | Γ∆ (x, w, a)  b0 } = min{a ∈ A | ϕ(x, w) ∗s a  b0 } = min{a ∈ A | s−1 (b0 )  s−1 (ϕ(x, w))  a}

(4.62)

−1 = min{a ∈ A | s−1 (b) ∗−1 u a  s (ϕ(x, w))} s = min{a ∈ A | ϕ(x, w)  b0 (∗−1 u ) a}  a0 .

But, by (2.14), (2.19), and (2.10), we have the equivalences ϕ(x, w) ∗s b0 = s(s−1 (ϕ(x, w)) ∗ b0 )  a0

⇔

s−1 (a0 )  s−1 (ϕ(x, w)) ∗ b0

⇔

−1 s−1 (a0 ) ∗−1 u b0  s (ϕ(x, w))

⇔

s ϕ(x, w)  a0 (∗−1 u ) b0 .

Thus, ϕ(x, w) is a regular value of ∗. (2) ⇒ (1) Assume that (2) holds and let x ∈ X, w ∈ W , and a, b ∈ A. Then, by (4.60) with c = ϕ(x, w), we have the equivalence s ϕ(x, w)  a(∗−1 u ) b

⇔

s ϕ(x, w)  b(∗−1 u ) a.

(4.63)

But, by (2.14), (2.19), and (2.10), we have s −1 −1 −1 ϕ(x, w)  a(∗−1 u ) b ⇔ s (a) ∗u b  s (ϕ(x, w))

s−1 (a)  s−1 (ϕ(x, w)) ∗ b

⇔

⇔

ϕ(x, w) ∗s b  a,

and, swapping a and b, s ϕ(x, w)  b(∗−1 u ) a

⇔

ϕ(x, w) ∗s a  b.

Hence, by (4.63), we obtain the equivalence ϕ(x, w) ∗s b  a

⇔

ϕ(x, w) ∗s a  b.

(4.64)

Consequently, by (4.49), (4.23), and (4.64), we obtain Γ∆
(w, x, b) = min{a ∈ A | Γ∆ (x, w, a)  b} = min{a ∈ A | ϕ(x, w) ∗s a  b}

(4.65)

= min{a ∈ A | ϕ(x, w) ∗ b  a} = ϕ(x, w) ∗ b. s

s

Hence, by (4.48) and (4.65),


g∆ (x) = sup Γ∆
(w, x, g(w)) = sup {ϕ(x, w) ∗s g(w)}, w∈W

w∈W

and, therefore, by Proposition 4.6,

∆

g ∈ AX , x ∈ X,

is a (∗, s)-duality with the associated coupling function (4.54).

4.5. Some examples.

.

.

Example 4.4 (the dual of a (⊗, s)-duality). As in Example 4.1, let A = G = (G, , ⊗, ⊗ . ) be the canon-

.

ical enlargement of a boundedly complete lattice ordered group G = (G, , ⊗), ∗ = ⊗, s(a) = a−1 , a ∈ A,

.

. −1

.

X and W be two sets, and ∆ : AX → AW be a (⊗, s)-duality (4.32). Then, by (3.17), (⊗u )s = ⊗ 2532

is commutative and associative. Hence, by Proposition 4.7, there exists a uniquely determined coupling function ϕ : W × X → A, namely, ϕ (w, x) = ϕ(x, w), x ∈ X, w ∈ W , such that we have

. −1




g∆ (x) = sup {ϕ (w, x) ⊗u g(w)}, w∈W

g ∈ AW , x ∈ X.

Hence, by (3.14), we obtain the representation


−1 g∆ (x) = sup {ϕ(x, w) ⊗ . g(w) }, x∈X

g ∈ AW , x ∈ X.

(4.66)

Example 4.5 (the dual of a (∨, s)-duality). Let A = (A, ) be a dual Brauerian complete lattice, ∗ = ∨, s : A → A be an antiautomorphism, X and W be two sets, and ∆ : AX → AW be a (∗, s)-duality (4.36). Then, by Propositions 4.3 and 4.4, the dual of ∆ is the (⊥, s−1 )-duality ∆ : AW → AX admitting the representation
−1 g ∈ AW , x ∈ X, g∆ (x) = sup {(s−1 ◦ g)(w) ∨−1 u s (ϕ(x, w))}, w∈W

i.e., by (3.26),


g∆ (x) = sup {(s−1 ◦ g)(w)s−1 (ϕ(x, w))},

g ∈ AW , x ∈ X.

