Probability over Plonka sums of Boolean algebras: states, metrics and topology
PPROBABILITY OVER PŁONKA SUMS OF BOOLEAN ALGEBRAS: STATES,METRICS AND TOPOLOGY
STEFANO BONZIO AND ANDREA LOIA bstract . The paper introduces the notion of state for involutive bisemilattices, a varietywhich plays the role of algebraic counterpart of weak Kleene logics and whose elementsare represented as Płonka sum of Boolean algebras. We investigate the relations betweenstates over an involutive bisemilattice and probability measures over the (Boolean) algebrasin the Płonka sum representation and, the direct limit of these algebras. Moreover, westudy completition of involutive bisemilattices, as pseudometric spaces, and the topologyinduced by the pseudometric. . I ntroduction Probability theory is grounded on the notion of “event”. Events are traditionally in-terpreted as elements of a ( σ -complete) Boolean algebra. Intrinsically, this means thatclassical propositional logic is the most suitable formal language to speak about events.The direct consequence of this standard assumption is that any event can either happen(to be the case) or not happen, and, thus, its negation is taking place. One could claimthat this criterion does not encompass all situations: certain events might not be eithertrue or false (namely, simply happen and not happen). Think about the coin toss to de-cide which one among two tennis players is choosing whether to serve or respond at thebeginning of a match. Although being statistically extremely rare, the coin may fall onthe edge, instead on one face. Pragmatically, the situation will be solved re-tossing thecoin (hoping to have it ending on one face). Theoretically, one should admit that thereare circumstances in which the event “head” (and so also its logical negation “tail”) couldbe indeterminate. Yet, this is not a good objection for renouncing to probability. We findit a good reason to look beyond classical probability, namely to render probability whenevents under consideration does not belong to classical propositional logic.The idea of studying probability in case events are non-classical but, contrarily, belong-ing to propositional Łukasiewicz logic is due to Mundici [ ]. Changing the structuresof the events led to the introduction of states , namely maps expressing probability, overMV-algebras (see also [ ]), the (equivalent) algebraic semantics of Łukasiewicz logic.The key difference with respect to the classical case is that such events are not simplyeither true or false, but assume a degree of truth in the real unit interval [
0, 1 ] . Thanks tothe introduction of states, it is possible to render the probability of certain fuzzy events. Mathematics Subject Classification.
Primary: B . Secondary E . Key words and phrases.
Involutive bisemilattice; Płonka sum; Boolean algebra; probability measure;pseudometric. a r X i v : . [ m a t h . L O ] J u l STEFANO BONZIO AND ANDREA LOI
Subsequently, the theory of states have attracted successful attention and have been stud-ied for the algebraic semantics of other fuzzy logics, such as G ¨odel-Dummet [ ], G ¨odel ∆ [ ], the logic of nilpotent minimum [ ] and product logic [ ]. Within the same strandof research, probability measures have been defined and studied also for other algebraicstructures (connected to logic), such as Heyting algebras [ ], De Morgan algebras [ ],orthomodular lattices [ ] and effect algebras [ ].The idea motivating the present work is to further extend the theory of states to non-classical events; in particular, to one of the three-valued logics in the weak Kleene family,whose algebraic counterpart is played by the variety of involutive bisemilattices (see [ ]).The peculiarity of such variety is that each of its members has a representtion in termsof Płonka sums of Boolean algebras. This abstract construction, originally introducedin universal algebra by J. Płonka [ , ], is performed over direct systems of algebraswhose index set is a semilattice. The axiomatization of states we propose, which ismotivated by the logic PWK (paraconsistent weak Kleene), allows to “break” a stateinto a family of (finitely additive) probability measure over the Boolean algebras in thePłonka sum representation of an involutive bisemilattice. In other words, our notion ofstate accounts for, and is strictly connected to all the Boolean algebras in the Płonka sum.Moreover, we show the relation between states of an involutive bisemilattice and finitelyadditive probability measures over the Boolean algebra which is the direct limit – whichwe refer to as the Booleanisation – of the algebras in the (semilattice) direct system of therepresentation. This allows to prove that each state corresponds to an integral over thedual space of the direct limit (the inverse limit of the dual spaces).The results obtained explore, on the one hand, the possibility of defining probabilitymeasures over the (non-equivalent) algebraic counterpart of certain Kleene logics. On theother, show how probability measures can be lifted from Boolean algebras to the Płonkasum of Boolean algebras.The paper is organized as follows. Section recaps all the necessary preliminary no-tions helpful for the reader to go through the entire paper. In Section , states overinvolutive bisemilattices are introduced. We show that each involutive bisemilattice B carries at least a state and show the correspondence with probability measures of thedirect limit of the Boolean algebras in the semilattice system of B . We dedicate Section to the analysis of strictly positive states, which we refer to as faithful states , in accordancewith the trend in MV-algebras. In particular, the presence of faithful states motives the in-troduction of the subclass of injective involutive bisemilattices, characterized by injectivehomomorphisms in the Płonka sum representation. In Section and , respectively, weapproach involutive bisemilattices carrying a state as pseudometric spaces and topologi-cal spaces (with the topology induced by the pseudometric), respectively. In the formerwe study completitions, while in the latter we insist on the relation between involutivebisemilattices and the correspondent direct limits (both as topological spaces). In partic-ular, we prove that the Boolean algebras of (open) regular sets are isomorphic and givea topological characterization of states. We close the paper with Section , introducingpossible further works and a short Appendix, discussing an alternative notion of stateand explaining why we discard it. . P reliminaries2 . . Płonka sums. A semilattice is an algebra I = (cid:104) I , ∨(cid:105) of type (cid:104) (cid:105) , where ∨ is a binarycommutative, associative and idempotent operation. Given a semilattice I , it is possibleto define a partial order relation between the elements of its universe, as follows i ≤ j ⇐⇒ i ∨ j = j ,for each i , j ∈ I . Definition . A semilattice direct system of algebras is a triple A = (cid:104){ A i } i ∈ I , I , p ij (cid:105) consistingof ( ) a semilattice I = (cid:104) I , ∨(cid:105) ;( ) a family of algebras { A i } i ∈ I of the same type with disjoint universes;( ) a homomorphism p ij : A i → A j , for every i , j ∈ I such that i ≤ j ,where ≤ is the order induced by the binary operation ∨ .Moreover, p ii is the identity map for every i ∈ I , and if i ≤ j ≤ k , then p ik = p jk ◦ p ij .Organising a family of algebras { A i } i ∈ I into a semilattice direct system means substan-tially requiring that the index set I forms a semilattice and that algebras whose indexesare comparable with respect to the order are “connected” by homomorphisms, whose“direction” is bottom up, namely from algebras whose index is lower to every algebraswith a greater index. The nomenclature in Definition is deliberately chosen to empha-size the presence of an index set equipped with the structure of semilattice . We will oftenrefer to a semilattice direct system simply as { A i } i ∈ I (instead of A = (cid:104){ A i } i ∈ I , I , p ij (cid:105) )and, in order to indicate homomorphisms, we sometimes write p i , j instead of p ij .The Płonka sum is a new algebra that is defined given a semilattice direct systems ofalgebras. Definition . Let A = (cid:104){ A i } i ∈ I , I , p ij (cid:105) be a semilattice direct system of algebras of type τ . The Płonka sum over A , in symbols P ł ( A ) or P ł ( A i ) i ∈ I , is the algebra such that( ) the universe of P ł ( A ) is the disjoint union (cid:71) i ∈ I A i ;( ) for every n -ary basic operation f (with n (cid:62)
1) in τ , and a , . . . , a n ∈ (cid:83) i ∈ I A i , weset f P ł ( A i ) i ∈ I ( a , . . . , a n ) : = f A j ( p i j ( a ) , . . . , p i n j ( a n )) where a ∈ A i , . . . , a n ∈ A i n and j = i ∨ · · · ∨ i n .In other words, operations on the elements a , . . . , a n of the Płonka sum are definedby computing the operation f in the algebra A j , whose index is the join of the indexescorresponding to the algebras where the elements a , . . . , a n live, respectively (this ideais clarified through Example ). With a slight abuse of notation, we identify semilattice I with its universe I . Systems of this kind are special cases of direct systems (of algebras), which differentiate with respect tothe index set which is assumed to be a directed preorder.
