Appl Categor Struct (2007) 15:243–257 DOI 10.1007/s10485-007-9061-z

Capacities with Values in Compact Hausdorff Lattices Oleh R. Nykyforchyn

Received: 11 May 2006 / Accepted: 5 January 2007 / Published online: 24 January 2007 © Springer Science + Business Media B.V. 2007

Abstract For a compact Hausdorff topological lattice L the set ML X of L-valued normed regular capacities on a compact Hausdorff topological space X is investigated. It is shown that the set ML X carries a compact Hausdorff topology, and ML extends to a weakly τ -normal functor in the category of compacta. If L is an upper and lower Lawson semilattice, then ML is the functorial part of two semimonads. These semimonads coincide and are a monad if and only if L is distributive, i.e., is a Lawson lattice. The obtained results have a natural interpretation if capacities are regarded as subjective estimates of likelihood of realization of events in conditions of uncertainty. Key words capacity · monad · category of compacta · Lawson lattice Mathematics Subject Classifications (2000) 18B30 · 06F30 · 54B30

1 Introduction Capacities were introduced by Choquet [1] and found numerous applications to mathematical physics, the theory of partial differential equations, geometry, and lately to mathematical economy [9–13] for simulation of decision making under uncertainty. Since capacities can be regarded as generalization of classical probabilities, they naturally appear in the theory of stochastic processes [2, 4, 5]. The set of regular normed capacities with nonnegative real values on a topological space X carries a topology [12] that for a compact Hausdorff X is compact and Hausdorff as well [2]. Any continuous mapping of compacta induces a continuous mapping between the respective spaces of capacities so that a functor of (real-valued) capacities in the category of compacta is obtained. This functor is the functorial part of the capacity

O. R. Nykyforchyn (B) Precarpathian National University, Ivano-Frankivsk, Ukraine e-mail: [email protected]

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monad that was investigated in detail in [18]. It turned out unexpectedly that this monad is related much more closely to the inclusion hyperspace monad, than to the probability measure monad. It is natural to study capacities with values in sets other than R, e.g. D-capacities for an idempotent semiring D (see [4]). The present paper aims in systematic investigation of functorial properties of spaces of capacities on compacta, i.e. on compact Hausdorff spaces, with values in compact Hausdorff topological lattices, first of all in Lawson lattices.

2 Preliminaries We write A ⊂ X (resp. A ⊂ X) if A is a closed (resp. open) subset of a topological cl

op

space X. The closure and the interior of arbitrary set A are denoted by Cl A and Int A respectively. A compactum is a compact Hausdorff topological space. The hyperspace of a compactum X is the set exp X of all nonempty closed subsets of X equipped by Vietoris topology [15]. The standard base of this topology consists of all sets of the form U 1 , U 2 , . . . , U n  = {G ∈ exp X | G ⊂ U 1 ∪ U 2 ∪ . . . U n , G ∩ U i  = ∅, i = 1, 2, . . . , n}, where sets U i are open in X. The hyperspace of a compactum is also a compactum. For a continuous mapping f : X → Y of compacta the mapping exp f : exp X → exp Y, that is defined by the formula exp f (A) = { f (x) | x ∈ A} for A ∈ exp X, is continuous [15]. Denote by C omp the category of compacta; its objects are compacta and arrows are continuous mappings of compacta. It is well known that exp is an endofunctor in C omp and satisfies the introduced by E.V. Shchepin definition of normal functor. An endofunctor in C omp is called normal [14] if it preserves empty set(=the initial object), singletons (=terminal objects), mono- and epimorphisms, intersections (=limits of diagrams that consist of monomorphisms with a common target and distinct sources), preimages (=pullbacks with one arrow being mono), codirected limits (also called “limits of inverse spectra” by topologists) and weight of infinite compacta. If preservation of weight is known to hold only for compacta of weight  τ , where τ is an infinite cardinal number, then the functor is called τ -normal [16]. If also preservation of preimages is not guaranteed, the functor is called weakly τ -normal. Codirected limits offers powerful methods for investigating of images of nonmetrizable compacta under the above mentioned classes of functors. By Theorem 1 from [19], for a functor in C omp, preservation of intersections and preservation of the classes of mono- and epimorphisms imply preservation of codirected limits, therefore this condition in the definition of weakly τ -normal functor is superfluous. We follow V.V. Fedorchuk [15] and call a triple F that consists of a functor F : C → C and natural transformations η : 1C → F and μ : F 2 → F a semimonad if the unit laws in the definition of monad (but not necessarily the associative law) are satisfied. We also extend in the obvious manner to semimonads the usual definitions of morphism and isomorphism of monads.