(4.67)

w∈W

4.6. (⊥, s)-Dualities revisited. Now we are ready to give a representation of (⊥, s)-dualities with the aid of a coupling function. Proposition 4.10. Let A = (A, ) be a complete lattice which is both Brauerian and dual Brauerian, ∗ = ⊥ (of (3.24)), s : A → A be an antiautomorphism, X and W be two sets, and σ : X × W → A be a coupling function. Then the mapping ∆ : AX → AW defined by f ∆ (w) = sup {(s ◦ f )(x)σ(x, w)}, x∈X

f ∈ AX , w ∈ W,

(4.68)

is a (⊥, s)-duality. Conversely, for every (⊥, s)-duality ∆ : AX → AW , there exists a coupling function σ : X × W → A such that we have (4.68). Proof. We define a mapping Θ : AW → AX by gΘ (x) := sup {(s−1 ◦ σ)(x, w) ∧ (s−1 ◦ g)(w)}, w∈W

g ∈ AW , x ∈ X.

(4.69)

Then, by Example 4.2, Θ is a (∨, s−1 )-duality. Indeed, the roles of X, W , ϕ, s, and f in (4.36) are now played by W , X, s−1 ◦ σ, s−1 , and g, respectively. Furthermore, by Example 4.5 (mutatis mutandis), the dual of the (∨, s−1 )-duality Θ is nothing else than Θ = ∆ in (4.68). Hence, by Proposition 4.3 and s −1 (3.27), ∆ is a ((∨−1 u ) , s )-duality, i.e., equivalently, a (⊥, s)-duality. Conversely, let ∆ : AX → AW be a (⊥, s)-duality. Then ∆ = (∆ ) , where, by Proposition 4.3 s −1 −1 and (3.37), ∆ : AW → AX is a ((⊥−1 u ) , s )-duality, i.e., equivalently, a (∨, s )-duality. Hence, by −1 Proposition 4.5 (applied to ∗ = ∨ and s replaced by s ) and (3.26), we obtain f ∆ (w) = sup {(s ◦ f )(x)s(ϕ(w, x))}, x∈X

f ∈ AX , w ∈ W,

(4.70)

where the coupling function ϕ : W × X → A is defined by


ϕ(w, x) = (χ{w},∨ )∆ (x),

w ∈ W, x ∈ X.

(4.71)

Consequently, for the coupling function σ : X × W → A defined by σ(x, w) := s(ϕ(w, x)),

x ∈ X, w ∈ W,

(4.72)

formula (4.70) yields (4.68). 2533

Remark 4.6. (a) By (3.26) and (2.8), formula (4.68) can be also written in the following equivalent form: f ∆ (w) = sup min{b ∈ A | (s ◦ f )(x)  b ∨ σ(x, w)}, x∈X

f ∈ AX , w ∈ W.

(4.73)

(b) In the particular case where A = (R, ≤) and s(a) = −a, a ∈ R, the coupling function σ of (4.68) is uniquely determined by ∆, and one can give an explicit formula for it (see [8, Theorem 3.1]). 4.7. s-Dualities associated to a set Ω ⊆ X × W . Now we study a class of dualities that are defined without using any binary operation ∗, but which turn out to be (∗, s)-dualities for all binary operations ∗ : A × A → A satisfying condition (inf-d). These dualities encompass some important particular cases. Definition 4.6. Let A = (A, ) be a complete lattice, s : A → A be an antiautomorphism, X and W be two sets, and Ω ⊆ X × W . We call the mapping ∆Ω : AX → AW defined by f ∆Ω (w) := s(

inf

x | (x,w)∈Ω

f (x)) =

sup

(s ◦ f )(x),

f ∈ AX , w ∈ W,

x | (x,w)∈Ω

(4.74)

the s-duality associated to Ω. This terminology is justified, since we have the following assertion. Proposition 4.11. Let A = (A, ) be a complete lattice, s : A → A be an antiautomorphism, X and W be two sets, and Ω ⊆ X × W . Then the mapping ∆Ω : AX → AW defined by (4.74) is a duality. Proof. We have to show that every mapping (4.74) satisfies (4.8). Let I be any index set and {fi }i∈I ⊆ AX . If I = ∅, then, by (2.1), (4.74), and (2.20), we have ∆Ω
inf fi (w) = +∞∆Ω (w) = s(+∞) = −∞ = sup f ∆Ω (w), w ∈ W. i∈∅

i∈∅

On the other hand, if I = ∅, then, by (4.74), ∆Ω


s ◦ inf fi (x) = inf fi (w) = sup i∈I

i∈I

x | (x,w)∈Ω

= sup

sup

sup

(s ◦ fi )(x) = sup fi∆Ω (w),

i∈I x | (x,w)∈Ω

sup(s ◦ fi )(x)

x | (x,w)∈Ω i∈I

w ∈ W.