STEFANO BONZIO AND ANDREA LOI
The theory of Płonka sums is intrinsically connected with the notion of partition func-tion which we recall in the following. Definition . Let A be an algebra of type τ . A function · : A → A is a partition function in A if the following conditions are satisfied for all a , b , c ∈ A , a , ..., a n ∈ A n and for anyoperation g ∈ τ of arity n (cid:62) ) a · a = a ,(PF ) a · ( b · c ) = ( a · b ) · c ,(PF ) a · ( b · c ) = a · ( c · b ) ,(PF ) g ( a , . . . , a n ) · b = g ( a · b , . . . , a n · b ) ,(PF ) b · g ( a , . . . , a n ) = b · a · ... · a n .The main connection between partition functions and Płonka sums is provided by thefollowing result. Theorem . [ , Thm. II] Let A be an algebra of type τ with a partition function · . The followingconditions hold:( ) A can be partitioned into { A i } i ∈ I where any two elements a , b ∈ A belong to the samecomponent A i exactly when a = a · b and b = b · a . ( ) The relation ≤ on I given by the rulei ≤ j ⇐⇒ there exist a ∈ A i , b ∈ A j s.t. b · a = bis a partial order and (cid:104) I , ≤(cid:105) is a semilattice .( ) For all i , j ∈ I such that i ≤ j and b ∈ A j , the map p ij : A i → A j , defined by the rulep ij ( x ) = x · b is a homomorphism.( ) A = (cid:104){ A i } i ∈ I , (cid:104) I , ≤(cid:105) , { p ij : i ≤ j }(cid:105) is a direct system of algebras such that P ł ( A ) = A . Theorem is enunciated in the more general form which allows the presence of con-stants in the type τ . In case the type τ contains constant operations, then the constructionof the Płonka is performed by assuming that the semilattice of indexes I contains a leastelement i and c P ł ( A i ) = c A i , for each constant in the type τ . Both the definition of par-tition function and Theorem are generalised in presence of constants (for details, see[ , ]).The following example may be useful for the reader to understand how the construc-tion of Płonka sum is performed over a semilattice direct system of Boolean algebras. Different definitions of partition function can be found in literature (see for instance [ , ]). We optfor recalling the one recurring to the minimal number of identities (see [ ]). With a slight abuse of notation we indicate here the semilattice of indexes I by the order ≤ and not bythe binary operation ∨ . Example . Consider the four-elements semilattice I = { i , i , j , k } whose order is givenas follows: I = ki ji Consider the family { A i , A i , A j , A k } of Boolean algebras, organised into a semilatticedirect system, whose index is I and homomorphisms are defined as extensions of thefollowing maps: p ik ( a ) = c , p jk ( b ) = e (thus, p ik ( a (cid:48) ) = c (cid:48) , p jk ( b (cid:48) ) = e (cid:48) ), p i m is defined inthe unique obvious way, for any m ∈ { i , j , k } . A k = k e (cid:48) c (cid:48) d d (cid:48) c e k A i = i a a (cid:48) i A j = j b b (cid:48) j A i = , the Płonka sum over the above introduced semilattice directsystem is the new algebra B , whose universe is B = A i (cid:116) A i (cid:116) A j (cid:116) A k . The constants in B are the constants of the Boolean algebra A i (which we have indeed indicated, not bychance, by 0, 1). We just give an example of how binary operations are computed in B (as should be clear that (cid:48) coincides with negation in each respective algebra). a ∧ B a (cid:48) = p ii ( a ) ∧ A i p ii (cid:48) ( a (cid:48) ) = a ∧ A i a (cid:48) = i .In words, a binary operation between elements belonging to the same algebra (e.g. A i )are computed as in that algebra. As for binary operations between elements “living” in STEFANO BONZIO AND ANDREA LOI (the universes of) two different algebras (e.g. A i , A j ) it is necessary to recur to the algebra A k (since k = i ∨ j ) and homomorphisms in the system: a ∨ B b = p ik ( a ) ∨ A k p jk ( b ) = c ∨ A k e = d (cid:48) . (cid:4) . . Involutive Bisemilattices.Definition . An involutive bisemilattice is an algebra B = (cid:104) B , ∧ , ∨ , (cid:48) , 0, 1 (cid:105) of type (
2, 2, 1, 0, 0 ) satisfying: I . x ∨ x ≈ x ; I . x ∨ y ≈ y ∨ x ; I . x ∨ ( y ∨ z ) ≈ ( x ∨ y ) ∨ z ; I . ( x (cid:48) ) (cid:48) ≈ x ; I . x ∧ y ≈ ( x (cid:48) ∨ y (cid:48) ) (cid:48) ; I . x ∧ ( x (cid:48) ∨ y ) ≈ x ∧ y ; I . 0 ∨ x ≈ x ; I . 1 ≈ (cid:48) .The class of involutive bisemilattices forms a variety which we denote by I BS L . Thisvariety coincides with the regularisation of the variety of Boolean algebras (firstly intro-duced by Płonka in [ ]), i.e. it satisfies all (and only) the regular identities holdingfor Boolean algebras. The importance of involutive bisemilattice is connected to logic, as(one of its subquasivariety) plays the role of the algebraic counterpart of paraconsistentweak Kleene logic (see [ ]). Example . The most prominent example of involutive bisemilattice is the -elementalgebra WK = (cid:104){
0, 1, n } , ∧ , ∨ , (cid:48) , 0, 1 (cid:105) , whose operations are defined via the weak Kleenetables : (cid:48) n n ∧ n
10 0 n n n n n n ∨ n
10 0 n n n n n n WK generates the variety of involutive bisemilattices [ ]. Observe that WK is the Płonkasum over a semilattice direct formed by the two-element Boolean algebra and a trivialalgebra. Weak Kleene logics – namely Bochvar [ ] and paraconsistent weak Kleene [ ] –are defined as induced by the matrices (cid:104) WK , { }(cid:105) and (cid:104) WK , { n }(cid:105) , respectively.Every involutive bisemilattice is the Płonka sum over a semilattice direct system ofBoolean algebras and, conversely, the Płonka sum over any semilattice direct system ofBoolean algebras (such as that in Example ) is an involutive bisemilattice (see [ , ]).For this reason, throughout the whole paper, when considering B ∈ I BS L we will al-ways identify it with its Płonka sum representation P ł ( A i ) , without explicitly mentioning An identity ϕ ≈ ψ is regular provided that Var ( ϕ ) = Var ( ψ ) . it (when clear from the contest). Notation: we denote by 1 i , 0 i the top and bottom element, respectively, of the Booleanalgebras A i (in the Płonka sum representation of B ∈ I BS L ). . . Finitely additive probability measures on Boolean algebras.
Let A be a Booleanalgebra. A finitely additive probability measure over A is a real-valued map m : A → [
0, 1 ] such that:( ) m ( ) = ) m ( a ∨ b ) = m ( a ) + m ( b ) , if a ∧ b = ”) todenote both the top element of a Boolean algebra and the unit element of R . A probabil-ity measure m over a Boolean algebra A is called regular (or, strictly positive) providedthat m ( a ) > a (cid:54) = ] and Panti [ ]), holds, morein general, for MV-algebras For the convenience of our reader, we opt to formulate itfor Boolean algebras, recalling that every Boolean algebra is a semisimple MV-algebra(where the operations ⊕ and (cid:12) coincide with ∨ and ∧ , respectively). The fact that aBoolean algebra A is semisimple, as MV-algebra, allows to represent it as an algebra of [
0, 1 ] -valued continuous functions defined over its (dual) Stone space A ∗ (see [ , Theo-rem . . ]). It follows that one can associate to each element a ∈ A , a unique continuousfunction a ∗ : A ∗ → [
0, 1 ] . Moreover, we refer to the space of all the (finitely additive)probability measures over a Boolean algebra A as S ( A ) . Theorem . Let A be a Boolean algebra and m : A → [
0, 1 ] a (finitely additive) probabilitymeasure. Then( ) There is a homeomorphism Ψ : S ( A ) → M ( A ∗ ) , where M ( A ∗ ) is the space of all regu-lar Borel probability measures on its dual space A ∗ .( ) For every a ∈ A, m ( a ) = (cid:90) A ∗ a ∗ ( M ) d µ s ( M ) , where a ∗ is the unique function associated to a, M ∈ A ∗ and d µ s = Ψ ( s ) . . . Booleanisation of an involutive bisemilattice.
