aa r X i v : . [ m a t h . L O ] S e p Lukasiewicz logic and Riesz spaces
Antonio Di NolaUniversita di Salerno,Via Ponte don Melillo 84084 Fisciano, Salerno, Italy [email protected]
Ioana Leu¸steanUniversity of Bucharest,Academiei 14, sector 1, C.P. 010014, Bucharest, Romania [email protected]
Abstract
We initiate a deep study of
Riesz MV-algebras which are MV-algebrasendowed with a scalar multiplication with scalars from [0 , ℓ -groups, we provethat Riesz MV-algebras are categorically equivalent with unit intervals inRiesz spaces with strong unit. Moreover, the subclass of norm-completeRiesz MV-algebras is equivalent with the class of commutative unital C ∗ -algebras. The propositional calculus R L that has Riesz MV-algebras asmodels is a conservative extension of Lukasiewicz ∞ -valued propositionalcalculus and it is complete with respect to evaluations in the standardmodel [0 , R L and werelate them with the analogue of de Finetti’s coherence criterion for R L . Keywords:
Riesz MV-algebra, Lukasiewicz logic, piecewise linear func-tion, quasi-linear combination.
MSC (2000):
MV-algebras are the algebraic structures of Lukasiewicz ∞ -valued logic. Thereal unit interval [0 ,
1] equipped with the operations x ∗ = 1 − x and x ⊕ y = min(1 , x + y )for any x, y ∈ [0 , , G, u ) such that A ≃ [0 , u ] G , where10 , u ] G = ([0 , u ] , ⊕ , ∗ , , u ] = { x ∈ G | ≤ x ≤ u } , x ⊕ y = ( x + y ) ∧ u and x ∗ = u − x for any x , y ∈ [0 , u ].If ( V, u ) is a Riesz space (vector-lattice) [22] with strong unit then the unitinterval [0 , u ] V is closed to the scalar multiplication with scalars from [0 , , u ] V = ([0 , u ] , · , ⊕ , ∗ , , u ] , ⊕ , ∗ ,
0) is the MV-algebra defined as above and · : [0 , × [0 , u ] V → [0 , u ] V satisfies the axioms of the scalar product is the fundamental example inthe theory of Riesz MV-algebras , initiated in [11] and further developed in thepresent paper.The study of Riesz MV-algebras is related to the problem of finding acomplete axiomatization for the variety generated by ([0 , , · , ⊕ , ∗ , , , ⊕ , ∗ ,
0) is the standard MV-algebra and · is the product of real numbers.The investigations led to the definition of product MV-algebras (PMV-algebras),which can be represented as unit intervals in lattice-ordered rings with strongunit [9]. A PMV-algebra is a structure ( P, · ), where P is an MV-algebra and · : P × P → P satisfies the equations of an internal product. PMV-algebras arean equational class, but the standard model [0 ,
1] generates only a quasi-varietywhich is a proper subclass of PMV-algebras [26]. In this context, it was naturalto replace the internal product with an external one: a
Riesz MV-algebra is astructure ( R, · ), where R is an MV-algebra and · : [0 , × R → R . Since we provethat the variety of Riesz MV-algebras is generated by [0 , R L , that has Riesz MV-algebras as models, is complete with respect toevaluations in [0 , ,
1] generates the variety of Riesz MV-algebras.In Section 5, the categorical equivalence is specialized to the class of norm-complete Riesz MV-algebras , which is dually equivalent with the category ofcompact Hausdorff spaces. Using the Gelfand-Naimark duality, this leads us toa connection with the theory of commutative unital C ∗ -algebras.Section 6 presents the propositional calculus R L which simplifies the oneintroduced in [11]. In Section 7 we prove a normal form theorem for formulas of R L . Since R L is a conservative extension of Lukasiewicz logic L , this theoremis a generalization of McNaughton theorem [24]. Our result asserts that for anycontinuous piecewise linear function f : [0 , n → [0 , there exists a formula ϕ of R L with n variables such that f is the term function associated to ϕ .In Section 8 we initiate the theory of quasi-linear combinations of formulasin R L . If f i : [0 , n → R are continuous piecewise linear functions and c i are real numbers for any i ∈ { , . . . , k } , then the normal form theorem guar-antees the existence of a formula Φ of R L , whose term function is equal to2( P ki =1 c i f i ) ∨ ∧ quasi-linear combination of f , · · · , f k . We prove de Finetti’s coherence criterion for R L and we providean equivalent characterization by the fact that a quasi-linear span contains onlyinvalid formulas.Some of results contained in this paper may overlap with the proceedingpaper [11]. Other results are proved in a more general setting in [10, 12, 21].For the sake of completeness we sketched the proofs that we consider importantfor the present development. An MV-algebra is a structure ( A, ⊕ , ∗ , A, ⊕ ,
0) is an abelian monoidand the following identities hold for all x, y ∈ A :(MV1) ( x ∗ ) ∗ = x ,(MV2) 0 ∗ ⊕ x = 0 ∗ ,(MV3) ( x ∗ ⊕ y ) ∗ ⊕ y = ( y ∗ ⊕ x ) ∗ ⊕ x .We refer to [7] for all the unexplained notions concerning MV-algebras and to[31] for advanced topics. On any MV-algebra A the following operations aredefined for any x, y ∈ A :1 = 0 ∗ , x ⊙ y = ( x ∗ ⊕ y ∗ ) ∗ , x → y = x ∗ ⊕ y x = 0, mx = ( m − x ⊕ x for any m ≥ ⊙ is more binding then ⊕ . Remark 2.1.
Any MV-algebra A is a bounded distributive lattice, with thepartial order defined by x ≤ y if and only if x ⊙ y ∗ = 0 and the lattice operations defined by x ∨ y = x ⊕ y ⊙ x ∗ and x ∧ y = x ⊙ ( x ∗ ⊕ y ) for any x , y ∈ A . Any MV-algebra A has an internal distance: d ( x, y ) = ( x ⊙ y ∗ ) ⊕ ( x ∗ ⊙ y ) for any x , y ∈ A . Lemma 2.2. [7, Proposition 1.2.5] In any MV-algebra A , the following proper-ties hold for any x , y , z ∈ A :(a) d ( x, y ) = d ( y, x ) ,(b) d ( x, y ) = 0 iff x = y ,(c) d ( x, z ) ≤ d ( x, y ) ⊕ d ( y, z ) . If ( A, ⊕ , ∗ ,
0) is an MV-algebra then an ideal is a nonempty subset I ⊆ A such that for any x , y ∈ A the following conditions are satisfied:3i1) x ∈ I and y ≤ x imply y ∈ I ,(i2) x and y ∈ I imply x ⊕ y ∈ I .An ideal I of A uniquely defines a congruence ∼ I by x ∼ I y iff x ⊙ y ∗ ∈ I and y ⊙ x ∗ ∈ I .We denote by A/I the quotient MV-algebra and we refer to [7] for more details.We recall that an ℓ -group is a structure ( G, + , , ≤ ) such that ( G, + ,
0) is agroup, ( G, ≤ ) is a lattice and any group translation is isotone [3]. In the followingthe ℓ -groups are abelian. For an ℓ -group G we denote G + = { x ∈ G | x ≥ } .An element u ∈ G is a strong unit if u ≥ x ∈ G there is a naturalnumber n such that x ≤ nu . An ℓ -group is unital if it posses a strong unit. If( G, u ) is a unital ℓ -group, we define [0 , u ] = { x ∈ G | ≤ x ≤ u } and x ⊕ y = ( x + y ) ∧ u , x ∗ = u − x for any x, y ∈ [0 , u ].Then [0 , u ] G = ([0 , u ] , ⊕ , ¬ ,
0) is an MV-algebra [7, Proposition 2.1.2].