i∈I

The next proposition shows that an s-duality is in fact a (∗, s)-duality, for every binary operation ∗ : A × A → A satisfying condition (inf-d). Proposition 4.12. Let A = (A, ) be a complete lattice, s : A → A be an antiautomorphism, ∗ : A×A → A be a binary operation satisfying condition (inf-d), X and W be two sets, and Ω ⊆ X × W . Then the mapping ∆Ω : AX → AW defined by (4.74) is a (∗, s)-duality. Proof. By Proposition 4.11, (4.74) is a duality. Furthermore, by (4.74), we have, for any f ∈ AX , c ∈ A, and w ∈ W ,

 
 sup (s ◦ f )(x) ∗ c inf (f ∗ c)(x) = s s−1 (f ∗ c)∆Ω (w) = s x | (x,w)∈Ω

x | (x,w)∈Ω

   = s s−1 ◦ f ∆Ω (w) ∗ c = f ∆Ω (w) ∗s c.

By Proposition 4.12, we may call ∆Ω the (∗, s)-duality associated to the set Ω. We have the following characterization of (∗, s)-dualities associated to a set Ω, among (∗, s)-dualities, in terms of the associated coupling function. 2534

Proposition 4.13. Let A = (A, ) be a complete lattice, s : A → A be an antiautomorphism, ∗ : A×A → A be a commutative binary operation satisfying condition (inf-d) and admitting a neutral element e, and X and W be two sets. For a (∗, s)-duality ∆ : AX → AW , the following statements are equivalent: (1) There exists a set Ω ⊆ X × W such that ∆Ω = ∆. (2) There exists a coupling function associated to ∆ (in the sense of Definition 4.4) such that ϕ(x, w) ∈ {s(e), −∞},

(x, w) ∈ X × W.

Proof. (1) ⇒ (2) Assume that (1) holds and let  s(e), if (x, w) ∈ Ω, ϕ(x, w) := −∞, if (x, w) ∈ / Ω.

(4.75)

(4.76)

Then, since ∗ satisfies condition (inf-d), ∗s satisfies condition (sup-d) (by Proposition 2.3(a)), whence −∞ ∗s f (x) = −∞. Hence, by (4.14), (4.76), (2.14), and (4.74), for any f ∈ AX and w ∈ W , we obtain f ∆(ϕ) (w) = sup {ϕ(x, w) ∗s f (x)} = x∈X

=

sup x | (x,w)∈Ω

{s(e) ∗s f (x)}

sup

x | (x,w)∈Ω

  s s−1 (s(e)) ∗ f (x) =

sup

(s ◦ f )(x) = f ∆Ω (w).

x | (x,w)∈Ω

(2) ⇒ (1) Assume that (2) holds and let Ω = {(x, w) ∈ X × W | ϕ(x, w) = s(e)}.

(4.77)

Then, as above, since ∗ satisfies condition (inf-d), ∗s satisfies −∞ ∗s f (x) = −∞. Hence, by (4.14), (4.75), (4.77), (2.14), and (4.74), for any f ∈ AX and w ∈ W , we obtain f ∆(ϕ) (w) = sup {ϕ(x, w) ∗s f (x)} = x∈X

=

sup

x | (x,w)∈Ω

sup

{s(e) ∗s f (x)}

x | ϕ(x,w)=s(e)

  s s−1 (s(e)) ∗ f (x) =

sup

(s ◦ f )(x) = f ∆Ω (w).

x | (x,w)∈Ω

Proposition 4.14. For the dual ∆Ω : AW → AX of the (∗, s)-duality ∆Ω in Proposition 4.13, we have


g∆Ω (x) =

sup

(s−1 ◦ g)(x),

w | (x,w)∈Ω

g ∈ AW , x ∈ X.