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If a poset contains a least (a greatest) element, we denote it by 0 (resp. 1). In this article we consider only (semi)lattices that contain at least two elements. Recall that an upper (lower) semilattice (L, ) is called topological if L carries a topology such that the supremum x ∨ y (respectively the infimum x ∧ y) is continuous with respect to x, y ∈ L. If the supremum and the infimum both exist for each x, y ∈ L and are continuous, call L a topological lattice. Call an upper (lower) semilattice (L, ) complete if for each nonempty set A ⊂ L there is sup A (resp. inf A). If (L, ) is a lattice and for each nonempty A ⊂ L there are sup A and inf A, call the lattice complete. A complete upper (lower) topological semilattice (L, , τ ) is called upper (resp. lower) Lawson semilattice, if the mapping sup : exp L → L (resp. inf : exp L → L), that sends each nonempty closed A ⊂ L to its supremum (infimum), is continuous with respect to the Vietoris topology. A Lawson lattice is a complete distributive topological lattice (L, , τ ) such that the mappings sup : exp L → L and inf : exp L → L are continuous. For an arbitrary subset A of a poset L denote: A↓ = {β ∈ L | β  α for some α ∈ A}, A↑ = {β ∈ L | β  α for some α ∈ A}. Lemma 1 Let L be a compact Hausdorff topological upper semilattice. If a set A ⊂ L is closed (or open), then the set A↓ ⊂ L is closed (resp. open). Proof Let A be closed, then the subset {β, α) | β ∨ α = α, α ∈ L, α ∈ A} of a compactum L × A is closed, therefore is a compactum. Its projection to the first factor is equal to the set A↓ and is compactum as well. This imply that A↓ is closed in L. If A is open, then for each β ∈ A↓ there is α ∈ A such that α ∨ β = α. For “∨” is continuous, there is an open set V  α such that α   α  ∨ β ∈ A for all α  ∈ V, that implies V ⊂ A↓. Thus A↓ is open.   Lemma 2 Let (L, ) be a compact topological upper semilattice. Then (L, ) is complete. Proof For A ⊂ L consider all sets {α}↑ for all α ∈ A and all {β}↓ for all upper bounds β of A. Then any finitely many of these sets have a nonempty intersection, because α1 ∨ α2 ∨ · · · ∨ αn belongs to {αi }↑ for all i = 1, 2, . . . , n and also to {β}↓ for every upper bound β of A. By the previous lemma all these sets are closed. Thus compactness of L implies that all of them have a nonempty intersection. But any point of this intersection must be a supremum of A.   Note that if a nonempty subset A ⊂ L is closed and x ∨ y ∈ A (resp. x ∧ y ∈ A) for all x, y ∈ A, then A is a subsemilattice itself and by the previous lemma A contains a greatest (resp. a least) element. In particular, this implies that any compact topological upper semilattice contains a greatest element, and any compact topological lower semilattice contains a least element. Lemma 3 Let L be a compact topological upper semilattice, A be a filtered by inclusion family of closed subsets of L such that the intersection of A equals {β}. Then the intersection of sets A↓ for all A ∈ A is equal to {β}↓.

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Proof It is obvious thatthe intersection of all A↓ contains {β}↓. To prove the converse, consider γ ∈ A∈A A↓. Then all sets {γ }↑ ∪ A are closed, nonempty and form a filtered family in a compact space. Thus there exist β  ∈ L such that   β  ∈ {γ }↑ ∪ A for all A ∈ A. Obviously β  = β and γ ∈ {β}↓. Remark 1 It is straightforward that in a compact Hausdorff lower semilattice L the statements dual to the last three lemmata hold: if a set A is closed (or open), then the set A↑ is closed (open) as well. A compact topological lower semilattice is complete. For a family A that satisfy the condition of Lemma 3 the intersection of all A↑ for A ∈ A is equal to {β}↑.

3 Notion of Capacity with Values in a (Semi)lattice 3.1 The Space of Capacities and the Functor of Capacities Let L be a compact Hausdorff topological upper semilattice that contains a least element. A function c : exp X ∪ {∅} → L is called an L-valued capacity on a compactum X if the following hold: (1) c(∅) = 0, c(X) = 1; (2) For each closed subsets F, G in X the inclusion F ⊂ G implies c(F)  c(G) (monotonicity); (3) For any open set V ⊂ L the set c−1 (V↓) \ {∅} ⊂ exp X is open with respect to the Vietoris topology. It is obvious that if conditions (1)–(2) hold, then (3) is equivalent to the following statement: (3 ) If the value c(F) for a closed set F ⊂ X belongs to an open set V ⊂ L, then there is an open set W ⊃ F such that c(Cl W) ∈ V↓. Moreover, we can formulate a condition that in the presence of (1) is equivalent to (2) and (3) together: (2 ) If F ⊂ X is closed and c(F) is in an open V ⊂ L, then there exists an open subset U ⊃ F such that c(G) ∈ V↓ for any closed G ⊂ X such that G ⊂ U (upper semicontinuity). Note that a capacity that complies to this definition is normed and outer regular [3]. Denote by ML X the set of all L-valued capacities on a compactum X. The subgraph of a capacity c ∈ ML X is a set sub c = {(F, α) | F ∈ exp X, α ∈ L, α  c(F)} ⊂ exp X × L. We omit an easy proof of the following Lemma 4 Let X be a compactum, L be a compact Hausdorff upper semilattice that contains a least element. A subset S ⊂ exp X × L is the subgraph of an L-valued capacity c if and only if the following conditions are satisfied for all closed nonempty subsets F, G of X: (1) If F ⊂ G, α  β, (F, α) ∈ S, then (G, β) ∈ S; (2) If (F, α), (G, β) ∈ S, then (F ∪ G, α ∨ β) ∈ S;

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(3) (4)

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S ⊃ {X} × L; S is closed.