Lemma 2.3. [7, Lemma 7.1.3] Let ( G, u ) be a unital ℓ -group, x ≥ in G and n ≥ a natural number such that x ≤ nu . Then x = x + · · · + x n , where x i = (( x − ( i − u ) ∨ ∧ u ∈ [0 , u ] for any i ∈ { , . . . , n } . We denote by MV the category of MV-algebras and by AG u the categoryof unital abelian lattice-ordered groups with unit-preserving morphisms. In [27]the functor Γ : AG u → MV is defined as follows:Γ( G, u ) = [0 , u ] G for any unital ℓ -group ( G, u ),Γ( f ) = f | [0 ,u ] for any morphsim f : ( G, u ) → ( G ′ , u ′ ) from AG u . Theorem 2.4. [7, Corollary 7.1.8] The functor Γ yields an equivalence between AG u and MV . Definition 2.5. If A and B are MV-algebras then a function ω : A → B iscalled additive if x ⊙ y = 0 implies ω ( x ) ⊙ ω ( y ) = 0 and ω ( x ⊕ y ) = ω ( x ) ⊕ ω ( y ) . Additivity was firstly studied in the context of states defined on MV-algebras[28]. The theory of states generalizes the boolean probability theory and reflectsthe theory of states defined on ℓ -groups. Definition 2.6. [28] If A is an MV-algebra then a function s : A → [0 , is a state if the following properties are satisfied for any x , y ∈ A :(s1) if x ⊙ y = 0 then s ( x ⊕ y ) = s ( x ) + s ( y ) ,(s2) s (1) = 1 . Proposition 2.7.
Assume ( G, u ) and ( H, v ) are unital ℓ -groups, A = Γ( G, u ) and B = Γ( H, v ) . Then for any additive function ω : A → B there exists aunique group morphism ω : G → H such that ω ( x ) = ω ( x ) for any x ∈ [0 , u ] .Proof. If x ∈ G and x ≥ x , . . . , x m ∈ [0 , u ] such that x = x + · · · + x m . Then we define ω ( x ) := ω ( x ) + · · · + ω ( x m ).The fact that ω ( x ) is well defined follows by Riesz decomposition property in ℓ -groups [3, 1.2.16]. Hence ω ( x ) is well defined for x ∈ G + and ω ( x + y ) = ω ( x ) + ω ( y ) for any x , y ∈ G + . By [3, 1.1.7] it follows that ω can be uniquelyextended to a group homomorphism defined on G . Lemma 2.8. If A and B are MV-algebras and ω : A → B is a function, thenthe following are equivalent:(a) ω is additive,(b) the following properties hold for any x , y ∈ A :(b1) x ≤ y implies ω ( x ) ≤ ω ( y ) ,(b2) ω ( x ⊙ ( x ∧ y ) ∗ ) = ω ( x ) ⊙ ω ( x ∧ y ) ∗ .Proof. (a) ⇒ (b) If x ≤ y then y = x ∨ y = x ⊕ y ⊙ x ∗ , so ω ( y ) = ω ( x ) ⊕ ω ( y ⊙ x ∗ )and ω ( x ) ≤ ω ( y ). Hence, ω is isotone. We remark that ω ( x ∧ y ) ≤ ω ( x ), so ω ( x ) ⊙ ω ( x ∧ y ) ∗ ⊕ ω ( x ∧ y ) = ω ( x ) ∨ ω ( x ∧ y ) = ω ( x ) = ω ( x ∨ ( x ∧ y )) = ω ( x ⊙ ( x ∧ y ) ∗ ) ⊕ ω ( x ∧ y ) . It follows that ω ( x ) ⊙ ω ( x ∧ y ) ∗ =( ω ( x ) ⊙ ω ( x ∧ y ) ∗ ) ∧ ω ( x ∧ y ) ∗ =( ω ( x ) ⊙ ω ( x ∧ y ) ∗ ⊕ ω ( x ∧ y )) ⊙ ω ( x ∧ y ) ∗ =( ω ( x ⊙ ( x ∧ y ) ∗ ) ⊕ ω ( x ∧ y )) ⊙ ω ( x ∧ y ) ∗ = ω ( x ⊙ ( x ∧ y ) ∗ ) ∧ ω ( x ∧ y ) ∗ = ω ( x ⊙ ( x ∧ y ) ∗ )(b) ⇒ (a) We remark that for x = 1 in (b2) we get ω ( y ∗ ) = ω (1) ⊙ ω ( y ) ∗ , so ω ( y ∗ ) ≤ ω ( y ) ∗ . Assume x ⊙ y = 0, so x ≤ y ∗ . Using (b1), we get ω ( x ) ≤ ω ( y ∗ ) ≤ ω ( y ) ∗ , so ω ( x ) ⊙ ω ( y ) = 0. In this case, using (b2) we get ω ( x ) = ω ( x ∧ y ∗ ) = ω (( x ⊕ y ) ⊙ y ∗ ) = ω ( x ⊕ y ) ⊙ ω ( y ) ∗ .It follows that: ω ( x ) ⊕ ω ( y ) = ω ( x ⊕ y ) ⊙ ω ( y ) ∗ ⊕ ω ( y ) = ω ( x ⊕ y ) ∨ ω ( y ).Using (b1), ω ( y ) ≤ ω ( x ⊕ y ), and we get ω ( x ⊕ y ) = ω ( x ) ⊕ ω ( y ).5 Riesz MV-algebras
Riesz MV-algebras are introduced in [11]. Below we give a simpler and moresuitable definition, which provides directly an equational characterization. Theequivalence between this definition and the one from [11] is proved in Theorem 3.17.
Definition 3.1. A Riesz MV-algebra is a structure ( R, · , ⊕ , ∗ , ,where ( R, ⊕ , ∗ , is an MV-algebra and the operation · : [0 , × R → R satisfiesthe following identities for any r , q ∈ [0 , and x , y ∈ R :(RMV1) r · ( x ⊙ y ∗ ) = ( r · x ) ⊙ ( r · y ) ∗ ,(RMV2) ( r ⊙ q ∗ ) · x = ( r · x ) ⊙ ( q · x ) ∗ ,(RMV3) r · ( q · x ) = ( rq ) · x ,(RMV4) · x = x .In the following we write rx instead of r · x for r ∈ [0 , and x ∈ R . Notethat rq is the real product for any r , q ∈ [0 , . Example 3.2. If X is a compact Hausdorff space then C ( X ) u = { f : X → [0 , | f continuous } is a Riesz MV-algebra, with all the operations defined componentwise. Thisexample will be further investigated in Section 5 Example 3.3. If G is an abelian ℓ -group, then R = Γ( R × lex G, (1 , is aRiesz MV-algebra, where R × lex G is the lexicographic product of ℓ -groups andthe scalar multiplication is defined by r ( q, x ) = ( rq, x ) for any r ∈ [0 , and ( q, x ) ∈ R . Lemma 3.4.