Proof. For ϕ : X × W → A defined by (4.76), we have   −1 (s(e)) = e, if (x, w) ∈ Ω, s s−1 (ϕ(x, w)) = = χΩ (x, w), /Ω s−1 (−∞) = +∞, if (x, w) ∈

(4.78)

(4.79)

where χΩ : X × W → {e, +∞} is the indicator function of the set Ω with respect to the neutral element e. Hence, by (4.43) and (4.79),


−1 g∆Ω (x) = sup {(s−1 ◦ g)(w) ∗−1 u s (ϕ(x, w))} w∈W

=

sup

{(s−1 ◦ g)(w) ∗−1 u χΩ (x, w)},

w | (x,w)∈Ω

g ∈ AW , x ∈ X.

(4.80)

2535

But, by (2.10) and the definition of χΩ (see (4.79)), for any g ∈ AW , x ∈ X, and w ∈ W with (x, w) ∈ Ω, we have −1 ◦ g)(w)  a ∗ χΩ (x, w)} (s−1 ◦ g)(w) ∗−1 u χΩ (x, w) = min{a ∈ A | (s

= min{a ∈ A | (s−1 ◦ g)(w)  a} = (s

−1

(4.81)

◦ g)(w).

Hence, from (4.80) and (4.81), we obtain (4.78). Let us now express the second dual. Proposition 4.15. For the (∗, s)-duality ∆Ω in Proposition 4.13, we have


f ∆Ω∆Ω (x) =

sup

inf

w | (x,w)∈Ω y | (y,w)∈Ω

f (y),

f ∈ AX , x ∈ X.

(4.82)

Proof. By (4.78) (applied to g = f ∆Ω ) and (4.74), for any f ∈ AX and x ∈ X, we have


∆Ω ∆Ω −1 ∆Ω −1 f s inf (x) = sup s (f (x)) = sup s f (x) , w | (x,w)∈Ω

w | (x,w)∈Ω

x | (x,w)∈Ω

i.e., (4.82). We have the following characterizations of s-dualities associated to a set Ω ⊆ X × W , in terms of some particular (∗, s)-dualities. Theorem 4.2. Let A = (A, ) be a complete lattice that is both Brauerian and dual Brauerian, s : A → A be an antiautomorphism, and X and W be two sets. For a mapping ∆ : AX → AW , the following statements are equivalent: (1) There exists a set Ω ⊆ X × W such that ∆ = ∆Ω . (2) ∆ is simultaneously a (∨, s)-duality and a (⊥, s)-duality. (3) ∆ is a (∨, s)-duality and its dual ∆ is a (∨, s−1 )-duality. (4) ∆ is a (⊥, s)-duality and its dual ∆ is a (⊥, s−1 )-duality. Proof. The implication (1) ⇒ (2) follows from Proposition 4.12 (applied to ∗ = ∨ and ∗ = ⊥). (2) ⇒ (1) Assume that (2) holds. Then, since ∆ is a (∨, s)-duality, by Example 4.2 there exists a unique coupling function ϕ : X × W → A satisfying (4.36), namely, the function ϕ(x, w) = χ{x},−∞ (w) in (4.37). Now, by (3.24) and (4.37), we have χ{x},−∞ = χ{x},−∞ ⊥a for all x ∈ X and a ∈ A\{−∞, +∞}. Hence, using also that ∆ is a (⊥, s)-duality and (4.35), we obtain, for each (x, w) ∈ X × W such that ϕ(x, w) > −∞, −∞ < ϕ(x, w) = (χ{x},−∞ )∆ (w) = (χ{x},−∞ ⊥a)∆ = χ∆ {x},−∞ s(a) = ϕ(x, w)s(a), and, therefore, by (3.24), ϕ(x, w) > s(a) for all a ∈ A\{−∞, +∞}; this inequality remains valid also for a = +∞, since s(+∞) = −∞. Consequently, since s : A → A is an antiautomorphism, we have ϕ(x, w) = +∞. This proves that ϕ(x, w) ∈ {−∞, +∞},

(x, w) ∈ X × W.

(4.83)

Ω := {(x, w) ∈ X × W | ϕ(x, w) = +∞}.

(4.84)

Let Then

2536

 +∞, if (x, w) ∈ Ω, ϕ(x, w) = −∞, if (x, w) ∈ / Ω.