If these conditions hold, then a capacity c is unique. Denote by sub the mapping ML X → exp(exp X × L) that sends each capacity to its subgraph. By the above lemma sub is injective. Let X be a compactum, L be a compact Hausdorff topological lattice. We briefly outline how to define a compact Hausdorff topology on ML X. The proofs are also omitted because they are straightforward and can be easily given. First we prove that subsets S ⊂ exp X × L that satisfy conditions (1)–(4) from the previous lemma, form a closed subset in exp(exp X × L). This implies that the image of a mapping sub : ML → exp(exp X × L) is a compactum. Next we define a topology on ML X by a subbase that consists of all sets of the form O+ (U, V) = {c ∈ ML X | there is F ⊂ U such that c(F)  α for some α ∈ V} cl

= {c ∈ ML X | there is F ⊂ U, c(F) ∈ V↑}, cl

where U ⊂ X, V ⊂ L, and op

op

O− (F, V) = {c ∈ ML X | c(F)  α for some α ∈ V} = {c ∈ ML X | c(F) ∈ V↓}, where F ⊂ X, V ⊂ L. It is easy to check that the above defined topology on ML X cl

op

is Hausdorff. Then we prove that the images of all sets of the form O+ (U, V) and O− (F, V), where U ⊂ X, F ⊂ X, V ⊂ L, are open in the image of this mapping. op

cl

op

Observe that the mapping sub−1 : sub(ML ) → ML is a continuous bijection of the compactum sub(ML ) onto the Hausdorff space ML and therefore a homeomorphism. Thus ML X is a compactum. Let f : X → Y be a continuous mapping of compacta. Define a mapping ML f : ML X → ML Y by the formula ML f (c)(F) = c( f −1 (F)). We obtain: Theorem 1 If L is a compact Hausdorff topological lattice of weight τ , then the assignment ML is a weakly τ -normal but not a τ -normal functor in the category C omp of compacta. The mapping sub : ML X → exp(exp X × L) is an embedding for each compactum X.

4 The Monad of Capacities Observe that the mapping η L X : X → ML X that is defined by a formula  1, if x ∈ F η L X(x)(F) = 0, if x ∈ /F is a component of a natural transformation η L : 1C omp → ML .

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We supplement η L with a natural transformation η L : ML 2 → ML such that a triple M L = (ML , η L , μ L ) is a semimonad and in certain cases even a monad. We omit a straightforward proof of the next lemma. Lemma 5 Let X be a compactum, L be a compact Hausdorff topological lattice. Then the set sub L X = {(F, c, α) | F ∈ exp X, c ∈ ML X, α ∈ L, c(F)  α} is closed in exp X × ML X × L. For α ∈ L and a closed set F in a compactum X denote by Fα the set {c ∈ ML X | c(F)  α}. Similarly denote by FA the set {c ∈ ML X | c(F) ∈ A↑} for F ⊂ X, A ⊂ L. cl

Lemma 6 Let L be a compact Hausdorff topological lattice, X be a compactum, F be closed in X and A be closed in L. Then the set FA is closed in ML X. Proof By Lemma 5 the set sub L X ∩ ({F} × ML X × A↑) is closed in the compactum exp X × ML X × L. Its projection to the second factor is a compactum as well and is exactly the set FA , thus FA ⊂ ML X.   cl

Consequently, the set Fα = F{α} is closed in ML X for each α ∈ L and each closed set F ⊂ X. Lemma 7 Let X be a compactum, L be a compact Hausdorff topological lattice, A be a filtered by inclusion set of closed subsets in L with the intersection that contains only one element α. Then the intersection of all sets of the form FA for A ∈ A is equal to Fα , and for each open neighborhood W of the set Fα in ML X there is a set A ∈ A such that FA ⊂ W. Proof Observe that by Lemma 3 and due to the fact that continuous mapping of compacta preserve intesections of filtered families of closed sets, the following holds:   FA = pr2 (sub L X ∩ ({F} × ML X × A↑)) A∈A

A∈A

=



pr2 

=



 (sub L X ∩ ({F} × ML X × A↑))

A∈A

pr2 sub L X ∩



 ({F} × ML X × A↑)

A∈A

=

pr2 (sub L X ∩ ({F} × ML X × {β}↑)) = Fα .

This implies that the sets (ML X \ W) ∩ FA for A ∈ A form a filtered family of compacta with empty intersection. Therefore there is A ∈ A such that (ML X \ W) ∩ FA = ∅, i.e., FA ⊂ W.   Call a closed set F ⊂ X α-reliable with respect to a capacity C ∈ ML 2 X, if C(Fα )  α. Obviously, F is α-reliable with respect to C ∈ ML 2 X if and only if there is a closed subset F ⊂ ML X such that C(F )  α and c(F)  α for all c ∈ F .

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Take C ∈ ML 2 X and define a mapping μ L X(C) : exp X ∪ {∅} → L by the formula μ L X(C)(F) = sup{α ∈ L | C(Fα )  α} for F ⊂ X. cl

The following lemma is obvious: Lemma 8 Let X be a compactum, L be a compact Hausdorff topological lattice, C ∈ ML 2 X, F ⊂ X, γ ∈ L. Then the inequality μ L X(C)(F)  γ is equivalent to existence cl

of a set A ⊂ L such that sup A  γ and C(Fα )  α for all α ∈ A. Lemma 9 Let X be a compactum, L be a compact Hausdorff topological lattice which is an upper Lawson semilattice, C ∈ ML 2 X. Then μ L X(C) is a capacity on X. Proof Obviously the condition (1) holds. Prove the upper semicontinuity. The subgraph sub C, being closed in exp ML X × L, is a compactum. A closed set RC ⊂ exp(exp X × ML X × L) defined by the equality RC = {D ⊂ sub L X | D = {F} × F × {α} for some F ∈ exp X, (F , α) ∈ sub C} cl

is a compactum as well. Deduce from this that the union of the projections of elements of the family RC to the first and the third factors is also a compactum:  rel C = exp pr13 (RC ) = {(F, α) | there exists F ⊂ ML X such that C(F )  α cl

and for all c ∈ F the inequality c(F)  α holds} ⊂ L × exp X. Observe that the set rel C consists of all pairs (F, α) such that F ∈ exp X is αreliable with respect to C. Denote c = μ L X(C) and assume that c(F) ∈ V, V ⊂ L. op