In any Riesz MV-algebra R the following properties hold for any r , q ∈ [0 , and x , y ∈ R :(a) x = 0 , r ,(b) x ≤ y implies rx ≤ ry ,(c) r ≤ q implies rx ≤ qx ,(d) rx ≤ x .Proof. (a) follows by (RMV1) and (RMV2) for x = y and, respectively, r = q .(b), (c) follow by Remark 2.1.(d) follows by (c) and (RMV4). Proposition 3.5.
The function ι : [0 , → R defined by ι ( r ) = r for any r ∈ [0 , is an embedding. Consequently, any Riesz MV-algebra R contains asubalgebra isomorphic with [0 , . roof. By Lemma 3.4 we get ι (0) = 0. If r , q ∈ [0 ,
1] then ι ( r ∗ ) = r ∗ · ⊙ ( r ∗ = ( r ∗ , ι ( r ⊙ q ) = ι ( r ⊙ q ∗∗ ) = ( r ⊙ q ∗∗ )1 = ( r ⊙ ( q ∗ ∗ = ( r ⊙ ( q ∗∗ = ( r ⊙ ( q Riesz space ( vector lattice ) [22] is a structure( V, · , + , , ≤ )such that ( V, + , , ≤ ) is an abelian ℓ -group, ( V, · , + ,
0) is a real vector space and,in addition,(RS) x ≤ y implies r · x ≤ r · y ,for any x , y ∈ V and r ∈ R , r ≥ unital if the underlaying ℓ -group is unital. Lemma 3.6. If ( V, u ) is a unital Riesz space, then [0 , u ] V = ([0 , u ] , · , ⊕ , ∗ , is a Riesz MV-algebra, where rx is the scalar multiplication of V for any r ∈ [0 , and x ∈ [0 , u ] .Proof. Assume r, q ∈ [0 ,
1] and x, y ∈ [0 , u ].(RMV1) r ( x ⊙ y ∗ ) = r (( x − y ) ∨
0) = ( rx − ry ) ∨ rx ) ⊙ ( ry ) ∗ .(RMV2) If r ≤ q then rx ≤ qx , so ( r ⊙ q ∗ ) x =(( r − q ) ∨ x = 0 = ( rx − qx ) ∨ rx ) ⊙ ( qx ) ∗ .If r > q then (( r − q ) ∨ x = ( r − q ) x = rx − qx = ( rx ) − ( qx ) ∨ rx ) ⊙ ( qx ) ∗ .We note that (RMV3) and (RMV4) hold in V , therefore they hold in [0 , u ]. Remark 3.7. If ( R, · , ⊕ , ∗ , is a Riesz MV-algebra then we denote its MV-algebra reduct by U( R ) = ( R, ⊕ , ∗ , . Assume I is an ideal of U( R ) . By Lemma3.4 (d) we infer that rx ∈ I whenever r ∈ [0 , and x ∈ I . It follows, by(RMV1), that rx ∼ I ry whenever r ∈ [0 , and x ∼ I y . As consequence, thequotient R/I has a canonical structure of Riesz MV-algebra.
Remark 3.8.
A Riesz MV-algebra R has the same theory of ideals (congru-ences) as its reduct U( R ) . If R is a Riesz MV-algebra and P ⊆ R an ideal thenit is straightforward that the following hold:(a) P is prime iff R/P is linearly ordered,(b) P is maximal iff R/P ≃ [0 , .Note that (b) holds since, for any maximal ideal P , the quotient R/P is an MV-subalgebra of [0 , . But the only subalgebra of [0 , which is a Riesz MV-algebrais [0 , by Proposition 3.5, so R/P ≃ [0 , . Lemma 3.9. If R is a Riesz MV-algebra, I ⊆ R an ideal and x ∈ R such that rx ∈ I for some r ∈ (0 , then x ∈ I .Proof. Let r ∈ (0 ,
1] such that rx ∈ I and let m be the integer part of r . Hence m +1 x ≤ rx , so m +1 x ∈ I . Since x = ( m + 1)( m +1 x ) we get x ∈ I .7 orollary 3.10. Any simple Riesz MV-algebra is isomorphic with [0 , . Anysemisimple Riesz MV-algebra is a subdirect product of copies of [0 , . In the sequel we investigate the morphisms of Riesz MV-algebras.
Corollary 3.11. If R and R are Riesz MV-algebras and f : U ( R ) → U ( R ) is a morphism of MV-algebras then f ( rx ) = rf ( x ) for any r ∈ [0 , and x ∈ R .Proof. Assume J is an ideal in R . Since f is an morphism of MV-algebras, itfollows that f − ( J ) is an ideal in R . If x ∈ R and r ∈ [0 ,
1] we have rf ( x ) ∈ J ⇒ f ( x ) ∈ J ⇒ x ∈ f − ( J ) ⇒ rx ∈ f − ( J ) ⇒ f ( rx ) ∈ J , f ( rx ) ∈ J ⇒ rx ∈ f − ( J ) ⇒ x ∈ f − ( J ) ⇒ f ( x ) ∈ J ⇒ rf ( x ) ∈ J .Note that we used Lemma 3.9 twice. We proved that, for any ideal J of R rf ( x ) ∈ J ⇔ f ( rx ) ∈ J. Therefore rf ( x ) ⊙ f ( rx ) ∗ ∈ J and f ( rx ) ⊙ ( rf ( x )) ∗ for any ideal J of R .This means that rf ( x ) ⊙ f ( rx ) ∗ = f ( rx ) ⊙ ( rf ( x )) ∗ = 0, so f ( rx ) = rf ( x ). Remark 3.12.
The above result asserts that a morphism of Riesz MV-algebrais simply a morphism between the corresponding MV-algebra reducts.
The following result is similar with Chang’s representation theorem for MV-algebras [6, Lemma 3].
Corollary 3.13.
Any Riesz MV-algebra is a subdirect product of linearly orderedRiesz MV-algebras.Proof. If R is a Riesz MV-algebra then, by Remark 3.8, T { P | P prime ideal of R } = { } and R/P is linearly ordered for any prime ideal P . As consequence, R is asubdirect product of the family { R/P | P prime ideal of R } .In order to prove that Riesz MV-algebras introduced in Definition 3.1 coin-cide with the ones defined in [11], we recall some results from [12]. Remark 3.14. [12] If Ω is a set of unary operation symbols, then an MV-algebra with Ω-operators is a structure ( A, Ω A ) where A is an MV-algebra andfor any ω ∈ Ω the operation ω A : A → A is additive. An additive function ω : A → A is an f -operator if x ∧ y = 0 implies ω ( x ) ∧ y = 0 for any x , y ∈ A . f ( A, Ω A ) is an MV-algebra with Ω -operators such that ω A is an f -operatorfor any ω ∈ Ω , then ( A, Ω A ) is a subdirect product of linearly ordered MV-algebras with Ω -operators [12, Corollary 5.6. ]. Remark 3.15.