(4.85)

But, since −∞ is the neutral element e for ∗ = ∨ and s(−∞) = +∞, formula (4.85) is nothing other than (4.76) for ∗ = ∨. Hence, by Proposition 4.13, we obtain f ∆ (w) =

sup

(s ◦ f )(x),

x | (x,w)∈Ω

f ∈ AX , w ∈ W,

i.e., (1). Finally, the equivalences (2) ⇔ (3) and (2) ⇔ (4) are immediate consequences of Proposition 4.3, taking into account (3.22), (3.35) (applied to s−1 ), (3.27), and (3.38). Remark 4.7. In the particular case where A = (R, ≤) and s(a) = −a (a ∈ R), there exist more complete results on dualities associated to Ω ⊆ X × W (see [8, Theorem 5.4]). 4.8. Some examples.

.

Example 4.6. Let A = R = (R, ≤, +, + . ) and s be as in Example 3.1(a), X and W be two sets, and

.

.

Ω ⊆ X ×W . Then + satisfies condition (inf-d) and hence ∆Ω of (4.74) is a (+, s)-duality. In the particular case where ϕ : X × W → R is a coupling function and Ω = {(x, w) ∈ X × W | ϕ(x, w) > 0},

(4.86)

this mapping becomes f ∆Ω (w) = −

inf

x∈X ϕ(x,w)>0

f (x),

X

f ∈ R , w ∈ W,

(4.87)

which is the “conjugate of type Lau of f with respect to ϕ” (see, e.g., [20]).

.

Example 4.7. Let A = R+ = (R+ , ≤, ×, × . ) and s be as in Example 3.1(b), X and W be two sets,

.

.

and Ω ⊆ X × W . Then × satisfies condition (inf-d) and hence ∆Ω of (4.74) is a (×, s)-duality. If ϕ : X × W → R+ is a coupling function and Ω = {(x, w) ∈ X × W | ϕ(x, w) > 1},

(4.88)

this mapping becomes f ∆Ω (w) = −

inf

x∈X ϕ(x,w)>1

f (x),

X

f ∈ R+ , w ∈ W.

(4.89)

n and ϕ(x, w) = min x w , x = (x ) ∈ Rn , w = (w ) ∈ Rn , the In the particular case where X = W = R+ i i i i 1≤i≤n

“conjugation” (4.89) was introduced by Rubinov and S ¸ im¸sek [16, 17] and further studied by Rubinov and Glover [15]. 5. (∗, s)-Subdifferentials Let X and W be two sets, A = (A, ) be a complete lattice, and ∆ : AX → AW be a duality represented as (4.18), with Γ∆ : X × W × A → A of (4.19), and let us consider the representation (4.48) of the dual ∆ : AW → AX of ∆ with Γ∆
of (4.49). We recall that, following [11], the subdifferential with respect to ∆, or, briefly, the ∆-subdifferential, of a function f : X → A at a point x0 ∈ X is the subset of W defined by ∂ ∆ f (x0 ) := {w ∈ W | Γ∆
(w, x0 , f ∆ (w)) = f (x0 )}.

(5.1)

For the dual ∆ : AW → AX of ∆, we have (∆ ) = ∆, whence, by (5.1) (mutatis mutandis),





∂ ∆ g(w0 ) = {x ∈ X | Γ∆ (x, w0 , g∆ (x)) = g(w0 )}.

(5.2)

In this section, we consider the particular case where ∆ is a (∗, s)-duality with ∗ : A × A → A and s : A → A as in the preceding sections. 2537

Proposition 5.1. Let X and W be two sets, A = (A, ) be a complete lattice endowed with a binary s operation ∗ : A × A → A satisfying condition (inf-d), s : A → A be an antiautomorphism such that (∗−1 u ) X W admits a left neutral element, and ∆ : A → A be a (∗, s)-duality. Then there exists a coupling function ϕ : W × X → A such that


g∆ (x) = sup {ϕ (w, x) ∗−1 u g(w)}, w∈W

g ∈ AW , x ∈ X,

(5.3)

and for any such ϕ and any f ∈ AX and w ∈ W , we have ∆ ∂ ∆ f (x0 ) = {w ∈ W | ϕ (w, x0 ) ∗−1 u f (w) = f (x0 )}.