Given a semilattice direct system ofalgebras, the Płonka sum is only one way to construct a new algebra. Another is the direct limit . We recall this construction in the special case of Boolean algebras.Consider an involutive bisemilattice B ∼ = P ł ( A i ) . The direct limit over a direct system { A i } i ∈ I of Boolean algebras is the Boolean algebra defined as the quotient:lim → i ∈ I A i : = (cid:71) i ∈ I A i / ∼ , Notice that this use of the term “regular” is different from above and refers to Borel measures overtopological spaces (see, for instance, [ ]). STEFANO BONZIO AND ANDREA LOI where a ∼ b if and only if there exist c ∈ A k , for some k ∈ I with i , j ≤ k such that p ik ( a ) = c = p jk ( b ) .It is always possible to associate to B ∈ I BS L , the Boolean algebra lim → A i , the directlimit of the algebras (in the system) { A i } i ∈ I , which we will call the Booleanisation of B ,and we will indicate it by A ∞ . Given an involutive bisemilattice B , we can define themap π : B → A ∞ , as π ( a ) : = [ a ] ∼ . Remark . The map π is a surjective homomorphism. We provide the details of one caseonly. π ( a ∧ b ) = [ a ∧ b ] ∼ = [ p ik ( a ) ∧ p jk ( b )] ∼ = [ p ik ( a )] ∼ ∧ [ p jk ( b )] ∼ = π ( a ) ∧ π ( b ) ,where the second last equality is justified by observing that c ∈ [ p ik ( a ) ∧ p jk ( b )] ∼ (foran arbitrary c ∈ A l and m = k ∨ l ) if and only if p lm ( c ) = p km ( p ik ( a ) ∧ p jk ( b )) = p km ( p ik ( a )) ∧ p km ( p jk ( b )) = p im ( a ) ∧ p jm ( b ) and this is equivalent to say c ∈ [ p ik ( a )] ∼ ∧ [ p jk ( b )] ∼ . Notation: to indicate elements of the Booleanisation we sometimes drop the subscript ∼ when no danger of confusion may arise. . S tates over I nvolutive B isemilattices Definition . Let B be an involutive bisemilattice. A state over B is a map s : B → [
0, 1 ] such that:( ) s( ) = ;( ) s ( a ∨ b ) = s ( a ) + s ( b ) , provided that a ∧ b ∈ (cid:83) i ∈ I { i } .Moreover, a state s : B → [
0, 1 ] is faithful if s ( a ) >
0, for every a (cid:54) = , ] to extend probability to certain non-classical events (belonging toan involutive bisemilattice). The choice of the two properties in Definition is moti-vated by the fact that the constants 1, 0 play the role of “true” and “false”, respectively, inthe logic PWK . Thus, logical truths and falsities are expected to have probability 1 and 0,respectively (see Proposition ). Moreover, another property characterizing probabilityis the additivity over the disjunction of logically incompatible. Due to the definition ofoperations in involutive bisemilattices (Płonka sum of Boolean algebra), we assume twoelements to be incompatible when the result of their conjunction is the bottom elementof the Boolean algebra where the operation is actually computed. Despite being ques-tionable (a different definition of state is discussed in the Appendix), this definition hasthe fortunate consequence that a state breaks into and can be constructed from a familyof probability measures over the Boolean algebras in the Płonka sum representation ofan involutive bisemilattice (see below).The following resumes the basic properties of a state over an involutive bisemilattice. Proposition . Let s be a state over an involutive bisemilattice B . Then( ) s ( ) = ; It is useful to recall that the present use of the term “Booleanisation” differs from other usages inliterature, in lattice theory (see [ ]) and in the theory of Boolean (inverse) semigroups (see [ ]). ( ) s ( i ) = and s ( i ) = , for every i ∈ I.( ) s ( a (cid:48) ) = − s ( a ) , for every a ∈ B.Proof. ( ) Since 0 ∧ = ∈ (cid:83) i ∈ I { i } , then s ( ) = s ( ∨ ) = s ( ) + s ( ) . Hence s ( ) = ) Observe that 0 i ∧ = i ∧ p i i ( ) = i ∧ i = i . Then s ( i ) + s ( ) = s ( i ∨ ) = s ( i ∨ i ) = s ( i ) + s ( i ) . Therefore s ( i ) = s ( ) = s ( i ) = i ∨ i = i .( ) Let a ∈ A i for some i ∈ I . Then a (cid:48) ∈ A i and a ∧ a (cid:48) = i . Therefore 1 = s ( i ) = s ( a ∨ a (cid:48) ) = s ( a ) + s ( a (cid:48) ) . (cid:4) Definition . Let B ∈ I BS L . For every i , j ∈ I and a family of finitely additive probabil-ity measures { m i } i ∈ I over { A i } i ∈ I (each A i carries the measure m i ), the homomorphism p ij preserves the measures if m j ( p ij ( a )) = m i ( a ) , for any a ∈ A i . Notation: we indicate the restriction of a state s on the Boolean components A i , A j A k , . . .of the Płonka sum representation of an involutive bisemilattice B , as m i , m j , m k , . . . in-stead of s | A i , s | A j , s | A k , . . . to make notation less cumbersome (the adoption of m , resem-bling “measure”, will be clear from the next result). Proposition . Let s be a map from B to [
0, 1 ] . The following are equivalent:( ) s is a state over B ;( ) m i : A i → [
0, 1 ] is a (finitely additive) probability measure over A i , for every i ∈ I, andp ij preserves the measures for each i ≤ j.Proof. ( ) ⇒ ( ). Assume that s is a state over B . Then, m i ( i ) = s ( i ) = i ∈ I , byProposition -( ). Moreover, let a , b ∈ A i be two elements such that a ∧ A i b = i . Observethat a ∧ B b = a ∧ A i b and a ∨ B b = a ∨ A i b . Then m i ( a ∨ b ) = s ( a ∨ b ) = s ( a ) + s ( b ) , using( ) in Definition . This shows that m i is a (finitely additive) probability measure over A i , for every i ∈ I . Now, let a ∈ A i , for some i ∈ I , and j ∈ I with i ≤ j . Observe that a ∧ B j = p ij ( a ) ∧ A j j = j , thus m j ( p ij ( a )) = s ( p ij ( a )) = s ( p ij ( a ) ∨ A j j ) = s ( a ∨ B j ) = s ( a ) + s ( j ) = s ( a ) = m i ( a ) ,where we have used the additivity of a state and Proposition -( ). This shows that anyhomomorphism p ij preserves the measures m i , for every i ≤ j .( ) ⇒ ( ) Assume that every Boolean algebra A i in the direct system carries a finitelyadditive probability measure m i and that each homomorphism p ij ( i ≤ j ) preserves themeasures. Let s : B → [
0, 1 ] be the map defined as s ( x ) : = m i ( x ) ,for x ∈ A i , with i ∈ I .( ) s ( ) = m i ( ) = ) Let a , b ∈ B such that a ∧ b ∈ (cid:83) i ∈ I { i } . W.l.o.g. let a ∈ A i and b ∈ A j , then a ∧ B b = p ik ( a ) ∧ A k p jk ( b ) = k , with k = i ∨ j . Consequently, s ( a ∨ B b ) = s ( p ik ( a ) ∨ A k p jk ( b )) = m k ( p ik ( a ) ∨ A k p jk ( b )) = m k ( p ik ( a )) + m k ( p jk ( b )) = m i ( a ) + m j ( b ) = s ( a ) + s ( b ) , where wehave essentially used the (finite) additivity of probability measures. (cid:4) STEFANO BONZIO AND ANDREA LOI
Example . Consider the involutive bisemilattice B introduced in Example and definethe map s : B → [
0, 1 ] as follows: s ( ) = s ( m ) =
1, for any m ∈ { i , j , k } , s ( ) = s ( m ) =
0, for any m ∈ { i , j , k } , s ( a ) = s ( a (cid:48) ) =
12 , s ( b ) =
13 , s ( b (cid:48) ) =
23 , s ( c ) = s ( c (cid:48) ) =
12 , s ( d ) =
16 , s ( e ) =
13 , s ( d (cid:48) ) =
56 , s ( e (cid:48) ) =
23 .It is not difficult to check that s is a faithful state over B . Indeed, it is immediate tosee that the restrictions of s to the Boolean components of the Płonka sum are finitelyadditive probability measures which are, moreover, preserved by homomorphisms of thePłonka sum of B . (cid:4) It then makes sense to ask under which circumstances an involutive bisemilatticescarries a state. Taking advantage of the fact that every Boolean algebras carries at leasta finitely additive) probability measure (see [ , Proposition . . ]), we can prove thefollowing. Lemma . Let (cid:104)(cid:104) I , ≤(cid:105) , { A i } i ∈ I , { p ij : i ≤ j }(cid:105) be a semilattice direct system of Boolean algebras.Then, for every i ≤ j (for i , j ∈ I) the algebras A i and A j carry two (finitely additive) probabilitymeasures m i : A i → [
0, 1 ] and m j : A j → [
0, 1 ] such that m i ( a ) = m j ( p ij ( a )) , for every a ∈ A i .Proof. Let a ∈ A i . Consider the ideal I p ij ( a ) generated by the element p ij ( a ) in the algebra A j . By the maximal ideal theorem (see [ , Theorem ]), there is a maximal ideal N extending I p ij ( a ) . Moreover, since p ij is a homomorphism, J = p − ij [ I p ij ( a ) ] is an idealof A i , containing a ; set M to be the maximal ideal extending J . Observe that A i / M ∼ = ∼ = A j / N (since M , N are maximal ideals) and thus, A i / M ( A j / N ) naturally embeds into[ , ] via the inclusion map ι . Then the map π i ◦ ι , where π i : A i → A i / M ( π j : A j → A j / N ) is the natural homomorphism onto the quotient, is a homomorphism and thus,a finitely additive probability measure. Define m i : = π i ◦ ι and m j : = π j ◦ ι . It followsby construction of M and N that the following diagram (where f is the isomorphismbetween A i / M and A j / N ) is commutative: A i / M A j / N fp ij A i A j π i π j It clearly follows that m i ( a ) = m j ( p ij ( a )) . (cid:4) Theorem . Let B ∈ I BS L with P ł ( A i ) its Płonka sum representation. The following areequivalent:( ) B carries a state;( ) { A i } i ∈ I contains no trivial algebra.Proof. ( ) ⇒ ( ). By Proposition , the restriction of a state over B to the Boolean com-ponents { A i } i ∈ I of the Płonka sum is a probability measure and each algebra A i (with i ∈ I ) carries a state if and only if it is not trivial.( ) ⇒ ( ). If { A i } i ∈ I contains no trivial algebra, then each algebra in the family carriesat least a state. Moreover, by Lemma , such states preserve the homomorphisms ( p ij ,for each i , j ∈ I such that i ≤ j ) of the direct system, i.e. by Proposition , B carries astate. (cid:4) The above result can be equivalently expressed by saying that the existence of (finitelyadditive) probability measures over the Boolean algebras involved in the (Płonka sum)representation of an involutive bisemilattice B is enough to grant the existence of a stateover B . It follows from ( )-Theorem that the algebra WK carries no state, as it is thePłonka sum of the two-element Boolean algebra with a trivial algebra.From now on, we will consider only involutive bisemilattices whose Płonka sum repre-sentation contains no trivial Boolean algebra. Hence, in view of Theorem , any involu-tive bisemilattice considered carries at least a state. We indicate by S ( B ) and S ( A ∞ ) thespaces of states and of (finitely additive) probability measures of an involutive bisemilat-tice B and of its Booleanisation A ∞ , respectively. Theorem . Let B ∈ I BS L . There exists a bijection Φ : S ( B ) → S ( A ∞ ) such that Φ is statepreserving, i.e. s ( b ) = Φ ( s )( π ( b )) .Proof. Consider Φ : S ( B ) → S ( A ∞ ) defined as follows: s ( b ) (cid:55)→ Φ ( s ) = m ∞ ([ b ] ∼ ) : = s ( b ) , ( )for every s ∈ S ( B ) and b ∈ B , a representative of the equivalence class [ b ] ∼ . Observe thatthe definition of Φ does not depend on the choice of the representative of [ a ] ∼ . Indeed,let a , b ∈ B with a (cid:54) = b , such that a , b ∈ [ a ] ∼ . W.l.o.g. assume that a ∈ A i and b ∈ A j , forsome i , j ∈ I . Since a ∼ b , then there exists k ∈ I with i , j ≤ k such that p ik ( a ) = p jk ( b ) .Then, by Proposition , s ( a ) = m i ( a ) = m k ( p ik ( a )) = m k ( p jk ( b )) = m j ( b ) = s ( b ) .Let us prove that Φ ( s ) = m ∞ is indeed a (finitely additive) probability measure over theBoolean algebra A ∞ . To this end, observe that [ ] ∼ ⊆ (cid:83) i ∈ I i and choose 1 j (for some j ∈ I ) as representative for [ ] ∼ . Then m ∞ ([ ] ∼ ) = s ( j ) =