The case F = ∅ is trivial, therefore consider F  = ∅, that implies sup{α ∈ L | (F, α) ∈ rel C} ∈ V. For L is an upper Lawson semilattice, there exists an open neighborhood V  , of the closed set {α ∈ L | (F, α) ∈ rel C}, such that sup A ∈ V↓ for all A ⊂ V  . Then sup A ∈ V↓ holds also for A ⊂ V  ↓. The set rel C \ exp X × V  ↓ is compact and does not intersect {F} × L, thus there exists an element U 1 , . . . , U n  of the standard base of the Vietoris topology, such that F ∈ U 1 , . . . , U n  and (U 1 , . . . , U n  × L) ∩ (rel C \ exp X × V  ↓) = ∅. This means that F ⊂ U 1 ∪ · · · ∪ U n , and for each G ∈ U 1 , . . . , U n  all α ∈ L, such that G is α-reliable with respect to C, are in V  ↓. This implies that c(G) ∈ V↓ for any closed set G that is included in U = U 1 ∪ · · · ∪ U n .   Lemma 10 Let X be a compactum, L be a compact Hausdorff topological lattice, that is both upper and lower Lawson semilattice. Then the mapping μ L X : ML 2 X → ML X is continuous. Proof It is sufficient to verify that the preimages under μ L X of all subbase elements of the forms O+ (U, V) and O− (F, V) are open. Let μ L X(C) ∈ O− (F, V). The set rel C constructed in the proof of Lemma 9, is closed, thus the set {α ∈ L | F is α-reliable with respect to C} is also closed. By the assumption, its supremum is in V↓. The semilattice L being upper Lawson, there exists an open neighborhood V  of the set {α ∈ L | F is α-reliable with respect to C}, such that sup Cl V  ∈ V↓. For each α ∈ / V  we have C(Fα )   α, and by the definition of topological lattice there are open neighborhoods Vα  α and Wα  C(Fα ) such that

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for each β ∈ Vα and β  ∈ Wα the inequality β  β  does not hold. By Lemma 7 and upper semicontinuity of the capacity C there is an open neighborhood Uα  α in L such that Cl Uα ⊂ Vα , C(FCl Uα ) ∈ (Wα )↓. Since sets Uα  α for α ∈ L \ V  form an open cover of the compactum L \ V  , we can choose a finite subcover Uα1 , . . . , Uαn . Let OC = O− (FCl Uα1 , Wα1 ) ∩ · · · ∩ O− (FCl Uαn , Wαn ). Assume that C ∈ OC and α ∈ L \ V  , then α ∈Uαi ⊂ Vαi for some i ∈ {1, 2, . . . , n}. Obtain C (Fα )  C (FCl Uαi ) ∈ (Wαi )↓, and due to the choice of Vαi and Wαi the inequality C (Fα )  α is impossible. Thus all α ∈ L such that C (Fα )  α, are in V  , and their supremum μ L X(C )(F) is in V. Therefore C ∈ OC ⊂ μ L X −1 (O− (F, V)), and the preimage of O− (F, V) is open in ML X. Let μ L X(C) ∈ O+ (U, V). Then the supremum α0 of the set {α ∈ L | there exists F ⊂ U, F ⊂ X, that is α-reliable with respect to C} cl

is in V↑. There is a finite set {α1 , . . . , αn } ⊂ L such that sup{α1 , . . . , αn } ∈ V↑ and for each its element αi there exists an αi -reliable with respect to C set Fi ⊂ U, Fi ⊂ X. cl

The set F = F1 ∪ · · · ∪ Fn is closed in X as well, is included in U and is αi -reliable for all i = 1, 2, . . . , n. By the definitions of upper and lower Lawson semilattices there are open neighborhoods V1  α1 , . . . , Vn  αn such that sup{inf V1 , . . . , inf Vn } ∈ V↑. Choose an open neighborhood OF ⊃ F such that its closure is contained in U, and put OC = O+ (O+ (OF, V1 ), V1 ) ∩ · · · ∩ O+ (O+ (OF, Vn ), Vn ). It is straightforward to verify that OC is an open neighborhood of the element C in ML 2 X, and for each capacity C ∈ OC there exist closed sets H1 , . . . , Hn ⊂ ML X and elements α1 ∈ V1 , . . . , αn ∈ Vn such that each set Hi is in O+ (OF, Vi ) and C (Hi )  αi holds for i = 1, 2, . . . , n. For a capacity c ∈ Hi we have c(Cl OF) ∈ Vi ↑, thus c(Cl OF)  inf Vi ↑ and the set Cl OF is αi -reliable with respect to C with αi = αi ∧ inf Vi = inf Vi . Therefore μ L X(C )(Cl OF)  sup{α1 , . . . , αn } = sup{inf V1 , . . . , inf Vn } ∈ V↑, and μ L X(C ) ∈ O+ (U, V). Thus C ∈ OC ⊂ μ L X −1 (O+ (U, V)), and the preimage μ L X −1 (O+ (U, V)) is open. This completes the proof of continuity of μ L X.   Theorem 2 Let L be a compact Hausdorff topological lattice that is both upper and lower Lawson semilattice. The mappings μ L X : ML 2 X → ML X for X ∈ Ob C omp form a natural transformation μ L : ML 2 → ML . The triple M L = (ML , η L , μ L ) is a semimonad. It is a monad if and only if the lattice L is distributive.1 Proof It is easy to verify that μ L is a natural transformation and the unit laws are satisfied. Let the lattice L be distributive, i.e., a Lawson lattice. We prove that the associative law holds. Recall that every compact Lawson semilattice L is completely distributive [6, 7], that is for each collection  of sets (Mt )t∈T in L the equality inf{sup Mt | t ∈ T} = sup{inf{αt | t ∈ T} | (αt )t∈T ∈ t∈T Mt } holds. Let C ∈ ML 3 X,C1 = ML μ L X(C ),

1 I.e.,

is a Lawson lattice.