(5.4)

s −1 Proof. By Proposition 4.3(a), the dual ∆ : AW → AX of ∆ is a ((∗−1 u ) , s )-duality. Furthermore, by −1 s Corollary 2.1(a), (∗u ) satisfies condition (inf-d). Hence, by Theorem 4.1(a) (with ∆, ∗, and s replaced s −1 −1 by ∆ , (∗−1 u ) , and s , respectively), and by (2.17) (applied to ∗ replaced by ∗u ), there exists a coupling function ϕ : W × X → A such that we have (5.3). On the other hand, by the result of [7] mentioned at the beginning of Remark 4.2(a), for the s −1  ((∗−1 u ) , s )-duality ∆ , there exists a function Γ∆
: W × X × A → A such that


g∆ (x) = sup Γ∆
(w, x, g(w)),

g ∈ AW , x ∈ X,

(5.5)

w∈W

where Γ∆
is uniquely determined by ∆ , namely (by (4.23), mutatis mutandis), −1

ss Γ∆
(w, x, b) = ϕ (w, x)(∗−1 b = ϕ (w, x) ∗−1 u ) u b,

w ∈ W, x ∈ X, b ∈ A,

(5.6)

with ϕ of (5.3). Hence, by (5.1) and (5.6), we obtain (5.4). Moreover, representation (5.4) of ∂ ∆ f (x0 ) does not depend on the function ϕ satisfying (5.3) and (5.6), since in definition (5.1) of ∂ ∆ f (x0 ), the function Γ∆
is uniquely determined by ∆ . Proposition 5.2. Let X and W be two sets, A = (A, ) be a complete lattice, ϕ : X × W → A be a coupling function, ∗ : A × A → A be a binary operation admitting a left neutral element, s : A → A be an antiautomorphism, and ∆ : AX → AW be a duality represented in the form (4.14). Then −1 ∂ ∆ f (x0 ) = {w ∈ W | (s−1 ◦ f ∆ )(w) ∗−1 u s (ϕ(x0 , w)) = f (x0 )}.

(5.7)

Proof. This follows from (5.1) and (4.50). Remark 5.1. From (5.7) and the inequality (4.27) of Corollary 4.2, it follows that −1 ∂ ∆ f (x0 ) = {w ∈ W | f (x0 )  (s−1 ◦ f ∆ )(w) ∗−1 u s (ϕ(x0 , w))}.

(5.8)

Hence, in particular, if f (x0 ) = −∞, then ∂ ∆ f (x0 ) = W . Definition 5.1. Let A = (A, ) be a complete lattice, ∅ = D ⊆ A, and ∗ : A × A → A be a binary operation satisfying condition (inf-d) and admitting a left neutral element. An element b ∈ A is called a (D, ∗−1 u )-interchangeable value if for all a ∈ A and d ∈ D, we have the equivalence a ∗−1 u b=d

⇔

a ∗−1 u d = b.

(5.9)

Some examples of (D, ∗−1 u )-interchangeable values in the particular case where (A, ) = (R, ≤) and ∗ = +, ∗ = ∨, and some applications can be found in [11].

.

Proposition 5.3. Let X and W be two sets, A = (A, ) be a complete lattice, ϕ : X × W → A be a coupling function, ∗ : A × A → A be a binary operation admitting a left neutral element, s : A → A be an antiautomorphism, and ∆ : AX → AW be a duality represented in the form (4.14). If f (x0 ) is a (s−1 (ϕ(X × W )), ∗−1 u )-interchangeable value, then we have the equivalence w ∈ ∂ ∆ f (x0 ) 2538

⇔

s f ∆ (w)(∗−1 u ) f (x0 ) = ϕ(x0 , w).

(5.10)

Proof. By (5.7), (5.9), and (2.14) (applied to ∗−1 u ), w ∈ ∂ ∆ f (x0 )

⇔

−1 (s−1 ◦ f ∆ )(w) ∗−1 u s (ϕ(x0 , w)) = f (x0 )

⇔ ⇔

−1 (s−1 ◦ f ∆ )(w) ∗−1 u f (x0 ) = s (ϕ(x0 , w))  −1  s (s ◦ f ∆ )(w) ∗−1 u f (x0 ) = ϕ(x0 , w)

⇔

s f ∆ (w) (∗−1 u ) f (x0 ) = ϕ(x0 , w).