1, by Proposition . Let [ a ] ∼ , [ b ] ∼ ∈ A ∞ two elements such that [ a ] ∼ ∧ [ b ] ∼ = [ ] ∼ . W.l.o.g. we can assume that STEFANO BONZIO AND ANDREA LOI a ∈ A i , b ∈ A j , for some i , j ∈ I , with i (cid:54) = j , and set k = i ∨ j . Then: m ∞ ([ a ] ∼ ∨ [ b ] ∼ ) = m ∞ ([ a ∨ b ] ∼ )= s ( a ∨ b )= m k ( p ik ( a ) ∨ A k p jk ( b ))= m k ( p ik ( a )) + m k ( p jk ( b )) (Prop. ) = m i ( a ) + m j ( b ) (Prop. ) = s ( a ) + s ( b )= m ∞ ([ a ] ∼ ) + m ∞ ([ b ] ∼ ) .To complete the proof, let us check that Φ is invertible. Let m : A ∞ → [
0, 1 ] be a (finitelyadditive) probability measure over A ∞ . Define Φ − ( m [ a ] ∼ ) = m i ( a ) : = m ([ a ] ∼ ) , forany [ a ] ∈ A ∞ , with a ∈ A i (for i ∈ I ). Let us check that m i : A i → [
0, 1 ] is a (finitelyadditive) probability measure (over A i ). m i ( i ) = m ([ i ] ∼ ) = m ([ ] ∼ ) =
1. Moreover, let a , b ∈ A i two elements such that a ∧ b = i . Then, [ a ] ∼ ∧ [ b ] ∼ = [ a ∧ b ] ∼ = [ i ] ∼ = [ ] ∼ .Therefore, m i ( a ∨ b ) = m ([ a ∨ b ] ∼ ) = m ([ a ] ∼ ∨ [ b ] ∼ ) = m ([ a ] ∼ ) + m ([ b ] ∼ ) = m i ( a ) + m i ( b ) . Observe that, the homomorphism p ij preserves the measures ( m i and m j ), forany i ≤ j . Indeed (for [ a ] ∼ ∈ A ∞ ) a ∼ p ij ( a ) , thus m i ( a ) = m ([ a ] ∼ ) = m ([ p ij ( a )] ∼ ) = m j ( p ij ( a )) . Therefore, since m i is a finitely additive probability measure over A i , foreach i ∈ I , and homomorphisms preserve the measures, by Proposition , we get that Φ − ( m ) is a state over B (it is immediate to check that Φ − is the inverse of Φ ). Finally,it follows from the definition that Φ is state preserving. (cid:4) Remark . The condition of existence of a state over an involutive bisemilattice B es-tablished in Theorem can be equivalently expressed, via Theorem , by the fact that A ∞ is non-trivial (and, therefore, carries at least a probability measure). It is not difficultto (directly) check that this latter condition is indeed equivalent to the fact that { A i } i ∈ I contains no trivial Boolean algebra (condition ( ) in Theorem ).The correspondence established in Theorem allows to provide an integral repre-sentation of states over an involutive bisemilattice. Let A ∞ be the Booleanisation of aninvolutive bisemilattice B and A ∗ ∞ the Stone space dually equivalent to A ∞ . It is usefulto recall that A ∗ ∞ corresponds to the inverse limit over the direct system whose elements { A ∗ i } i ∈ I are the dual spaces of the Boolean algebras { A } i ∈ I in the representation of B (see [ ]). Theorem (Integral representation of states) . Let B ∈ I BS L and s : B → [
0, 1 ] be a state.Then( ) There is a bijection χ : S ( B ) → M ( A ∗ ∞ ) , where M ( A ∗ ∞ ) is the space of all regular Borelprobability measures on A ∗ ∞ .( ) For every b ∈ B, s ( b ) = (cid:90) A ∗ ∞ (cid:99) [ b ] ∼ ( M ) d µ s ( M ) , where (cid:99) [ b ] ∼ is the unique function associated to [ b ] ∼ ∈ A ∞ , M ∈ A ∗ ∞ and d µ s = χ ( s ) . Proof.
It follows from the bijective correspondence between states of an involutive bisemi-lattices and of its Booleanisation (Theorem ) and integral representation for probabilitymeasures over Boolean algebras (Theorem ). (cid:4) One may wonder whether it is possible to provide a different integral representationof states which makes use of the dual space of an involutive bisemilattice (for instance,the Płonka product described in [ ]) instead of the inverse limits of the dual spaces ofthe Boolean algebras involved in the Płonka sum representation. This is a question thatwe do not address in the present paper. . F aithful states Recall that a state s over an involutive bisemilattice B is faithful (cfr. Definition )when s ( a ) >
0, for any a (cid:54)∈ { i } i ∈ I . Similarly, a (finitely additive) probability measure m over a Boolean algebra C is regular provided that m ( a ) >
0, when a (cid:54) = s over an involutive bisemilattice B has a non-trivialconsequence on the structure of its Płonka sum representation, as expressed in the fol-lowing. Proposition . Let s be a state over B ∈ I BS L . The following are equivalent:( ) s is faithful;( ) m i : A i → [
0, 1 ] is a regular (finitely additive) probability measure over A i , for everyi ∈ I, and p ij is an injective homomorphism preserving the measures, for each i ≤ j.Proof. We just show the non-trivial direction ( ) ⇒ ( ). By Proposition , we only haveto prove that m i is regular and that p ij is an injective homomorphism, for every i ≤ j . Theformer is immediate, indeed for a ∈ A i , (with i ∈ I ) such that a (cid:54) = i , then m i ( a ) = s ( a ) >
0. As for the latter, let i ≤ j for some i , j ∈ I . Let a ∈ ker ( p ij ) , then p ij ( a ) = j . Then, m j ( p ij ( a )) = s ( p ij ( a )) = s ( j ) =
0, by Proposition . By Proposition , p ij preserves themeasures (and m i , m j are probability measures), hence m i ( a ) = m i is regular(as shown above), then a = i . This shows that ker ( p ij ) = { i } , i.e. p ij is injective. (cid:4) Definition . Let B ∈ I BS L . We say that B is injective if, for every i , j ∈ I such that i ≤ j , the homomorphism p ij : A i → A j is injective.We refer to the class of injective involutive bisemilattices as I − I BS L . It is not dif-ficult to see that
I − I BS L is closed under subalgebras and products but not underhomomorphic images.Recall that the variety of involutive bisemilattices (as any regularization of a stronglyirregular variety) admits a partition function · (see Definition ), which, in this peculiarcase, can be defined as x · y : = x ∧ ( x ∨ y ) (see [ ]). Proposition . Let B ∈ I BS L with partition function · . The following are equivalent:( ) B ∈ I − I BS L ;( ) B | = x · y ≈ x & y · x ≈ y & x · z ≈ y · z ⇒ x ≈ y.Proof. ( ) ⇒ ( ) Suppose B ∈ I − I BS L , i.e. p ij is an embedding for each i ≤ j . Suppose,in view of a contradiction, that B does not satisfy condition ( ). Then, there exist elements STEFANO BONZIO AND ANDREA LOI a , b , c ∈ B such that a · b = a and b · a = b and a · c = b · c but a (cid:54) = b . By Theorem -( ), a · b = a and b · a = b imply that a , b ∈ A i , for some i ∈ I . W.l.o.g. assume that c ∈ A j ,for some j ∈ I and set k = i ∨ j . Then, applying Theorem and the assumption that a · c = b · c , we have p ik ( a ) = p ik ( a ) · p jk ( c ) = a · c = b · c = p ik ( b ) · p jk ( c ) = p ik ( b ) . Since B ∈ I − I BS L , p ik is injective, namely a = b , in contradiction with our hypothesis.( ) ⇒ ( ) Suppose B satisfies condition ( ) and, by contradiction, that B (cid:54)∈ I − I BS L ,namely there exists a homomorphism p ij (for some i ≤ j ) which is not an embedding.Hence, there exists element a (cid:54) = b such that p ij ( a ) = p ij ( b ) . Clearly a , b ∈ A i (otherwise p ij is not well defined), so, by Theorem -( ), a · b = a and b · a = b . Let c = p ij ( a ) = p ij ( b ) ,then a · c = p ij ( a ) · p ij ( a ) = p ij ( a ) = p ij ( b ) = p ij ( b ) · p ij ( b ) = b · c . Then, by condition ( ),we have that a = b , a contradiction. (cid:4) It follows from the above Proposition that
I − I BS L is quasi-variety, which can beaxiomatized by the identities defining an involutive bisemilattice (see Definition ) plusthe quasi-identity x · y ≈ x & y · x ≈ y & x · z ≈ y · z ⇒ x ≈ y (see [ ] for a study of thesubquasivarieties of a regularised variety). Theorem . Let B ∈ I − I BS L . Then there is a bijective correspondence between faithfulstates over B and regular measures over A ∞ .Proof. The correspondence is given by the map Φ , defined in ( ). Indeed, let B ∈I − I BS L and s : B → [
0, 1 ] a faithful state. Assume that [ a ] ∼ ∈ A ∞ with [ a ] ∼ (cid:54) = [ ] ∼ .Then a (cid:54) =
0, hence s ( a ) > m ∞ ([ a ] ∼ ) = s ( a ) >
0, which shows that Φ ( s ) is reg-ular. For the other direction, it is immediate to check that, given a regular measure m : A ∞ → [
0, 1 ] then m i ( a ) = Φ − ( m ([ a ] ∼ )) (for a ∈ A i ) is also a regular measure andsince B is injective, Proposition guarantees that Φ − ( m ) is a faithful state over B . (cid:4) Combining [ , Proposition . . ] and Theorem we directly get the following. Corollary . The space S ( B ) of states of an involutive bisemilattice B can be identified (via Φ )with a non-empty compact subspace of [