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C2 = μ L ML X(C ), c1 = μ L X(C1 ), c2 = μ L X(C2 ), F ⊂ X. Then Lemma 8 implies that the inequality c1 (F)  γ is equivalent to:

cl

(1) There is a subset A ⊂ L such that sup A  γ and for any α ∈ A there exists a subset Tα ⊂ ML 2 X such that C (Tα )  α and for any capacity C ∈ Tα there is cl

BC ⊂ L such that sup BC  α and C(Fβ )  β for all β ∈ BC . Analogously by Lemma 8 the inequality c2 (F)  γ means that (2) There is a subset A ⊂ L such that sup A  γ and for each α  ∈ A there exists a subset Bα ⊂ L such that sup Bα  α  and C ((Fα )β )  β for any element β ∈ Bα .  Let (1) hold. Fix α ∈ A and put Bα = {α ∧ inf{βC | C ∈ Tα } | (βC )C∈Tα ∈ C∈Tα BC }. Then for each β = α ∧ inf{βC | C ∈ Tα } ∈ Bα , C ∈ Tα the inequality β  βC is true, thus C(Fβ )  C(FβC )  βC  β, therefore Tα ⊂ (Fβ )β and C ((Fβ )β )  C (Tα )  α  β. Using complete distributivity of the lattice L, we obtain sup Bα = sup{α ∧  inf{βC | C ∈ Tα } | (βC )C∈Tα ∈ C∈Tα BC } = α ∧ inf{sup BC | C ∈ Tα } = α, resting on sup BC  α for each  C ∈ Tα . Now put A = α∈A Bα and Bα = {α  } for each α  ∈ A . Then sup A  sup A  γ , and β = α  for any β ∈ Bα . Thus obtain C ((Fα )β ) = C ((Fβ )β )  β, i.e., (2) holds. Now assume (2). Put A = {α  ∧ β | α  ∈ A , β ∈ Bα }, and let Tα = (Fα )α for each α ∈ A. If C ∈ Tα , put BC = {α}. Then sup A = sup{α  ∧ sup Bα | α  ∈ A } = sup A  γ , and for each α ∈ A we have α = α  ∧ β for some α  ∈ A , β ∈ Bα . Thus C (Tα ) = C ((Fα )α )  C ((Fα )β )  β  α. If C ∈ Tα and β ∈ BC , then C(Fα )  α and β = α, that imply C(Fβ )  β. Therefore (1) is true. This implies c1 = c2 , i.e., μ L X ◦ μ L ML X(C ) = μ L X ◦ ML μ L X(C ), and the triple M L = (ML , η L , μ L ) is a monad. Now assume that L is not distributive, i.e., there are elements α, β1 , β2 such that α ∧ (β1 ∨ β2 )  = (α ∧ β1 ) ∨ (α ∧ β2 ). Then α ∧ (β1 ∨ β2 )  (α ∧ β1 ) ∨ (α ∧ β2 ), 0 < α, β1 , β2 < 1, and neither of the inequalities β1  β2 , β2  β1 is true. We show that the equality μ L X ◦ μ L ML X = μ L X ◦ ML μ L X fails. Observe that for any compactum X, closed nonempty subsets B1 , B2 , . . . , Bn ⊂ X and elements α1 , α2 , . . . , αn ∈ L the following mapping is an L-valued capacity:  1, if F = X, δ(B1 , α1 ; B2 , α2 ; . . . ; Bn , αn )(F) = sup{αi | 1  i  n, Bi ⊂ F} otherwise. In these denotation we drop braces around singletons in X, e.g., δ({x}, α; B, β) = δ(x, α; B, β). Let X = {x1 , x2 , x3 }, F = {x1 , x2 }, γ = α ∧ (β1 ∨ β2 ). Consider the capacity C = δ({C , C }, γ ) ∈ ML 3 X where C , C ∈ ML 2 X, C = δ(δ(x1 , β1 ), β1 ; δ(x2 , β2 ), β2 ), C = δ(δ(F, α), α). We prove that ML μ L X(C )(F)  γ . Put A = {γ }, T = {C , C }, BC = {β1 , β2 }, BC = {α}. Then C (T) = γ , sup BC = β1 ∨ β2  γ , sup BC = α  γ . Since δ(x1 , β1 ) ∈ Fβ1 , we have C (Fβ1 )  β1 . Similarly C (Fβ2 )  β2 , C (Fα ) = α. Thus (1) holds, that implies ML μ L X(C )(F)  γ . Assume that μ L ML X(C )(F)  γ for the chosen C , F and γ , i.e., (2) is true. Let us investigate α  , β ∈ L \ {0} such that the inequality C ((Fα )β )  β is possible. Since (Fα )β  = ML 2 X, we obtain C ((Fα )β )  γ and consequently β  γ and C , C ∈