Proposition 5.4. Let X and W be two sets, A = (A, ) be a complete lattice, ϕ : X × W → A be a coupling function, ∗ : A × A → A be a binary operation admitting a left neutral element, s : A → A be an antiautomorphism, and ∆ : AX → AW be a duality represented in the form (4.14). Then, for an element w0 ∈ W , the following statements are equivalent: (1) w0 ∈ ∂ ∆ f (x0 ); (2) There exists b ∈ A such that −1 s−1(b) ∗−1 u s (ϕ(x, w0 ))  f (x),

s

−1

(b) ∗−1 u

s

−1

x ∈ X,

(ϕ(x0 , w0 )) = f (x0 ).

(5.11) (5.12)

Moreover, in this case f ∆ (w0 ) = min{b ∈ A | b satisfies (5.11) and (5.12)}.

(5.13)

Proof. By [11, Theorem 1.1], which remains valid, with the same proofs, for any complete lattice A = (A, ) (as has been observed in [12, Remark 3.2] and [20, p. 454]), if ∆ : AX → AW is a duality represented as (4.18), with Γ∆ : X × W × A → A of (4.19) and whose dual ∆ : AW → AX is represented as (4.48), with Γ∆
: W × X × A → A of (4.49), then for an element w0 ∈ W , we have w0 ∈ ∂ ∆ f (x0 ) if and only if there exists b ∈ A such that Γ∆
(w0 , ·, b)  f,

(5.14)

Γ∆
(w0 , x0 , b) = f (x0 ),

(5.15)

f ∆ (w0 ) = min{b ∈ A | b satisfies (5.14) and (5.15)}.

(5.16)

and in this case we have

Consequently, by (4.50), we obtain the desired conclusion.


Finally, let us consider ∂ ∆ g(w0 ). Proposition 5.5. Let X and W be two sets, A = (A, ) be a complete lattice endowed with a binary operation ∗ : A × A → A admitting a left neutral element, s : A → A be a bijection, and ∆ : AX → AW be a (∗, s)-duality represented in the form (4.14), with a coupling function ϕ : X × W → A. Then for the dual ∆ : AW → AX of ∆, we have





∂ ∆ g(w0 ) = {x ∈ X | ϕ(x, w0 ) ∗s g∆ (x) = g(w0 )},

g ∈ AW , w0 ∈ W.

(5.17)

Proof. This follows from (5.2) and (4.23). Proposition 5.6. Let X and W be two sets, A = (A, ) be a complete lattice endowed with a binary operation ∗ : A × A → A satisfying condition (inf-d) and admitting a neutral element, s : A → A be a bijection, and ∆ : AX → AW be a (∗, s)-duality represented in the form (4.14), with a coupling function ϕ : X × W → A. Given f ∈ AX and x0 ∈ X, for any w0 ∈ W we have the implication ϕ(x0 , w0 ) ∗s f (x0 ) = f ∆ (w0 )

⇒




x0 ∈ ∂ ∆ f ∆ (w0 ).

(5.18) 2539

Proof. For any f ∈ AX , w0 ∈ W , and g = f ∆ , set (5.17) becomes





∂ ∆ f ∆ (w0 ) = {x ∈ X | ϕ(x, w0 ) ∗s f ∆∆ (x) = f ∆ (w0 )}.

(5.19)

ϕ(x0 , w0 ) ∗s f (x0 ) = f ∆ (w0 ).

(5.20)

Let x0 ∈ X be such that


Then, by the relation f ∆∆  f and the isotony of ∗ (see Remark 2.1(c)), we have


s−1 (ϕ(x0 , w0 )) ∗ f ∆∆ (x0 )  s−1(ϕ(x0 , w0 )) ∗ f (x0 ), whence, by (5.20), (2.19), and (2.14), f ∆ (w0 ) = ϕ(x0 , w0 ) ∗s f (x0 ) = s(s−1 (ϕ(x0 , w0 )) ∗ f (x0 ))


 s(s−1 (ϕ(x0 , w0 )) ∗ f ∆∆ (x0 )) = ϕ(x0 , w0 ) ∗ f s

∆∆


(5.21)

(x0 ).


But, by inequality (4.25) of Corollary 4.2 (applied to f ∆∆ ), we have





ϕ(x0 , w0 ) ∗s f ∆∆ (x0 )  f ∆∆ ∆ (w0 ) = f ∆ (w0 ).