0, 1 ] A ∞ . It is natural to wonder under which conditions an involutive bisemilattice B (whosePłonka sum representation contains no trivial algebra) carries a faithful state. Theorem suggests that this might be the case provided that B is injective and its Booleanisation A ∞ actually carries a regular probability measure. In general, as observed in [ ], notevery (non-trivial) Boolean algebra carries a regular probability measure (necessary andsufficient conditions for a Boolean algebra to carry a regular measure are stated in [ ,Theorem ].It is possible to define the categories of injective involutive bisemilattices and Booleanalgebras “with faithful state”, “with regular probability measure”, respectively. Objectsare pairs ( B , s ) and ( A , m ), where B ∈ I BS L , A is a Boolean algebra, s is a state over B and m is a (finitely additive) probability measure over A . A morphism between twoobjects ( B , s ) and ( B , s ) is a homomorphism (between the corresponding algebras inthe first component) which preserves the measures, namely s ( b ) = s ( h ( b )) , for every b ∈ B and every homomorphism h : B → B . It is immediate to check that involutivebisemilattices carrying a faithful state (Boolean algebras carrying a probability measure,resp.) form a category, which we indicate by the pair (cid:104)I BS L , S ( B ) (cid:105) ( (cid:104)BA , S ( A ) (cid:105) , resp.). Let us define functor F : (cid:104)I BS L , S ( B ) (cid:105) → (cid:104)BA , S ( A ) (cid:105) , which associate to each object ( B , s ) the object F ( B , s ) = ( A ∞ , Φ ( s )) , with Φ defined as in ( ) and to each morphism h : ( B , s ) → ( B , s ) , the morphism F ( h ) = h with h : A ∞ → A ∞ , defined as h [ a ] =[ h ( a )] . Theorem . F is a covariant functor between the categories of injective involutive bisemilatticeswith faithful states and Boolean algebras with regular probability measures.Proof. Let ( B , s ) an object in (cid:104)I BS L , S ( B ) (cid:105) . Then, by Theorem , F ( B , s ) = ( A ∞ , Φ ( s )) is an object in (cid:104)BA , S ( A ) (cid:105) . We have to prove that, for every homomorphism h : ( B , s ) → ( B , s ) preserving states, the map h : ( A ∞ , Φ ( s )) → ( A ∞ , Φ ( s )) is a Boolean homo-morphism, which preserves the (finitely additive) probability measures. To see that h isindeed a Boolean homomorphism, we show just one case (all the others are proved anal-ogously). Let [ a ] , [ b ] ∈ A ∞ , then h ([ a ] ∧ [ b ]) = h ([ a ∧ b ]) = [ h ( a ∧ b )] = [ h ( a ) ∧ h ( b )] =[ h ( a )] ∧ [ h ( b )] = h ( a ) ∧ h ( b ) . Moreover, h preserves the states. Indeed let [ a ] ∈ A ∞ and Φ ( s ) ∈ S ( A ∞ ) . Then Φ ( s )([ a ]) = s ( a ) = s ( h ( a )) = Φ ( s )([ h ( a )]) = Φ ( s )( h [ a ]) ,where we have used the fact that h preserves states. Finally, it follows from the definitionof h that the following diagram is commutative and this concludes the proof of our claim. ( A ∞ , Φ ( s )) ( A ∞ , Φ ( s )) hh ( B , s ) ( B , s ) ( π , Φ ) ( π , Φ ) (cid:4) The functor F admits many adjoints, depending on the number of injective involutivebisemilattices having the same Booleanisation. Problem:
Characterise all the injective involutive bisemilattices having the same Booleani-sation.The above problem does not reduce to “knowing” the lattice of subalgebras (even inthe case that I is finite, and then A ∞ = A j , with j the top element in I ) of the Booleanisa-tion A ∞ of an injective involutive bisemilattice B , as the structure of B depends also theinjective homomorphisms between the (possibly infinite number of) Boolean algebras inthe system . For the lattice of subalgebras of a Boolean algebra, see [ , ]. A study of this kind for a specific subclass of involutive bisemilattices can be found in [ ]. STEFANO BONZIO AND ANDREA LOI . S tates , metrics and completition Definition . Let X be a set. A pseudometric on X is a map d : X × X → R such that:( ) d ( x , y ) ≥ ) d ( x , x ) = ) d ( x , y ) = d ( y , x ) ,( ) d ( x , z ) ≤ d ( x , y ) + d ( y , z ) (triangle inequality),for all x , y , z ∈ X .A pair ( X , d ) given by a set with a pseudometric d is called pseudometric space . Apseudometric d , over X , is a metric if d ( x , y ) = x = y .Let us recall that in any Boolean algebra A it is possible to define the symmetric difference (cid:77) : A × A → A , a (cid:77) b : = ( a ∧ b (cid:48) ) ∨ ( a (cid:48) ∧ b ) . The presence of a (finitely additive) probabilitymeasure m over a Boolean algebra A allows to define a pseudo-metric d : = m ◦ (cid:77) on A ,which becomes a metric, in case m is regular (see [ ] for details).The symmetric difference can be (analogously) defined also for an involutive bisemi-lattice B . Let a , b ∈ B , with B ∼ = P ł ( A i ) , a ∈ A i and b ∈ A j , for some i , j ∈ I . Then a (cid:77) b = ( a ∧ B b (cid:48) ) ∨ B ( a (cid:48) ∧ B b ) = p ik ( a ) (cid:77) A k p jk ( b ) , where k = i ∨ j . Clearly, in case B carries a state s , then one can define the map d s : B → [
0, 1 ] as: d s : = s ◦ (cid:77) . ( ) Proposition . Let B be an involutive bisemilattice carrying a state s. Then d s is a pseudo-metric on B.Proof. In order to check the (validity of the) properties in Definition , we consider thePłonka sum representation P ł ( A i ) of B .( ) obviously holds, since s : B → [
0, 1 ] .( ) Let a ∈ A i , for some i ∈ I . Then d s ( a , a ) = s (( a ∧ a (cid:48) ) ∨ ( a (cid:48) ∧ a )) = s ( i ∨ i ) = s ( i ) = .( ) holds since ∨ and ∧ are commutative operations.( ) Let a ∈ A i , b ∈ A j and c ∈ A k with i (cid:54) = j (cid:54) = k . Preliminarily, observe that d s ( a , c ) = s (( a ∧ B c (cid:48) ) ∨ B ( a (cid:48) ∧ B c ))= s (( p i , i ∨ k ( a ) ∧ A i ∨ k p k , i ∨ k ( c ) (cid:48) ) ∨ A i ∨ k ( p i , i ∨ k ( a ) (cid:48) ∧ A i ∨ k p k , i ∨ k ( c )))= s i ∨ k (( p i , i ∨ k ( a ) ∧ A i ∨ k p k , i ∨ k ( c ) (cid:48) ) ∨ A i ∨ k ( p i , i ∨ k ( a ) (cid:48) ∧ A i ∨ k p k , i ∨ k ( c )))= s i ∨ k ∨ j ( p i ∨ k , i ∨ k ∨ j (( p i , i ∨ k ( a ) ∧ A i ∨ k p k , i ∨ k ( c ) (cid:48) ) ∨ A i ∨ k ( p i , i ∨ k ( a ) (cid:48) ∧ A i ∨ k p k , i ∨ k ( c ))) (Prop. ) = s i ∨ k ∨ j ( a (cid:77) A i ∨ k ∨ j c ) .Similarly, it is possible to show that d s ( a , b ) = s i ∨ k ∨ j ( a (cid:77) A i ∨ k ∨ j b ) and d s ( b , c ) = s i ∨ k ∨ j ( b (cid:77) A i ∨ k ∨ j c ) . Hence d s ( a , c ) = s i ∨ k ∨ j ( a (cid:77) A i ∨ k ∨ j c ) ≤ s i ∨ k ∨ j ( a (cid:77) A i ∨ k ∨ j b ) + s i ∨ k ∨ j ( b (cid:77) A i ∨ k ∨ j c ) = s ◦ d ( a , b ) + s ◦ d ( b , c ) = d s ( a , b ) + d s ( b , c ) . (cid:4) Remark . Recall that, for Boolean algebras (and, more in general, MV-algebras [ ]),the choice of a regular probability measure m (or, a faithful state in the case of MV-algebras) implies that the induced pseudometric d m is indeed a metric. This fact does not hold for involutive bisemilattices, where the choice of a faithful state s is not enough toguarantee that d s is a metric. Indeed, consider any B ∈ I BS L and two distinct elements a , b ∈ B , a ∈ A i , b ∈ B j , such that p i , i ∨ j ( a ) = p j , i ∨ j ( b ) . Then d s ( a , b ) = s ( a (cid:77) B b ) = s ( p i , i ∨ j ( a ) (cid:77) A i ∨ j p j , i ∨ j ( b )) = s ( i ∨ j ) =
0; however, a (cid:54) = b . Notice, moreover, that a ∼ b , i.e.they belong to the same equivalence class in the Booleanisation A ∞ . Indeed, it is easy tocheck that d s is a metric over B if and only if B = A ∞ .We decide to give explicit proofs of all the following results in the particular case ofpseudometric (injective) involutive bisemilattices (and relative Booleanisations), althoughsome of them could be derived from the general theory of pseudometric spaces (see forinstance [ ]).In analogy to what is done in [ ] for MV-algebras, in the remaining part of this sectionwe study the completition for involutive bisemilattices. The completition is a standardconstruction for metric spaces (see [ ]), which can be analogously applied to pseudo-metric spaces.Recall that a (psudo)metric space is complete if every Cauchy sequence is convergent.Given a pseudometric space ( X , d ) , a completition ( (cid:98) X , (cid:98) d ) such that:( ) ( (cid:98) X , (cid:98) d ) is complete;( ) there exists an isometric injective map j : X → (cid:98) X such that j ( X ) is dense in (cid:98) X .Observe that the second condition means that, for every (cid:98) x ∈ (cid:98) X , there exists a sequence x n ∈ X such that j ( x n ) → (cid:98) x .In the sequel, we can assume, up to isometry, that the embedding j is the naturalinclusion X (cid:44) → (cid:98) X . Notice that the completition ( (cid:98) X , (cid:98) d ) is uniquely determined, by theabove conditions ( )-( ), up to isometries.We proceed in the same way for involutive bisemilattices. In detail, given ( B , d s ) apair where B is an involutive bisemilattice (carrying a state s ) and d s the pseudometricinduced by the state s , we can associate to it, on the one hand, the completition ( (cid:98) B , (cid:98) d s ) . Onthe other hand, we can consider its Płonka sum representation P ł ( A i ) ( B ∼ = P ł ( A i )) . ByProposition , for every i ∈ I , ( A i , d s i ) is a pseudo-metric space (since s i is a probabilitymeasure over A i ), which is metric in case s is faithful (see Proposition ). Therefore,it makes sense to consider the pseudo-metric space ( (cid:101) B , (cid:101) d ) , where (cid:101) B = P ł ( (cid:98) A i ) i ∈ I (thePłonka sum of the completitions of the Boolean algebras in the system representing B ) and (cid:101) d ( (cid:101) a , (cid:101) b ) = lim n → ∞ d s ( a n , b n ) , where (cid:101) a ∈ (cid:98) A i , (cid:101) b ∈ (cid:98) A j (for some i , j ∈ I ) and a n , b n are sequences (of elements) in A i , A j , respectively, such that a n → (cid:101) a , b n → (cid:101) b . We aregoing to show (see Theorem below) that (cid:101) B ∈ I BS L and, moreover, that ( (cid:101) B , (cid:101) d ) is thecompletition of ( B , d s ) . Lemma . Let ( A , d ) , ( A , d ) be two Boolean algebras with distance (induced by a proba-bility measure) and h : A → A be a distance preserving homomorphism. Then there exists adistance preserving homomorphism (cid:98) h : (cid:98) A → (cid:98) A such that (cid:98) h | A = h. The fact that the completition of a Boolean algebra is still a Boolean algebra is a routine exercise (see[ ], for MV algebras). STEFANO BONZIO AND ANDREA LOI
Proof.