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(Fα )β . Thus C (Fα )  β, C (Fα )  β. The inequality C (Fα ) > 0 holds, therefore at least one of the inequalities C (Fα )  β1 and C (Fα )  β2 is true, and we have δ(x1 , β1 ) ∈ Fα or δ(x2 , β2 ) ∈ Fα . This is equivalent to α   β1 or α   β2 . Similarly C (Fα ) > 0 implies α  α  . The element α  ∈ A to be nonzero, it is necessary that sup Bα  = 0, i.e., Bα  ⊂ {0}. By the above-mentioned arguments this is possible only if α   α and also α   β1 or α   β2 , that implies α   (α ∧ β1 ) ∨ (α ∧ β2 ) for each element α  ∈ A . Thus sup A  (α ∧ β1 ) ∨ (α ∧ β2 ) < α ∧ (β1 ∨ β2 ) = γ , and (2) is impossible. Since μ L X ◦ ML μ L X(C )(F)  = μ L X ◦ μ L ML X(C )(F), the associative law fails, and triple M L = (ML , η L , μ L ) is not a monad.  

5 Dual Capacities and Morphisms of Monads ˜ the lattice obtained by reversing the order. Since both For a lattice L denote by L ˜ i.e., lattices contain the same elements, we add tildes to denotations that concern L, ˜, ˜ , ∧, ˜ coincides with , sup ˜ A↓˜ etc. Obviously  ˜ ∨, ˜ sup, we write  ˜ inf, ˜ A↑, ˜ with inf ˜ Thus we can and so forth. If L is a compact Hausdorff topological lattice, then so is L. ˜ also consider L-valued capacities, and all properties that are proved for the functor ML : C omp → C omp hold for the functor ML˜ : C omp → C omp. If L is an upper and ˜˜ = L. ˜ Obviously L lower Lawson semilattice or a Lawson lattice, then so is L. ˜ ˜ For c ∈ ML X we define a mapping c˜ : exp X ∪ {∅} → L by the formula c(F) = ˜ = sup{c(G) | G ⊂ X, G ∩ F = inf{c(G) ˜ | G ⊂ X, G ∩ F = ∅}, or, equivalently, c(F) cl

∅}. We omit a proof of the following lemma.

cl

Lemma 11 Let X be a compactum, L be a compact Hausdorff lattice, c be an L˜ valued capacity. Then c˜ is a L-valued capacity. Call the capacity c˜ dual to the capacity c. The relation of duality for capacities is symmetric, which is expressed by the following lemma. Lemma 12 If X is a compactum, L is a compact Hausdorff topological lattice, c is an L-valued capacity, then c˜˜ = c. ˜˜ = inf{sup{c(H) | H ∩ G = ∅} | G ∩ F = ∅}. If Proof For any F ⊂ X we have c(F) cl

G ∩ F = ∅, then c(F) ∈ {c(H) | H ∩ G = ∅}, and c(F)  sup{c(H) | H ∩ G = ∅}. ˜˜ Thus c(F)  inf{sup{c(H) | H ∩ G = ∅} | G ∩ F = ∅} = c(F). Assume that c(F) < ˜˜ By (3 ) there ˜ ˜c(F). ˜ is open and c(F) ∈ V = V↓   c. ˜ By Lemma 1 the set V = L \ {c}↑ exist W ⊂ X such that F ⊂ W and c(Cl W) ∈ V↓. Let G = X \ W. Then G ⊂ X, op

cl

G ∩ F = ∅, and for any H ⊂ X, H ∩ G = ∅ we have H ⊂ W ⊂ Cl W and c(H)  cl

˜˜ = inf{sup{c(H) | c(Cl W), thus sup{c(H) | H ∩ G = ∅}  c(Cl W) ∈ V↓, and c(F) ˜˜ H ∩ G = ∅} | G ∩ F = ∅} ∈ V↓, that is a contradiction. Therefore only c(F) = c(F) is possible.   For a compactum X and a compact Hausdorff topological lattice L we define a pair of mutually inverse mappings κ L X : ML X → ML˜ X and κ L˜ X : ML˜ X → ˜ It is straightforward ML X by the same formulae κ L X(c) = c˜ and κ L˜ X(c) = c.

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to check that these mappings are continuous and the collections of maps κ L = (κ L X) X∈Ob C omp and κ L˜ = (κ L˜ X) X∈Ob C omp are inverse natural transformations ML → ML˜ and ML˜ → ML . Thus the functors ML and ML˜ are naturally isomorphic [17]. This implies that if L is a compact Hausdorff topological lattice that is an upper and lower Lawson ˜ L = (ML , η˜ L , μ˜ L ) with the semilattice, then there exists a unique semimonad M functorial part ML , such that the natural isomorphism κ L : ML → ML˜ is a morphism ˜ L is a monad if and only if L is ˜ L → M ˜ . The semimonad M of the semimonads M L distributive. Now we consider components of the natural transformations η˜ L and μ˜ L . Obviously η˜ L = η L . Let C ∈ ML 2 X, C = κ L ML X(C) ∈ ML˜ ML X, C = ML˜ κ L X(C ) ∈ ˜ G ⊂ X. We will clarify when G is α-reliable with respect to C . It ML˜ 2 X, α ∈ L, cl

˜ α})  ˜ α holds, i.e., C ({c ∈ ¯ is necessary that the inequality C ({c¯ ∈ ML˜ X | c(G)  ML X | κ L X(c)(G)  α})  α. The last inequality is equivalent to C ({c ∈ ML X | c(H)  α for each H ⊂ X, H ⊂ X \ G})  α, cl

that is, C(F )  α for each closed family F ⊂ ML X that is included in ML X \ {c ∈ ML X | c(H)  α for any H ⊂ X, H ⊂ X \ G} cl