Assume that ( R, ⊕ , ∗ , is an MV-algebra and let · : [0 , × R → R such that (RMV2) , (RMV3) , (RMV4) hold and the function ω r : R → R , ω r ( x ) = r · x is additive for any r ∈ [0 , . By (RMV2) and (RMV4) we get ω r ( x ) ≤ x for any r ∈ [0 , and x ∈ R , so ω r is an f -operator for any r ∈ [0 , . If Ω = { ω r | r ∈ [0 , } then, by Remark 3.14, ( R, Ω) is an MV-algebra with Ω -operators that can be represented as subdirect product of linearly ordered MV-algebras with Ω -operators. Lemma 3.16.
Assume that ( R, ⊕ , ∗ , is an MV-algebra. If · : [0 , × R → R then the following are equivalent:(RMV2) ( r ⊙ q ∗ ) · x = ( r · x ) ⊙ ( q · x ) ∗ for any r , q ∈ [0 , and x ∈ R , (RMV2 ′ ) r ⊙ q = 0 then ( r · x ) ⊙ ( q · x ) = 0 and ( r ⊕ q ) · x = ( r · x ) ⊕ ( q · x ) for any r , q ∈ [0 , and x ∈ R .Proof. For x ∈ R define ω x : [0 , → R by ω x ( r ) = rx for any r ∈ [0 , ω x satisfies (RMV2) then the condition (b) from Lemma 2.8 is satisfied, so ω x satisfies also (RMV2 ′ ). Conversely, if ω x satisfies (RMV2 ′ ) then, by Lemma 2.8,we also get ω x (0) = 0. Assume r , q ∈ [0 ,
1] such that r ≤ q . Hence r ⊙ q ∗ = 0and rx ≤ qx , so ( r ⊙ q ∗ ) x = 0 x = 0 = ( rx ) ⊙ ( qx ) ∗ . If r , q ∈ [0 ,
1] such that r > q then (RMV2) coincides with the equation (b2) from Lemma 2.8.
Theorem 3.17.
Assume that ( R, ⊕ , ∗ , is an MV-algebra and · : [0 , × R → R . Then ( R, · , ⊕ , ∗ , is a Riesz MV-algebra if and only if the following proper-ties are satisfied for any x , y ∈ R and r , q ∈ [0 , : (RMV1 ′ ) if x ⊙ y = 0 then ( r · x ) ⊙ ( r · y ) = 0 and r · ( x ⊕ y ) = ( r · x ) ⊕ ( r · y ) , (RMV2 ′ ) if r ⊙ q = 0 then ( r · x ) ⊙ ( q · x ) = 0 and ( r ⊕ q ) · x = ( r · x ) ⊕ ( q · x ) , (RMV3) r · ( q · x ) = ( rq ) · x , (RMV4) 1 · x = x .Proof. By Lemma 3.16, if ( R, ⊕ , ∗ ,
0) is an MV-algebra and · : [0 , × R → R , then the algebra ( R, · , ⊕ , ∗ ,
0) satisfies (RMV2), (RMV3) and (RMV4) ifand only if it satisfies (RMV2 ′ ), (RMV3) and (RMV4). Assume now that( R, · , ⊕ , ∗ ,
0) satisfies (RMV2), (RMV3) and (RMV4). We have to prove that(RMV1) is satisfied if and only if (RMV1 ′ ) is satisfied. By Corollary 4.1 andRemark 3.15, it suffices to prove the equivalence for linearly ordered structures.In this case, by Lemma 2.8, the equivalence of (RMV1) and (RMV1 ′ ) is straight-forward. 9ote that in [11] a Riesz MV-algebra is defined by (RMV1 ′ ), (RMV2 ′ ),(RMV3) and (RMV4), so we proved that Definition 3.1 is equivalent with theinitial one. By Theorem 3.17, Riesz MV-algebras are exactly the MV-modules [10] over[0 , Proposition 4.1.
For any Riesz MV-algebra R there is a unital Riesz space ( V, u ) such that R ≃ [0 , u ] V .Proof. By Theorem 2.4, there exists a unital ℓ -group ( V, u ) such that R and[0 , u ] V are isomorphic MV-algebras. For any λ ∈ R and x ∈ V we have to definethe scalar multiplication λx . We can safely assume that R = [0 , u ] ⊆ V .If r ∈ [0 ,
1] then x rx is an additive function from [0 , u ] V to [0 , u ] V so, byProposition 2.7, it can be uniquely extended to a group morphism ω r : V → V .Hence we define rx = ω r ( x ) for any x ∈ V . We note that x ≥ rx ≥ q ∈ [0 ,
1] then ω rq = ω r ◦ ω q since they coincide on the positive cone, so r ( qv ) = ( rq ) v .Note that v = ( v ∨ − (( − v ) ∨ v = 1( v ∨ − − v ) ∨
0) = ( v ∨ − (( − v ) ∨
0) = v .If λ ≥ v ∈ V , then there are r , . . . , r m ∈ [0 ,
1] such that λ = r + · · · + r m .Then we define λv = r v + · · · + r m v. One can prove that λv is well-defined using the Riesz decomposition property[3, 1.2.16]. If µ ≥
0, then µ = q + · · · + q n for some q , . . . , q n ∈ [0 ,
1] and λ ( µv ) = λ ( P nj =1 q j v ) = P mi =1 r i (cid:16)P nj =1 q j v (cid:17) = P mi =1 P nj =1 r i ( q j v ) = P mi =1 P nj =1 ( r i q j ) v = (cid:16)P mi =1 P nj =1 ( r i q j ) (cid:17) v = ( λµ ) v .If λ ≤ R then we set λv = − ( | λ | v )), where | λ | is the module of λ in R .It is straightforward that λ ( µv ) = ( λµ ) v for another µ ∈ R .We know that ( V, u ) is a unital ℓ -group and we defined the scalar product λv for any λ ∈ R and v ∈ V such that λv ≥ λ ≥ v ≥ V, u ) is a unital vector lattice.We denote by
RMV the category of Riesz MV-algebras and by RS u thecategory of unital Riesz spaces with unit-preserving morphisms.Following this construction we get a functor10 R : RS u → RMV defined as follows:Γ R ( V, u ) = [0 , u ] V for any unital Riesz space ( V, u ),Γ R ( f ) = f | [0 ,u ] for any morphism f : ( V, u ) → ( V ′ , u ′ ) from RS u . Theorem 4.2. [11, 10] The functor Γ R yields an equivalence between RS u and RMV .Proof.