Let (cid:98) a ∈ (cid:98) A and define (cid:98) h : (cid:98) A → (cid:98) A as (cid:98) h ( (cid:98) a ) : = lim n → ∞ h ( a n ) ,where a n → (cid:98) a is a Cauchy sequence of (elements of) A convergent to (cid:98) a . Observe that (cid:98) h is well defined. Indeed, suppose that a (cid:48) n → (cid:98) a , i.e. a (cid:48) n is a different Cauchy sequenceconvergent to the element (cid:98) a . Then lim n → ∞ (cid:98) d ( h ( a n ) , h ( a (cid:48) n )) = lim n → ∞ d ( h ( a n ) , h ( a (cid:48) n )) = lim n → ∞ d ( a n , a (cid:48) n ) = (cid:98) d ( (cid:98) a , (cid:98) a ) =
0, where the second equality is justified by the fact that h isan isometry (distance preserving map). This shows that lim n → ∞ h ( a n ) = lim n → ∞ h ( a (cid:48) n ) .Moreover, it follows by construction that (cid:98) h | A = h . It only remains to show that (cid:98) h isdistance preserving and a Boolean homomorphism. Let (cid:98) a , (cid:98) b ∈ (cid:98) A . Then (cid:98) d ( (cid:98) h ( (cid:98) a ) , (cid:98) h ( (cid:98) b )) = (cid:98) d ( lim n → ∞ h ( a n ) , h ( b n ))= lim n → ∞ (cid:98) d ( h ( a n ) , h ( b n )) ( (cid:98) d is continuous )= lim n → ∞ d ( h ( a n ) , h ( b n ))= lim n → ∞ d ( a n , b n ) ( h isometry )= (cid:98) d ( (cid:98) a , (cid:98) b ) .To see that (cid:98) h is a Boolean homomorphism, we only show the case of a binary operationhere (the others are checked analogously). (cid:98) h ( (cid:98) a ∧ (cid:98) b ) = (cid:98) h ( lim n → ∞ a n ∧ b n )= lim n → ∞ (cid:98) h ( a n ∧ b n ) ( (cid:98) h is continuous )= lim n → ∞ h ( a n ∧ b n ) ( (cid:98) h | A = h )= lim n → ∞ h ( a n ) ∧ h ( b n ) ( h homomorphism )= lim n → ∞ h ( a n ) ∧ lim n → ∞ h ( b n )= (cid:98) h ( (cid:98) a ) ∧ (cid:98) h ( (cid:98) b ) . (cid:4) Notation: given the pseudo-metric space ( B , d s ) , where B ∈ I BS L and d s is the pseudo-metric induced by a state s , we indicate by d ∞ (instead of d Φ ( s ) ) the pseudo-metric on itsBooleanisation, obtained via the bijection in Theorem . Lemma . Let B ∈ I BS L carrying a state s. Then ( B , d s ) is complete if and only if ( A ∞ , d ∞ ) is complete.Proof. Observe that a sequence { x n } n ∈ N of elements in B is a Cauchy sequence if and onlyif (the sequence) { [ x n ] ∼ } n ∈ N is a Cauchy sequence in A ∞ . Indeed, if, for each ε >
0, thereis a n ∈ N such that d s ( x n , x m ) < ε , for every n , m > n , then also d ∞ ([ x n ] ∼ , [ x m ] ∼ ) < ε ,since d s ( x n , x m ) = d ∞ ([ x n ] ∼ , [ x m ] ∼ ) by Theorem . The result follows by observing thatlim n → ∞ x n = x if and only if lim n → ∞ [ x n ] ∼ = [ x ] ∼ . (cid:4) Theorem . Let ( B , d s ) an involutive bisemilattice with the pseudo-metric d s induced by a states such that B ∼ = P ł ( A i ) . Then:( ) (cid:101) B = P ł ( (cid:98) A i ) is an involutive bisemilattice;( ) ( (cid:98) B , (cid:98) d s ) is isometric to ( (cid:101) B , (cid:101) d ) .Proof. ( ) (cid:101) B = (cid:70) i ∈ I (cid:98) A i , where (cid:98) A i , for each i ∈ I , is a Boolean algebra (since Boolean alge-bras are closed under completitions). Moreover, by Lemma , the system (cid:104){ (cid:98) A i } i ∈ I , I , { (cid:98) h ij } i ≤ j (cid:105) is a semilattice direct systems of Boolean algebras (given that (cid:104){ A i } i ∈ I , I , { h ij } i ≤ j (cid:105) is such)and this is enough to conclude that (cid:101) B is an involutive bisemilattice.( ) Preliminarily observe that (cid:101) A ∞ = (cid:98) A ∞ (the Booleanisations of (cid:101) B and (cid:98) B coincide). Then,by Lemma , it follows that (cid:101) B is complete, as its Booleanisation is complete. We claimthat ( B , d s ) is a dense subset of ( (cid:101) B , (cid:101) d ) . Indeed, let (cid:101) b ∈ (cid:101) B , then (cid:101) b ∈ (cid:98) A i , for some i ∈ I (since (cid:101) B = (cid:70) i ∈ I (cid:98) A i ). Thus, there exists a sequence { a n } n ∈ N ∈ A i such that lim n → ∞ d s ( a n , (cid:101) b ) = (cid:101) d , (cid:101) d ( a n , (cid:101) b ) →
0, which implies that a n (as element of B ) convergesto (cid:101) b , i.e. B is dense in (cid:101) B . It follows (from the previous claim) that there is an isometricbijection f : ( (cid:98) B , (cid:98) d s ) → ( (cid:101) B , (cid:101) d ) . (cid:4) . T he topology of involutive bisemilattices An involutive bisemilattice B carrying a state s can be topologised with the topologyinduced by the pseudo-metric d s , defined in ( ). In virtue of Theorem (and Theorem for faithful states), also the Booleanisation A ∞ (of B ) can be topologised via the corre-sponding probability measure which we indicate as d ∞ = Φ ( s ) ◦ (cid:77) (where Φ is the mapdefined in ( )). In this section, we confine ourselves only to faithful states over (injective)involutive bisemilattices and when referring to B and A ∞ as topological spaces, we thinkthem as equipped with the topologies T d s and T d ∞ induced by the respective (pseudo)metric. Recall that, in this topology, a subset U ⊂ B is open if and only if for every x ∈ U ,there exists r > D r ( x ) ⊂ U , where D r ( x ) = { y ∈ B | d s ( x , y ) < r } is the opendisk centered in x with radius r . Moreover, one base of both topologies is given by thefamily of all open disks with respect to d s and d ∞ , respectively. Definition . Let X be a topological space and let ≡ ⊆ X × X the equivalence relationdefined as x ≡ y if and only if x and y have the same open neighbourhoods. Then, thespace X / ≡ is the Kolmogorov quotient of X .In words, two points x , y belonging to the same equivalent class with respect to ≡ are topologically indistinguishable . Proposition . Let B ∈ I − I BS L . Then the Kolmogorov quotient of B is its Booleanisation A ∞ .Proof. We have to show that two elements a , b ∈ B are topologically indistinguishable(namely, d s ( a , b ) =
0) if and only if a ∼ b . Assume w.l.o.g. that a ∈ A i and b ∈ A j with i (cid:54) = j and set k = i ∨ j .( ⇒ ) Let d s ( a , b ) = a , b are indistinguishable). Then s ( a (cid:77) B b ) = s is STEFANO BONZIO AND ANDREA LOI faithful, a (cid:77) B b = p ik ( a ) (cid:77) A k p jk ( b ) = k . Then p ik ( a ) = p jk ( b ) , i.e. a ∼ b .( ⇐ ) Let a ∼ b . It follows that there exists l ∈ I such that i , j ≤ l and p il ( a ) = p jl ( b ) . Then s ( a (cid:77) b ) = s ( p ik ( a ) (cid:77) A k p jk ( b ))= s k ( p ik ( a ) (cid:77) A k p jk ( b ))= s l ( p kl ( p ik ( a ) (cid:77) A k p jk ( b ))) (Proposition ) = s l ( p kl ◦ p ik ( a ) (cid:77) A l p kl ◦ p jk ( b ))= s l ( p il ( a ) (cid:77) A l p jl ( b ))= s ( l )= a , b are indeed two topologically indistinguishable points. (cid:4) The proof of Proposition shows another interesting fact which it worths to be high-lighted: while B is a pseudo-metric space, its Booleanisation A ∞ becomes a metric space.By combining Proposition and Theorem we immediately get the following. Corollary . Let B ∈ I − I BS L with a faithful state s. Then (cid:98) B / ≡ = (cid:98) A ∞ , i.e. the Kolmogorovquotient of the completition ( B , d s ) is the completition of the metric space ( A ∞ , d ∞ ) . Remark . Observe that, given a (finitely additive) probability measure m over a Booleanalgebra A and the (psudo)metric d m , it holds that m ( a ) = d m ( a , 0 ) , for any a ∈ A . ABoolean algebra A topologised with T d m , where d m is the discrete metric (defined viaa probability measure m ), coincides with the two-elements Boolean algebra. Indeed,suppose that A contains an element a (cid:54)∈ {
0, 1 } . Then 1 = d m ( a , a (cid:48) ) = m ( a (cid:77) a (cid:48) ) = m (( a ∧ a ) ∨ ( a (cid:48) ∧ a (cid:48) )) = m ( a ∨ a (cid:48) ) = m ( a ) + m ( a (cid:48) ) = d m ( a , 0 ) + d m ( a (cid:48) , 0 ) = +