= {c ∈ ML X | there is H ⊂ X, H ⊂ X \ G, c(H) ∈ L \ {α}↓} = O+ (X \ G, L \ {α}↓).

cl

Thus μ L˜ (C )(G) = inf{α ∈ L | C(F )  α for each F ⊂ ML X, F ⊂ O+ (X \ G, L \ {α}↓)}. cl

Denote U = X \ G and obtain for F ⊂ X the equality cl



μ˜ L (C )(F) = (κ L˜ X ◦ μ L˜ X(C ))(F) = inf{inf{α ∈ L | C(F )  α for each

F ⊂ ML X, F ⊂ O+ (U, L \ {α}↓)} | U ⊂ X, U ⊃ F} op

cl

= inf{α ∈ L | there is U ⊂ X, U ⊃ F such that C(F )  α op

for each F ⊂ ML X, F ⊂ O+ (U, L \ {α}↓)}. cl

Denote the latter set by AC . Observe that AC is a subset of the set

BC = {α ∈ L | C(F )  α for each F ⊂ ML X, F ⊂ F L\{α}↓ }, cl

therefore inf AC  inf BC . To prove the inverse inequality, we show that each open neighborhood V of an element β ∈ BC contains an element α ∈ BC . Assume the contrary for an open neighborhood V and choose an open neighborhood V   β such that V  ⊂ V and sup V  = α ∈ V. Then for each U ⊂ X, U ⊃ F there is a op

family F ⊂ ML X, F ⊂ O+ (U, L \ {α}↓) ⊂ (Cl U) L\V  ↓ such that C (F )   α. We have cl

C ((Cl U) L\V  ↓ )  α, which implies C ((Cl U) L\V  ↓ ) ∈/ V  ↓. But the intersection of the filtered family of sets (Cl U) L\V  ↓ with U running through all open neighborhoods of

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the set F is equal to F L\V  ↓ ⊂ F L\{β}↓ . The inclusion β ∈ BC implies C (F L\V  ↓ )  β, thus C (F L\V  ↓ ) ∈ V  ↓. By upper semicontinuity there should be an open neighborhood U ⊃ F such that C ((Cl U) L\V  ↓ ) ∈ V  ↓. This contradiction shows that each point of the set BC is a touch point of the set AC . These sets are contained in a topological lattice, therefore inf AC  inf BC , thus obtaining inf AC = inf BC . If for α ∈ L, F ⊂ X, C ∈ ML 2 X the inequality C(F )  α holds for each family cl

F ⊂ ML X such that F ⊂ F L\{α}↓ , then call the set F α-safe with respect to the cl

capacity C . Summing up, we obtain a statement that clarifies the description of the (semi)mo˜ L. nad M Proposition 1 Let X be a compactum, L be a compact Hausdorff topological lattice that is an upper and lower Lawson semilattice, C ∈ ML 2 X, F be a closed set in X. Then μ˜ L X(C )(F) equals to the greatest lower bound of all α ∈ L such that F is α-safe with respect to C . Lemma 13 Let X be a compactum, L be a compact Hausdorff topological lattice that is an upper and lower Lawson semilattice, C ∈ ML 2 X, F be a closed set in X. Then μ L X(C )(F)  μ˜ L X(C )(F). Proof Let α, β ∈ L and the set F be α-reliable and β-safe. Assume that α   β. Then the family Fα is closed in ML X and lies in F L\{β}↓ . The inequality C (Fα )  α implies C (Fα )  β, that contradicts to the assumption that F is β-safe. Therefore: μ L X(C )(F) = sup{α ∈ L | F is α-reliable with respect to C }

 μ˜ L X(C )(F) = inf{β ∈ L | F is β-safe with respect to C }.

 

We use a lemma that is dual to a statement that was first published in [8]. For an element x of a complete lattice L denote x+ = inf{y ∈ L | y   x} and put x∗ = sup{s+ | s ∈ L, s   x}. Then: Lemma 14 L is completely distributive if and only if x∗ = x for each x ∈ L. Proposition 2 Let a compact Hausdorff topological lattice L be an upper and lower Lawson semilattice. The multiplications μ L X and μ˜ L X coincide for each compactum X if and only if L is distributive. Proof Let C ∈ ML 2 X and F be a closed set X. If the lattice L is distributive, then it is completely distributive. Denote γ = μ˜ L X(C )(F). If β   γ , then F is not β-safe, i.e., there exists a family F ⊂ ML X, F ⊂ F L\{β}↓ , such that C (F )   β. Thus C (F )  β+ , cl

F ⊂ Fβ+ , that implies C (Fβ+ )  β+ , and F is β+ -reliable. Therefore: μ L X(C )(F) = sup{α ∈ L | F is α-reliable with respect C }

 sup{β+ | β ∈ L, β  γ } = γ∗ = γ = μ˜ L X(C )(F).