It follows from Theorem 2.4, Corollary 3.11 and Proposition 4.1.It is straightforward that the following diagram is commutative, where Uare forgetful functors: RS u Γ R −→ RMV U ↓ ↓ U AB u Γ −→ MV The standard Riesz MV-algebra is ([0 , , · , ⊕ , ∗ , · : [0 , × [0 , → [0 ,
1] is the product of real numbers and ([0 , , ⊕ , ∗ ,
0) is the standard MV-algebra. In the sequel we prove that the variety of Riesz MV-algebras is gen-erated by [0 , , R functor. The first-order theory of Riesz MV-algebras, aswell as the theory of Riesz spaces, are obtained considering for each scalar r an unary function ρ r which denotes in a particular model the scalar mul-tiplication by r , i.e. x ρ r rx . In the following, the language of Riesz MV-algebras is L RMV = {⊕ , ∗ , , { ρ r } r ∈ [0 , } and the language of Riesz spaces is L Riesz = {≤ , + , − , ∨ , ∧ , , { ρ r } r ∈ R } .Let t ( v , . . . , v k ) be a term of L RMV and v a propositional variable differentfrom v , . . . , v k . We define t as follows:- if t = 0 then 0 is 0,- if t = v then t is v - if t = t ∗ then t is v − t ,- if t = t ⊕ t then t is ( t + t ) ∧ v ,- if t = ρ r ( t ) then t is ρ r ( t ).Let ϕ ( v , . . . , v k ) be a formula of L RMV such that all the free and boundvariables of ϕ are in { v , . . . , v k } and v a propositional variable different from v , . . . , v k . We define ϕ as follows:- if ϕ is t = t then ϕ is t = t ,- if ϕ is ¬ ψ then ϕ is ¬ ψ ,- if ϕ is ψ ∨ χ then ϕ is ψ ∨ χ and similarly for ∧ , → , ↔ ,11 if ϕ is ( ∀ v i ) ψ then ϕ is ∀ v i ((0 ≤ v i ) ∧ ( v i ≤ v ) → ψ ),- if ϕ is ∃ v i ψ then ϕ is ∃ v i ((0 ≤ v i ) ∧ ( v i ≤ v ) → ψ ).Thus to any formula ϕ ( v , . . . , v k ) of L RMV we associate a formula ϕ ( v , . . . , v k , v )of L Riesz . As a consequence, to any sentence σ of L RMV corresponds a formulawith only one free variable σ ( v ) of L Riesz . Proposition 4.3.
Let ( V, u ) be a Riesz space with strong unit and R = Γ R ( V, u ) .If σ is a sentence in the first-order theory of Riesz MV-algebras then R | = σ if and only if V | = σ [ u ] .Proof. By structural induction on terms it follows that t [ a , . . . , a n ] = t [ a , . . . , a n , u ]whenever t ( v , . . . , v n ) is a term of L RMV and a , . . . , a n ∈ R . The rest of theproof is straightforward. Theorem 4.4.
An equation σ in the theory of Riesz MV-algebras holds in allRiesz MV-algebras if and only if it holds in the standard Riesz MV-algebra [0 , .Proof. One implication is obvious. To prove the other one, let R be a RieszMV-algebra such that R = σ . Since R ≃ Γ R ( V, u ) for some Riesz space withstrong unit (
V, u ), we have that Γ R ( V, u ) = σ . Using Proposition 4.3, we inferthat V = σ [ u ] in the theory of Riesz spaces. Since the order relation in anylattice can be expressed equationally, we note that σ ( v ) is a quasi-identity. By[20, Corollary 2.6] a quasi-identity is satisfied by all Riesz spaces if and only ifit is satisfied by R . Hence there exists a real number c ≥ R = σ [ c ].Since R | = σ [0], we get c >
0. If follows that f : R → R defined by f ( x ) x/c is an automorphism of Riesz spaces. We infer that R = σ [1], so [0 , = σ . Remark 4.5.
The variety of Riesz MV-algebras is generated by the standardmodel [0 , in the language of MV-algebras enriched with unary operations x rx for any r ∈ [0 , . This features are reflected by the propositional calculus R L presented in Section 6, whose Lindenbaum-Tarski algebra is a Riesz MV-algebra.Since the class of Riesz MV-algebras is a variety, free structures exist. Thefree Riesz MV-algebra with n free generators is characterized in Corollary 7.8. Let (
V, u ) be a unital Riesz space and define k · k u : V → R by k x k u = inf { α ≥ | | x | ≤ αu } for any x ∈ V .Then k · k u is a seminorm [25, Proposition 1.2.13] and | x | ≤ | y | implies k x k u ≤ k y k u for any x , y ∈ V . Remark 5.1. If ( V, u ) is a unital Riesz space, then k x k u = inf { α ∈ [0 , | x ≤ αu } for any x ∈ [0 , u ] . Definition 5.2. [23] If R is a Riesz MV-algebra then the unit seminorm k · k : R → [0 , is defined by k x k = inf { r ∈ [0 , | x ≤ r } for any x ∈ R . Remark 5.3. If R and R are Riesz MV-algebras and f : R → R is a mor-phism, then x ≤ r in R implies f ( x ) ≤ r in R , so k f ( x ) k ≤ k x k for any x ∈ R . If f is injective then k f ( x ) k = k x k for any x ∈ R .This fact allows us to infer properties of the unit seminorm in Riesz MV-algebras directly from the properties of the unit seminorm in Riesz spaces. Lemma 5.4.
In any Riesz MV-algebra R , the following properties hold for any x , y ∈ R and r ∈ [0 , .(a) k k = 0 , k k = 1 ,(b) k x ⊕ y k ≤ k x k + k y k ,(c) x ≤ y implies k x k ≤ k y k ,(d) k rx k = r k x k ,(e) if ( m − x ≤ x ∗ then k mx k = m k x k for any natural number m ≥ .Proof. By Theorem 4.2 and Remark 5.3 we can safely assume that R is [0 , u ] V for some unital Riesz space ( V, u ). Hence (a)-(d) follow from the properties ofthe unit seminorm in Riesz spaces [13, 25H].(e) Note that ( m − x ≤ x ∗ implies x ⊕ · · · ⊕ x | {z } m = x + · · · + x | {z } m ,where + is the group addition of V , so the desired equality is straightforward. Example 5.5. If X is a compact Hausdorff space, then C ( X ) u = { f : X → [0 , | f continuous } is a Riesz MV-algebra and, for any f ∈ C ( X ) u , we have k f k = inf { r ∈ [0 , | f ( x ) ≤ r ∀ x ∈ X } = sup { f ( x ) | x ∈ X } = k f k ∞ . Recall that an
M-space is a unital Riesz space (
V, u ) that is norm-completewith respect to the unit norm.
Example 5.6. If X is a compact Hausdorff space and C ( X ) = { f : X → R | f continuous } ,then ( C ( X ) , ) is an M-space, where is the constant function ( x ) = 1 forany x ∈ X . The above example is fundamental, as proved by Kakutani’s representationtheorem.
Theorem 5.7. [17] For any M-space ( V, u ) there exists a compact Hausdorffspace X such that ( V, u ) is isomorphic with ( C ( X ) , ) . MU the category of M-spaces with unit-preserving mor-phisms and by KH aus S p the category of compact Hausdorff spaces with con-tinuous maps. Theorem 5.8. [2, 16] The category KH aus S p is dual to the category MU . We characterize in the sequel those Riesz MV-algebras that are, up to isomor-phism, unit intervals in M-spaces. Note that, on any Riesz MV-algebra R , wecan define δ k·k ( x, y ) = k d ( x, y ) k for any x , y ∈ R .By Lemmas 2.2 and 5.4 it follows that δ k·k is a pseudometric on R . Definition 5.9.