(5.22)

∆


Hence, from (5.21), (5.22), and (5.17) for g = f ∆ , we obtain x0 ∈ ∂ f ∆ (w0 ). Corollary 5.1. Under the assumptions of Proposition 5.6, given f ∈ AX and x0 ∈ X, for any w0 ∈ W
with f (x0 ) = f ∆∆ (x0 ), we have the equivalence ϕ(x0 , w0 ) ∗s f (x0 ) = f ∆ (w0 )

⇔




x0 ∈ ∂ ∆ f ∆ (w0 ).

(5.23)

Proof. The implication ⇒ holds for any f ∈ AX , by Proposition 5.6. The reverse implication ⇐ follows
from f (x0 ) = f ∆∆ (x0 ) and (5.19). Acknowledgment. The work of the second author was partially supported by DGICYT (Spain), Project PB98-0867, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya, Grant 2000SGR-154. REFERENCES 1. F. Baccelli, G. Cohen, G. J. Olsder, and J.-P. Quadrat, Synchronization and Linearity, Wiley Series in Probability and Mathematical Statistics, Wiley (1992). 2. T. S. Blyth and M. F. Janowitz, Residuation Theory, Pergamon Press, Oxford (1972). 3. G. Birkhoff, Lattice Theory (3rd ed.), Amer. Math. Soc. Coll. Publ. 25, Providence (1967). ´ 4. G. Cohen, “Residuation and applications,” In: 26-`eme Ecole de printemps d’informatique th´eorique. Alg`ebres max-plus et applications en informatique et automatique (Ile de Noirmoutier), INRIALIAFA-IRCyN (1998), pp. 203–233. 5. L. Fuchs, Partially Ordered Algebraic Systems, Pergamon Press (1963). 6. A. I. Kokorin and V. M. Kopytov, Fully Ordered Groups [in Russian], Nauka, Moscow (1972); English translation: Wiley (1974). 7. J.-E. Mart´ınez-Legaz and I. Singer, “Dualities between complete lattices,” Optimization, 21, 481–508 (1990). 8. J.-E. Mart´ınez-Legaz and I. Singer, “∨-Dualities and ⊥-dualities,” Optimization, 22, 483–511 (1991). 9. J.-E. Mart´ınez-Legaz and I. Singer, “∗-Dualities,” Optimization, 30, 295–315 (1994). 10. J.-E. Mart´ınez-Legaz and I. Singer, “Dualities associated to binary operations on R,” J. Convex Anal., 2, 185–209 (1995). 11. J.-E. Mart´ınez-Legaz and I. Singer, “Subdifferentials with respect to dualities,” Math. Methods Oper. Res., 42, 109–125 (1995). 2540

12. J.-E. Mart´ınez-Legaz and I. Singer, “On conjugations for functions with values in extensions of complete ordered groups,” Positivity, 1, 193–218 (1997). 13. J.-E. Mart´ınez-Legaz and I. Singer, “An extension of d.c. duality theory, with an Appendix on ∗subdifferentials,” Optimization, 42, 9–37 (1997). 14. J.-J. Moreau, “Inf-convolution, sous-additivit´e, convexit´e des fonctions num´eriques,” J. Math. Pures Appl., 49, 109–154 (1970). 15. A. M. Rubinov and B. M. Glover, “Duality for increasing positively homogeneous functions and normal sets,” Recherche Op´erationnelle/Oper. Res., 32, 105–123 (1998). 16. A. M. Rubinov and B. S¸im¸sek, “Dual problems of quasiconvex maximization,” Bull. Aust. Math. Soc., 51, 139–144 (1995). 17. A. M. Rubinov and B. S ¸ im¸sek, “Conjugate quasiconvex nonnegative functions,” Optimization, 35, 1–22 (1995). 18. I. Singer, “Conjugation operators,” In: Selected Topics in Operations Research and Mathematical Economics, Lect. Notes Econ. Math. Syst., Vol. 226, Springer-Verlag, Berlin (1984), pp. 80–97. 19. I. Singer, “Infimal generators and dualities between complete lattices,” Ann. Mat. Pura Appl. (4), 148, 289–358 (1987). 20. I. Singer, Abstract Convex Analysis, Wiley-Interscience, Wiley, New York (1997). 21. U. Zimmermann, Linear and Combinatorial Optimization in Ordered Algebraic Structures, North Holland, Amsterdam (1981).

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