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Thus μ L X(C )(F) = μ˜ L X(C )(F) for any C and F, and μ L X = μ˜ L X. Now let L be non-distributive, i.e., there are elements α, β1 , β2 such that α ∧ (β1 ∨ β2 )  = (α ∧ β1 ) ∨ (α ∧ β2 ). Then α ∧ (β1 ∨ β2 )  (α ∧ β1 ) ∨ (α ∧ β2 ), 0 < α, β1 , β2 < 1, and neither of the inequalities β1  β2 , β2  β1 holds. Put X = {x1 , x2 , x3 }, F = {x1 , x2 } and consider the capacity

C = δ({δ(x1 , β1 ), δ(x2 , α)}, α ∧ (β1 ∨ β2 ); {δ(x1 , β2 ), δ(x2 , α)}, α ∧ (β1 ∨ β2 )). If F is γ -reliable for γ ∈ L, γ > 0, then Fγ ⊃ {δ(x1 , β1 ), δ(x2 , α)} or Fγ ⊃ {δ(x1 , β2 ), δ(x2 , α)}, thus γ  α ∧ β1 or γ  α ∧ β2 , that implies γ  (α ∧ β1 ) ∨ (α ∧ β2 ). This results in μ L X(C )(F)  (α ∧ β1 ) ∨ (α ∧ β2 ). Let F be γ -safe for γ ∈ L, γ < 1. If γ   α ∧ (β1 ∨ β2 ), then the set F L\{γ }↓ cannot contain either of the sets {δ(x1 , β1 ), δ(x2 , α)} and {δ(x1 , β2 ), δ(x2 , α)}. Thus at least one of the inequalities γ  α, γ  β1 and at least one of the inequalities γ  α and γ  β2 hold. If γ   α, then the inequalities γ  β1 and γ  β2 both hold. Thus obtain γ  α or γ  β1 ∨ β2 , and γ  α ∧ (β1 ∨ β2 ) again. Therefore μ˜ L X(C )(F)  α ∧ (β1 ∨ β2 ) and μ˜ L X(C )(F)  = μ L X(C )(F), thus μ˜ L X  = μ L X.   The following results is a consequence of the above statements. Theorem 3 Let L be a compact Hausdorff topological lattice that is an upper and lower Lawson semilattice. If L is distributive, then the natural transformations μ˜ L and μ L from ML 2 to ML coincide and the natural transformation κ L : ML → ML˜ is an isomorphism of the monad M L = (ML , η L , μ L ) and the monad M L˜ = (ML˜ , η L˜ , μ L˜ ). If L is not distributive, there exist distinct semimonads M L = (ML , η L , μ L ) and ˜ L = (ML , η L , μ˜ L ) with the functorial part ML . They are not monads and the natM ural transformation κ L : ML → ML˜ is simultaneously an isomorphism of semimon˜ L = (ML , η L , μ˜ L ) → M ˜ = (M ˜ , η ˜ , μ ˜ ) and an isomorphism of semimonads ads M L L L L ˜ ˜ = (M ˜ , η ˜ , μ˜ ˜ ). M L = (ML , η L , μ L ) → M L L L L

6 “Practical” Interpretation In the spirit of [9, 12, 13] etc. we can interpret the value c(F) of an L-valued capacity as subjective estimate of the likelihood of realization of one of elementary events that belong to a closed subset F of a compact Hausdorff space X of elementary events. Then ML X is a set of all estimates (expert opinions) that satisfy certain natural requirements like monotonicity and upper semicontinuity. Closed subset F ⊂ ML X is a thought of a group of experts, and the capacity C (F ) for C ∈ ML 2 X describes the level of confidence in this expert group. A set F is α-reliable with respect to C , if there is a group of experts with level of confidence  α, and each of them estimates the likelihood of realization of the event F as  α. Thus μ L X(C )(F) is an “optimistic” (the highest) estimate of the likelihood of a desirable event, that relies on its level of reliability with respect to different expert groups and the estimate C of competence of these groups. A set F is β-safe with respect to C if each group of experts such that all of them does not agree with the estimate of likelihood of event F as  β, has the level of confidence  β. Thus μ˜ L X(C )(F) is an “optimistic” (the lowest) estimate of the

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likelihood of a undesirable event, if we neglect opinions of “pessimistic” groups of experts with low level of confidence with respect to C . ˜ L are semimonads. The unit laws μ L X ◦ Consider the fact that M L and M ηML X = 1 ML X and μ˜ L X ◦ ηML X = 1 ML X for each compactum X have obvious interpretation: if C = ηML X(c), then we completely believe in a single expert opinion c. It is natural to expect that μ L X(C) = μ˜ L X(C) = c, i.e., final estimate completely coincide with the estimate of this expert. The two other equalities μ L X ◦ ML ηX = 1 ML X and μ˜ L X ◦ ML ηX = 1 ML X also have straightforward meaning: if C = ML η(c), then all our belief is restricted to experts that have unambiguous prognosis of the form ηML (x), i.e., they are absolutely sure that some elementary event x ∈ X will take place. Thus we estimate the likelihood of F ⊂ X as common cl

level of competence of all experts that predict some event x ∈ F. This interpretation justifies use of such an uncanonical and defective from pure categorical point of view construction as semimonad. The associative law also can be interpreted this way, but even in its absence the unit laws are useful and meaningful. The last theorem specifically means that for compact Hausdorff topological lattices, that are upper and lower Lawson semilattices, estimates of likelihood of desirable and undesirable events coincide exactly for distributive lattices (i.e., Lawson lattices).

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17. Mac Lane, S.: Categories for the Working Mathematician, 2nd ed. Springer, Berlin Heidelberg New York (1998) 18. Zarichnyi, M.M., Nykyforchyn, O.R.: Funktor yemkostey v kategorii kompaktov (Capacity functor in the category of compacta). Submitted to Mat. Sb. 33 pp. (2006) (in Russian) 19. Nykyforchyn, O.R.: On the axiom of continuity, openness and bicommutativity. Methods Funct. Anal. Topology 4, 82–85 (1998)