We say that a Riesz MV-algebra R is norm-complete if ( R, δ k·k ) is a complete metric space. Theorem 5.10. If ( V, u ) is a unital Riesz space then the following are equiva-lent:(i) ( V, u ) is an M-space,(ii) Γ R ( V, u ) is a norm-complete Riesz MV-algebra.Proof. We denote R = Γ R ( V, u ).(i) ⇒ (ii) By Remark 5.1, k x k = k x k u for any x ∈ [0 , u ]. In consequence, anyCauchy sequence w.r.t k · k from R Γ R ( V, u ) is a Cauchy sequence w.r.t k · k u in( V, u ) and we use the fact that (
V, u ) is norm-complete.(ii) ⇒ (i) Let ( v n ) n be a Cauchy sequence in V w.r.t. k · k u such that v n ≥ n . It follows that it is bounded, i.e. there is y ∈ V and k v n k u ≤ k y k u for any n . We get v n ≤ k v n k u u ≤ k y k u u for any n , so there exists a naturalnumber k such that v n ≤ ku for any n . By Lemma 2.3, v n = v n + · · · + v n k , where v n i = (( v n − ( i − u ) ∨ ∧ u ∈ [0 , u ]for any i ∈ { , . . . , k } . Since v n i ∈ [0 , u ] we get k v n i k u = k v n i k for any n and i ∈ { , . . . , k } .One can easily see that ( v n i ) n is a Cauchy sequence in R = [0 , u ] for any i ∈{ , . . . , k } . Since R is norm-complete it follows that, for any i ∈ { , . . . , k } thereis w i ∈ R such that lim n d ( v n i , w i ) = lim n k v n i − w i k u = 0. If w = w + · · · + w k then k v n − w k u ≤ k v n − w k u + · · · + k v n k − w k k u for any n , so lim n k v n − w k = 0 and ( v n ) n is convergent w.r.t. k · k u in V .Recall that v = ( v ∨ − (( − v ) ∨
0) for any v ∈ V , so the convergenceof arbitrary Cauchy sequences reduces to the convergence of positive Cauchysequences.Denote by URMV the category of norm-complete Riesz MV-algebras, whichis a full subcategory of
RMV . By Remark 5.3, the norm-preserving morphismscoincide with the monomorphisms of
URMV .Using Theorem 5.10, the functor Γ R yields the following categorical equiva-lence. 14 orollary 5.11. The categories
URMV and MU are equivalent. Corollary 5.12.
The categories
URMV and KH aus S p are dually equivalent. Remark 5.13.
Following [16, Chapter IV] and [2], the functors establishingthe above equivalences are defined on objects as follows: R M ax ( R ) and X C ( X ) u for any norm-complete Riesz space R and compact Hausdorff space X , where M ax ( R ) is the set of all maximal ideals of R . In the sequel, we connect our result with the Gelfand-Naimark duality forC ∗ -algebras [14]. Recall that MV-algebras are related with AF C ∗ -algebras in[27], but in this case the K-theory is used.Denote C ∗ the category whose objects are commutative unital C ∗ -algebrasand whose morphisms are unital C ∗ -algebra morphisms. Theorem 5.14. [18, Chapter 1.1] The categories C ∗ and KH aus S p are duallyequivalent. As a corollary we infer immediately that the categories of commutative unitalC ∗ -algebras and norm-complete Riesz MV-algebras are equivalent. Corollary 5.15.
The categories C ∗ and URMV are equivalent. R L We denote by L ∞ the ∞ -valued propositional Lukasiewicz logic. Recall that L ∞ has ¬ (unary) and → (binary) as primitive connectives and, for any ϕ and ψ we have: ϕ ∨ ψ := ( ϕ → ψ ) → ψ , ϕ ∧ ψ := ¬ ( ¬ ϕ ∨ ¬ ψ ), ϕ ↔ ψ := ( ϕ → ψ ) ∧ ( ψ → ϕ ).The language of R L contains the language of L ∞ and a family of unary connec-tives {∇ r | r ∈ [0 , } . We denote by F orm ( R L ) the set of formulas, which aredefined inductively as usual. Definition 6.1. An axiom of R L is any formula that is an axiom of L ∞ andany formula that has one of the following forms, where ϕ , ψ , χ ∈ F orm ( R L ) and r , q ∈ [0 , :(RL1) ∇ r ( ϕ → ψ ) ↔ ( ∇ r ϕ → ∇ r ψ ) ;(RL2) ∇ ( r ⊙ q ∗ ) ϕ ↔ ( ∇ q ϕ → ∇ r ϕ ) ;(RL3) ∇ r ∇ q ϕ ↔ ∇ ( rq ) ϕ ;(RL4) ∇ ϕ ↔ ϕ ,The deduction rule of R L is modus ponens and provability is defined as usual. emark 6.2. { ϕ } ⊢ ∇ r ϕ is a derived deduction rule for any r ∈ [0 , . We recall the usual construction of the Lindenbaum-Tarski algebra. Theequivalence relation ≡ is defined on F orm ( R L ) as follows: ϕ ≡ ψ iff ⊢ ϕ → ψ and ⊢ ψ → ϕ .We denote by [ ϕ ] the equivalence class of a formula ϕ and we define on F orm ( R L ) / ≡ the following operations:[ ϕ ] ∗ = [ ¬ ϕ ],[ ϕ ] ⊕ [ ψ ] = [ ¬ ϕ → ψ ], [ ϕ ] ⊙ [ ψ ] = [ ¬ ( ϕ → ¬ ψ )],0 = [ ¬ ( v → v )], 1 = 0 ∗ = [ v → v ].In order to define the scalar multiplication we introduce new connectives:∆ r ϕ := ¬ ( ∇ r ¬ ϕ )and we set r [ ϕ ] = [∆ r ϕ ] for any r ∈ [0 ,
1] and ϕ formula of R L . Proposition 6.3.
The Lindenbaum-Tarski algebra RL = ( F orm ( R L ) / ≡ , · , ⊕ , ∗ , is a Riesz MV-algebra.Proof. The axioms (RL1)-(RL4) are logical expressions of the duals of (RMV1)-(RMV4). We prove in detail that RL satisfies (RMV1). If ϕ and ψ are twoformulas and r ∈ [0 ,
1] then, by (RL1), we get[ ∇ r ( ¬ ϕ → ¬ ψ )] = [ ∇ r ¬ ϕ → ∇ r ¬ ψ ].It follows that: [ ∇ r ¬ ( ¬ ϕ ⊙ ψ )] = [ ¬ ( ∇ r ¬ ϕ ⊙ ¬∇ r ¬ ψ )][ ¬∇ r ¬ ( ¬ ϕ ⊙ ψ )] = [ ∇ r ¬ ϕ ⊙ ¬∇ r ¬ ψ ][∆ r ( ¬ ϕ ⊙ ψ )] = [ ¬ ∆ r ϕ ⊙ ∇ r ψ ] r ([ ϕ ] ∗ ⊙ [ ψ ]) = ( r [ ϕ ]) ∗ ⊙ ( r [ ψ ]),so (RMV1) holds in RL .Let R be an Riesz MV-algebra. An evaluation is a function e : F orm ( R L ) → R which satisfies the following conditions for any ϕ , ψ ∈ F orm ( R L ) and r ∈ [0 , e ( ϕ → ψ ) = e ( ϕ ) ∗ ⊕ e ( ψ ),(e2) e ( ¬ ϕ ) = e ( ϕ ) ∗ ,(e3) e ( ∇ r ϕ ) = ( re ( ϕ ) ∗ ) ∗ .As a consequence of Theorem 4.4, the propositional calculus R L is completewith respect to [0 , Theorem 6.4.