1, which isa contradiction. It follows that the involutive bisemilattice B topologised with T d s , where d s is the pseudometric (defined via a certain state s ) obtained in case d m i is the discretemetric over the Boolean algebra A i in the Płonka sum, is decomposed into the Płonkasum of two-element Boolean algebras ( B = (cid:70) i ∈ I i ). Remark . Let the Booleanisation A ∞ of an involutive bisemilattice be finite. Then T d ∞ is the discrete topology (this does not imply that d ∞ is the discrete metric). Indeed, itholds in general for finite metric spaces ( X , d ) , that the induced topology T d is discrete.It follows that any space ( A i , T s i ) is discrete.Given a surjective map f : X → Y between topological spaces X and Y , a section of f isa continuous map g such that f ◦ g = id . Theorem . The following facts hold for the topological spaces B and A ∞ :( ) There exists a section σ : A ∞ → B of π such that σ ( A ∞ ) is dense in B ;( ) σ preserves states, namely s ◦ σ = Φ ( s ) . Recall that the discrete metric d on a set X is defined as d ( x , y ) = (cid:40)
1, if x (cid:54) = y ,0, otherwise. . Proof. ( ) Consider σ : A ∞ → B defined as follows: σ ([ a ] ∼ ) = a ,where a is a rapresentative of the equivalence class [ a ] ∼ (a choice that can be alwaysdone recurring to the Axiom of choice). We first show that σ is continuous. Let D r ( b ) an the open disk of radius r centered in b, for some b ∈ B . σ − ( D j ) = { [ y ] ∼ ∈ A ∞ | σ ([ y ] ∼ ) ∈ D j } = { [ y ] ∼ ∈ A ∞ | y ∈ D j } = { [ y ] ∼ ∈ A ∞ | d s ( y , b ) < r } = { [ y ] ∼ ∈ A ∞ | d ∞ ([ b ] ∼ , [ y ] ∼ ) < r } = D r ([ b ] ∼ ) , the open disk of A ∞ centered in [ b ] ∼ . This shows thecontinuity of σ .The fact that π ◦ σ = id immediately follows from the definition of σ .To show that σ ( A ∞ ) is dense in B , we have to check that σ ( A ∞ ) ∩ U (cid:54) = ∅ , for everynon-void open set U ⊂ B . Since U is non empty, there exists b ∈ U . Moreover, since U is open, then, for some r > D r ( b ) ⊂ U (where D r ( b ) = { y ∈ B | d s ( b , y ) < r } ). W.l.o.g.we can assume that b ∈ A i , for some i ∈ I ( A i is a Boolean algebra in the Płonka sumrepresentation of B ). Observe that, for each j ∈ I with i ≤ j , we have that p ij ( b ) ∈ U .Indeed d s ( b , p ij ( b )) = s ( b (cid:77) B p ij ( b )) = s j ( p ij ( b ) (cid:77) A j p ij ( b )) =
0, which implies that p ij ( b ) ∈ D r ( b ) ⊂ U . If σ ([ b ]) = b , then we have finished. So, assume that σ ([ b ]) = a ,with a (cid:54) = b . W.l.o.g. let a ∈ A j (for some j ∈ I ). By definition of σ , a ∈ [ b ] ∼ , i.e. thereexists some k ∈ I , such that i , j ≤ k and p ik ( b ) = p jk ( a ) . Since p ik ( b ) ∈ U (for the aboveobservation), then also p jk ( a ) ∈ U . Reasoning as above, one checks that d s ( a , p jk ( a )) = a ∈ U . This shows that σ ( A ∞ ) ∩ U (cid:54) = ∅ , for every non-void open set U ⊂ B , i.e. σ ( A ∞ ) is dense in B .( ) follows from the definition of σ and Theorem . (cid:4) Remark . Observe that the projection π admits many sections (depending on the car-dinality of the fiber π − ([ b ]) , for b ∈ B ) and all of them are topological embeddings byconstruction.Let f : X → Y be a continuous map between two topological spaces ( X and Y ). U ⊆ X is f -saturated (or saturated with respect to f ) if U = f − ( f ( U )) . Lemma . Every open and closed set of an involutive bisemilattice B is saturated with respectto the projection π : B → A ∞ . In particular, π is (continuous) open and closed.Proof. Since the basis of the topology (over B ) is the family of open disks, then, withrespect to open sets, it is enough to check that π − ( π ( D r )) = D r . Let D r ( b ) an open disk(of radius r ) centered in b , for some b ∈ B . Then π − ( π ( D r ( b ))) = { x ∈ B | π ( x ) ∈ π ( D r ( b )) } = { x ∈ B | d ∞ ([ x ] , [ b ]) < r } = { x ∈ B | d s ( x , b ) < r } = D r ( b ) , where thesecond last equality holds by Theorem .Let C ⊆ B a closed set. Then C = B \ U , for some open set U . Observe that C ⊆ π − ( π ( C )) holds in general, so we have to show only the other inclusion. To this end π − ( π ( C )) = π − ( π ( B \ U )) ⊆ π − ( π ( B ) \ π ( U )) = π − ( π ( B )) \ π − ( π ( U )) = B \ U = C , where the second last equality holds since open sets are saturated with respect to π . (cid:4) STEFANO BONZIO AND ANDREA LOI
Remark . In the proof of the following results we will use some well-known facts ingeneral topology that we will briefly recap (see, for instance, [ ]). Let f : X → Y , be anopen and closed continuous function between topological spaces. Then( ) f − ( Int ( B )) = Int ( f − ( B )) , for every B ⊆ Y ;( ) f ( A ) = f ( A ) , for every A ⊆ X . Lemma . Let C ⊆ B a closed set of an involutive bisemilattice B . Then π ( Int ( C )) = Int ( π ( C )) .Proof. Let C be a closed set in B . Observe that, from Lemma , we have that π is anopen and closed continuous map. Hence π ( Int ( C )) = π ( Int ( π − ( π ( C )))) = π ( π − ( Int ( π ( C )))) = Int ( π ( C )) ,where we have applied Lemma and the properties of open and closed continuousmaps (see Remark ). (cid:4) Observe that the statement of Lemma is in general false (see [ ] for details). Re-call that a map f : X → Y (between two topological spaces) preserves the interiors ifInt ( f ( A )) = f ( Int ( A )) , for all A ⊆ X . Interior preserving maps are studied in [ ].One can wonder whether the statement of Lemma could be extended to any subsetof involutive bisemilattice (instead of confining to closed subsets). Interestingly enough,the next results shows that the projection π is interior preserving if and only if B is aBoolean algebra. Theorem . Let B an involutive bisemilattice. The following facts are equivalent:( ) B = A ∞ ;( ) π : B → A ∞ is an interior preserving map;( ) σ ( A ∞ ) is open (closed, saturated) in B , for every section σ : A ∞ → B .Proof. ( ) ⇒ ( ) is trivial (as π = id ).( ) ⇒ ( ). We reason by contraposition, and suppose that B (cid:54) = A ∞ . This implies thatthere exists an element [ a ] ∈ A ∞ such that | π − ([ a ]) |≥
2. Let b ∈ π − [ a ] . Observethat A ∞ \ { [ a ] } is open (as { [ a ] } is closed), thus, since π is continuous, B \ { π − ([ a ]) } isopen. This implies that Int ( B \ { b } ) = B \ { π − ([ a ]) } . Then, π ( Int ( B \ { b } )) = π ( B \{ π − ([ a ]) } ) = A ∞ \ { [ a ] } . On the other hand, Int ( π ( B \ { b } )) = A ∞ , since | π − ([ a ]) |≥
2, which shows that π does not preserve interiors.( ) ⇒ ( ) is obvious.( ) ⇒ ( ). Let σ ( A ∞ ) be open (closed, saturated) in B . Then, by Lemma , σ ( A ∞ ) is π -saturated, i.e. σ ( A ∞ ) = π − ( π ( σ ( A ∞ ))) = π − ( A ∞ ) = B , so π is a bijection being σ its inverse. (cid:4) Recall that, for a topological space ( X , T ) , an open set U ⊆ X is an open regular set if U = Int ( U ) (where U indicates the closure of U ). To keep in mind the difference betweenan open and an open regular set, consider R topologised (as usual) with the Euclidiantopology. Then (
0, 1 ) is an example of an open regular set, while U = (
0, 1 ) ∪ (
1, 2 ) is an open set which is not regular, as Int ( U ) = (