For a formula ϕ of R L the following are equivalent:(i) ϕ is provable in R L ,(ii) e ( ϕ ) = 1 for any Riesz MV-algebra R and for any evaluation e : F orm ( R L ) → R ,(iii) e ( ϕ ) = 1 for any evaluation e : F orm ( R L ) → [0 , . emark 6.5. The system R L is a conservative extension of L ∞ , i.e. a formula ϕ of L ∞ is a theorem of L ∞ if and only if it is a theorem of R L . Since anyproof in L ∞ is also a proof in R L , one implication is obvious. To prove the otherone, assume that ϕ is a formula of L ∞ which is not a theorem of L ∞ . Hencethere exists an evaluation e ′ : F orm ( L ∞ ) → [0 , such that e ′ ( ϕ ) = 1 . Let e : F orm ( R L ) → [0 , the unique evaluation in R L such that e ( v ) = e ′ ( v ) forany propositional variable v . By structural induction on formulas one can provethat e ( ψ ) = e ′ ( ψ ) for any ψ ∈ F orm ( L ∞ ) . It follows that e ( ϕ ) = e ′ ( ϕ ) = 1 , so ϕ is not a theorem of R L . Remark 6.6.
A formula ϕ with variables from { v , . . . , v n } uniquely defines a term function : e ϕ : [0 , n → [0 , , e ϕ ( x , . . . , x n ) = e ( ϕ ) ,where e is an evaluation such that e ( v i ) = x i for any i ∈ { , . . . , n } . By Theorem6.4 it follows that [ ϕ ] = [ ψ ] if and only if e ϕ = e ψ . In the following, we characterize the class of functions that can be defined byformulas in R L . Definition 7.1.
Let n > be a natural number. A piecewise linear function isa function f : R n → R for which there exists a finite number of affine functions q , . . . , q k : R n → R and for any ( x , . . . , x n ) ∈ R n there is i ∈ { , . . . , k } such that f ( x , . . . , x n ) = q i ( x , . . . , x n ) . We say that q , . . . , q k are the components of f .We denote by P L n the set of all continuous functions f : [0 , n → [0 , thatare piecewise linear.For the rest of the paper, all piecewise linear functions are continuous. Theorem 7.2. If ϕ is a formula of R L with propositional variables from { v , · · · , v n } then e ϕ ∈ P L n .Proof. We prove the result by structural induction on formulas.If ϕ is v i for some i ∈ { , . . . , n } then e ϕ = π i (the i -th projection).If ϕ is ¬ ψ and q , . . . , q s are the components of e ψ , then 1 − q , . . . , 1 − q s are the components of e ϕ .Assume ϕ is ψ → χ . If q , . . . , q m are the components of e ψ and p , . . . , p k arethe components of e χ , then e ϕ is defined by { } ∪ { s ij } i,j , where s ij = 1 − q i + p j for any i ∈ { , . . . , s } and j ∈ { , . . . , r } .If ϕ is ∆ r ψ for some r ∈ [0 ,
1] and q , . . . , q s are the components of e ψ , then1 − r + rq , . . . , 1 − r + rq s are the components of e ϕ .17 emark 7.3. The continuous piecewise linear functions f : [0 , n → [0 , with integer coefficients are called McNaughton functions andthey are in one-one correspondence with the formulas of Lukasiewicz logic by Mc-Naughton theorem [24]. The continuous piecewise linear functions with rationalcoefficients correspond to formulas of Rational Lukasiewicz logic, a propositionalcalculus developed in [15] that has divisible MV-algebras as models. In Theorem7.7 we prove that any continuous piecewise linear function with real coefficients f : [0 , n → [0 , is the term function of a formula from R L . For now on we define ̺ : R → [0 ,
1] by ̺ ( x ) = ( x ∨ ∧ x ∈ R . Lemma 7.4.
For any x , y ∈ R the following hold:(a) ( x ∨
0) + ( y ∨ ≥ ( x + y ) ∨ ,(b) x ≥ iff ̺ ( − x ) = 0 ,(c) ̺ ( x ) = ̺ ( x ∨ .Proof. (a) ( x ∨
0) + ( y ∨
0) = ( x + y ) ∨ x ∨ y ∨ ≥ ( x + y ) ∨ ̺ ( − x ) = 0 iff (( − x ) ∨ ∧ − x ) ∨ − x ≤ x ≥ ̺ ( x ∨
0) = ( x ∨ ∨ ∧ x ∨ ∧ ̺ ( x ).In the following we generalize some results from [7, Lemma 3.1.9]. Lemma 7.5. If g : [0 , n → R and h : [0 , n → [0 , then the followingproperties hold.(a) ̺ ◦ ( g + h ) = (( ̺ ◦ g ) ⊕ h ) ⊙ ( ̺ ◦ ( g + 1)) .(b) ̺ ◦ (1 − g ) = 1 − ( ̺ ◦ g ) Proof.
Let x = ( x , · · · , x n ) be an element from [0 , n .(a) If g ( x ) > g ( x ) + 1 > g ( x ) + h ( x ) >
1. It follows that ̺ ( g ( x )) = ̺ (( g + 1)( x )) = ̺ (( g + h )( x )) = 1,so the intended identity is obvious.If g ( x ) ∈ [0 ,
1] then ̺ ( g ( x )) = g ( x ) and ̺ (( g + 1)( x )) = 1 for any x ∈ [0 , n ,so ̺ (( g + h )( x )) = h ( x ) ⊕ g ( x )= h ( x ) ⊕ ̺ ( g ( x )) = (( ̺ ◦ g ) ⊕ h )( x )= ((( ̺ ◦ g ) ⊕ h ) ⊙ x )= ((( ̺ ◦ g ) ⊕ h ) ⊙ ( ̺ ◦ ( g + 1)))( x ) . Assume that g ( x ) <
0, so ̺ ( g ( x )) = 0. We have to prove that ̺ (( g + h )( x )) = h ( x ) ⊙ ̺ (( g + 1)( x )).If g ( x ) ≤ − ̺ ( g ( x )) = ̺ (( g + 1)( x )) = 0 and g ( x )+ h ( x ) ≤ − h ( x ) ≤
0, so ̺ (( g + h )( x )) = 0 = h ( x ) ⊙ h ⊙ ( ̺ ◦ ( g + 1)))( x ).18f g ( x ) ∈ ( − ,
0) then ̺ (( g + 1)( x )) = ( g + 1)( x ) = g ( x ) + 1, so we get( ̺ ◦ ( g + h ))( x ) = 0 ∨ (1 ∧ ( h ( x ) + g ( x )))= 0 ∨ ( h ( x ) + g ( x ))= 0 ∨ ( h ( x ) + g ( x ) + 1 − h ( x ) ⊙ ( g ( x ) + 1)= h ( x ) ⊙ ̺ (( g + 1)( x ))= ( h ⊙ ( ̺ ◦ ( g + 1)))( x ) . (b) If g ( x ) < ̺ ( g ( x )) = 0 and ̺ ((1 − g )( x )) = 1 = 1 − − ̺ ( g ( x )) . If g ( x ) ∈ [0 ,
1] then (1 − g )( x ) ∈ [0 , ̺ ((1 − g )( x )) = (1 − g )( x ) = 1 − g ( x ) = 1 − ̺ ( g ( x )) . If g ( x ) > ̺ ( g ( x )) = 1 and ̺ ((1 − g )( x )) = 0 = 1 − − ̺ ( g ( x )). Proposition 7.6.