0, 2 ) . The set of open regular sets Reg ( X ) of a topological space ( X , T ) can be turned into a (complete) Boolean algebra (see, for instance, [ ]) Reg ( X ) = (cid:104) Reg ( X ) , ∩ , ∨ , (cid:114) , ∅ , X (cid:105) , where U ∨ V : = Int ( A ∪ B ) .Moreover, the Boolean algebra of Clopen ( X ) (of the clopen sets of X ) is a subalgebra of Reg ( X ) . Despite the fact that an involutive bisemilattice B and its Booleanisation A ∞ arenot homeomorphic (except in the trivial case B = A ∞ ), surprisingly enough, the Booleanalgebras of regular sets arising from B and A ∞ are isomorphic, as shown in the following. Theorem . The projection π : B → A ∞ induces a bijection between Open ( B ) and Open ( A ∞ ) ,the open sets of B and A ∞ , respectively.Moreover, the Boolean algebras Reg ( B ) and Reg ( A ∞ ) are isomorphic.Proof. The fact that π is a bijection between Open ( B ) and Open ( A ∞ ) follows from Lemma . The isomorphism between Reg ( B ) and Reg ( A ∞ ) is given by the projection π , re-stricted to Reg ( B ) . We first show that the map is well defined, i.e. that given an openregular set U ∈ Reg ( B ) , then π ( U ) ∈ Reg ( A ∞ ) . To show regularity, observe that π ( U ) = π ( Int ( U )) ( U is regular )= Int ( π ( U )) ( Lemma )= Int ( π ( U )) ( π is continuous, open and closed ) To conclude the proof, we only need to check that π is a homomorphism (with respect tothe Boolean operations of Reg ( B ) and Reg ( A ∞ ) ). With respect to the constants, observethat π ( ∅ ) = ∅ and, since π is surjective, π ( B ) = A ∞ . Now, let U , V ∈ Reg ( B ) , then π ( U ) ∩ π ( V ) = π ◦ π − ( π ( U ) ∩ π ( V )) = π ( π − ( π ( U )) ∩ π − ( π ( V ))) = π ( U ∩ V ) ,where the last equality follows from Lemma . Moreover, π ( U ∨ V ) = π ( Int ( U ∪ V ))= Int ( π ( U ∪ V )) ( Lemma )= Int ( π ( U ∪ V )) ( π is continuous, open and closed )= Int ( π ( U ) ∪ π ( V ))= π ( U ) ∨ π ( V ) .Since we have shown that π preserves the constants and the binary operations, it followsthat it preserves also the unary operation \ , hence we are done. (cid:4) Theorem (Topological characterization of states) . Let s be a state over B and t : B → [
0, 1 ] a continuous map such that t ◦ σ = Φ ( s ) , for any section σ : A ∞ → B . Then t = s.Proof. By assumption, t ◦ σ = Φ ( s ) , for any section σ : A ∞ → B . This implies that the twocontinuous maps s and t coincide over a dense subset σ ( A ∞ ) of B (in virtue of Theorem -( )). Therefore, since [
0, 1 ] is Hausdorff, t = s . (cid:4) Remark . As we have seen, in general, the spaces B and A ∞ are not homeomorphic.However, it follows from the general theory of Kolmogorov quotients (see [ , Theorem STEFANO BONZIO AND ANDREA LOI . ]) that, under the assumption that they are both Alexandrov discrete they are homo-topically equivalent. Obviously, the equivalence holds in the particular case whether B is finite. . C onclusion and further work In this work, we have shown how to define a notion of state on Płonka sums of Booleanalgebras, with the aim of avareging the probability for elements of an involutive bisemi-lattice, a variety associated to the logic PWK. In particular, we have exploited the con-nections between such notion, the probability measures carried by Boolean algebras ina Płonka sum and the Booleanisation of an involutive bisemilattices. These connectionsare crucial in the study of the completition and the topology induced by a state over aninvolutive bisemilattices.This work sheds a further light on the possibility of developing the theory of probabil-ity beyond the boundaries of classical events, namely elements of a Boolean algebra. Tothe best of our knowledge, this consists of the first attempt to lift probability measuresfrom Boolean algebras to Płonka sums of Boolean algebras. For this reason, many theo-retical problems as well as potential applications of this theory are not examined in thepresent work. At first, it shall be noticed that there is nothing special behind the choice ofBoolean algebras, a part the fact that Płonka sums of Boolean algebra play the importantrole to characterize the algebraic counterpart of paraconsistent weak Kleene logic. Theideas developed here could be used, in principle, to define states for varieties that arerepresented as Płonka sums of classes of algebras admitting states, such as MV-algebras,Goedel algebras, Heyting algebras, just to mention some for which a theory of statesexists. The connection between Płonka sums and certain logics has been explored in [ ].A relevant question that we leave for further investigations is the possibility of charac-terizing states over involutive bisemilattices as coherent books over a (finite) set of eventsof the logic PWK. Coherent books have been introduced, in the classical case, by deFinetti [ , ], via a specific (reversible) betting game and are shown to be one-to-onecorrespondence with (finitely additive) probability measures over the Boolean algebragenerated by the events considered. This kind of abstract betting scenario has been usedalso to characterize states over non-classical structures [ ].We have shown (see Theorem ) that states over involutive bisemilattices correspondto integrals on the dual space of the Booleanisation. It makes sense to ask whether thiscorrespondence can be extended to faithful states, relying on the integral representationproved for faithful states over free MV-algebras in [ ].The theory of states we developed could find potential applications in the field of knowledge representation . This is mainly due to the fact that states break into probabilitymeasures over the Boolean algebras in the Płonka sum representation. One may inter-pret the semilattice of indexes, involved in the representation, to model, for instance,situations of branching time (as the index set is, in general, not a chain). A state, then,encapsulates information related to the probabilities of classical events (Boolean algebras) A topological space ( X , T ) is Alexandrov discrete when the arbitrary intersection of open sets is anopen set. A similar idea is developed from the construction of horizontal sums in [ ]. located in every point (indexes) of the structure. This might be used, in principle, also toanalyse conditional bettings or counterfactual situations, under the assumption, for in-stance, that events are related when there is a homomorphism connecting they algebrasthey belong to. A ppendix Our definition of state relies (see Definition ) on the assumption that two elements a , b ∈ B of an involutive bisemilattice B are logically incompatible provided that a ∧ B b = i , where 0 i is the bottom element of the Boolean algebra (in the Płonka sum represen-tation of B ) where the operation ∧ is computed. One could question this principle andintend two elements a , b ∈ B as incompatible when a ∧ B b =
0. This leads to a differentdefinition of state obtained, by replacing condition ( ) in Definition with the following: s ( a ∨ b ) = s ( a ) + s ( b ) provided that a ∧ b =
0. ( )However, since the element 0 of an involutive bisemilattice always belongs to theBoolean algebra (in the Płonka sum) whose index is the least element in the semilat-tice (cid:104) I , ≤(cid:105) of indexes, this latter choice leads to the following consequence. Proposition . Let B and involutive bisemilattice. Then the following are equivalent:( ) s : B → [
0, 1 ] satisfies s ( ) = and condition ( ) ;( ) m i is a (finitely additive) probability measure over the Boolean algebra A i where i is theminimum element in I.Proof. ( ) ⇒ ( ). Immediate by observing that 1 ∈ A i and that, for any arbitrary pair ofelements a , b ∈ B , a ∧ b = a , b ∈ A i .( ) ⇒ ( ). Let m i : A i → [
0, 1 ] be any finitely additive probability measure over A i .Then, the map s : B → [
0, 1 ] s ( x ) : = (cid:40) m i ( x ) if x ∈ A i , α otherwise,for any α ∈ (
0, 1 ) , satisfies that s ( ) = s ( a ∨ b ) = s ( a ) + s ( b ) , when a ∧ b = m i is the restriction of s over A i . (cid:4) In words, the above result suggests that, this different notion of state, obtained byreplacing ( ) in Definition with ( ), implies that only the elements belonging to theBoolean algebra A i are actually measured following the standard rules of probability.R eferences [ ] S. Aguzzoli, M. Bianchi, B. Gerla, and D. Valota. Probability Measures in G ¨odel ∆ Logic. In A. An-tonucci, L. Cholvy, and O. Papini, editors,
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Formal Logic , ( ): – , .S tefano B onzio , D ipartimento di F ilosofia e S cienze dell ’E ducazione , U niversit ` a di T orino ,I taly . E-mail address : [email protected] A ndrea L oi , D ipartimento di M atematica e I nformatica , U niversit ` a di C agliari , I taly . E-mail address ::