For any affine function f : [0 , n → R there exists a formula ϕ of R L such that ̺ ◦ f = e ϕ .Proof. Let f : [0 , n → R be an affine function, i.e. there are c , . . . , c n ∈ R such that f ( x , . . . , x n ) = c n x n + · · · + c x + c for any ( x , . . . , x n ) ∈ [0 , n . Note that for any c ∈ R there is a natural number m such that c = r + · · · + r m where r , . . . , r m ∈ [ − , f ( x , . . . , x n ) = r m y m + · · · r p +1 y p +1 + r p + · · · + r where m ≥ ≤ p ≤ m are natural numbers, r j ∈ [ − , \ { } for any j ∈ { , . . . , m } and y j ∈ { x , . . . , x n } for any j ∈ { p + 1 , · · · , m } .We prove the theorem by induction on m ≥
1. Let us denote x = ( x , · · · , x n )an element from [0 , n Initial step m = 1. We have f ( x ) = r for any x ∈ [0 , n or f ( x ) = rx i forany x ∈ [0 , n , where r ∈ [ − , \ { } and i ∈ { , . . . , n } . If r ∈ [ − ,
0) then ̺ ◦ f = 0 so ̺ ◦ f = e ϕ for ϕ = v ⊙ ¬ v . If r ∈ (0 ,
1] then f = ̺ ◦ f . It followsthat f = e ϕ where ϕ = ∇ r ( v → v ) if f ( x ) = r for any x ∈ [0 , n and ϕ = ∇ r v i if f ( x ) = rx i for any x ∈ [0 , n . Induction step . We take f = g + h where ̺ ◦ g = e ϕ for some formula ϕ and thereare r ∈ [ − , \ { } and i ∈ { , . . . , n } such that h ( x ) = r for any x ∈ [0 , n ,or h ( x ) = rx i for any x ∈ [0 , n . We consider two cases. Case 1. If r ∈ (0 ,
1] then h : [0 , n → [0 ,
1] so19 ◦ f = (( ̺ ◦ g ) ⊕ h ) ⊙ ( ̺ ◦ (1 + g ))by Lemma 7.5 (a). Following the initial step, there is a formula ψ such that h = e ψ . Note that 1 + g = 1 − ( − g ) and, since the induction hypothesis holds for( − g ), there is a formula χ such that ̺ ◦ ( − g ) = e χ . In consequence, by Lemma7.5 (b), ̺ ◦ (1 + g ) = 1 − e χ = f ¬ χ . We get ̺ ◦ f = e θ where θ = ( ϕ ⊕ ψ ) ⊙ ¬ χ . Case 2. If r ∈ [ − , g + h = ( g −
1) + (1 + h ) and 1 + h : [0 , n → [0 , ̺ ◦ f = (( ̺ ◦ ( g − ⊕ (1 + h )) ⊙ ( ̺ ◦ g ).Following the initial step, there is a formula ψ such that − h = e ψ , so 1 + h = 1 − ( − h ) = f ¬ ψ .In the sequel we have to find a formula χ that corresponds to ̺ ◦ ( g − g ( x ) = r m y m + · · · r p +1 y p +1 + r p + · · · + r with r j ∈ [ − , \ { } for any j ∈ { , · · · , m } and y j ∈ { x , . . . , x n } for any j ∈ { p + 1 , · · · , m } . Case 2.1. If r j ≤ j ∈ { , · · · , m } then g − ≤
0, so ̺ ◦ ( g −
1) = 0 = e χ with χ = v ⊙ ¬ v . Case 2.2.
If there is j ∈ { , · · · , p } such that r j >
0, then it follows that( g − x ) = r m y m + · · · r p +1 y p +1 + r p + · · · + ( r j −
1) + · · · + r and r j − ∈ [ − , g −
1. In consequence,there exists a formula χ such that ̺ ◦ ( g −
1) = e χ . Case 2.3.
If there is j ∈ { p + 1 , · · · , m } such that r j >
0, then we set h ( x ) = r j y j and g ( x ) = g ( x ) − r j y j − . It follows that g − g + h such that g satisfies the induction hypothesisand h : [0 , n → [0 , Case 1 , so there exists aformula χ such that ̺ ◦ ( g −
1) = e χ .Summing up, we get ̺ ◦ ( g + h ) = e θ with θ = (( χ ⊕ ¬ ψ ) ⊙ ϕ ). Theorem 7.7.
For any f : [0 , n → [0 , from P L n there is a formula ϕ of R L such that f = e ϕ .Proof. Let f : [0 , n → [0 ,
1] be in
P L n . Using the Max-Min representationfrom [32], there are finite sets I and J such that f = _ i ∈ I ^ j ∈ J f ij , where f ij : [0 , n → R are affine functions. We note that20 = ̺ ◦ f = _ i ∈ I ^ j ∈ J ( ̺ ◦ f ij ) . By Proposition 7.6, for any i ∈ I and j ∈ J there is a formula ϕ ij such that ̺ ◦ f ij = f ϕ ij . In consequence, if we set ϕ = W i ∈ I V j ∈ J ϕ ij then f = e ϕ .For any n ≥
1, the set
P L n is a Riesz MV-algebra with the operationsdefined componentwise. If RL n is the Lindenbaum-Tarski algebra of R L definedon formulas with variables from { v , . . . , v n } , then RL n is the free Riesz MV-algebra with n free generators by standard results in universal algebra [4]. Sincethe function [ ϕ ] e ϕ is obviously an isomorphism between RL n and P L n thefollowing corollary is straightforward. Corollary 7.8.
P L n is the free Riesz MV-algebra with n free generators. We recall in the beginning de Finetti’s coherence criterion for boolean events.If S = { ϕ , . . . , ϕ k } is a set of classical events then a book is a set { ( ϕ i , r i ) | i ∈ { , . . . , k }} ,where r i ∈ [0 ,
1] is a ”betting odd” assigned by a bookmaker for ϕ i for any i ∈ { , . . . , k } . The book is coherent if there is no system of bets { c , · · · , c k } which causes the bookmaker a sure loss. This means that for any real numbers { c , · · · , c k } there exists an evaluation e : S → { , } such that P ki =1 c i ( r i − e ( ϕ i )) ≥
0. De Finetti’s coherence criterion [8] states that the book { ( ϕ i , r i ) | i ∈{ , . . . , k }} is coherent if there is a boolean probability µ defined on the algebraof events generated by S such that µ ( ϕ i ) = r i for any i ∈ { , . . . , k } . Whenthe underlying logic is Lukasiewicz logic [30], the events belongs to an MV-algebra and they are evaluated in [0 , R L . In order to accomplish this task, we initiate thestudy of linear combinations of formulas in R L .The next result characterizes the states defined on Riesz MV-algebras. Lemma 8.1. If R is a Riesz MV-algebra then any state s : U ( R ) → [0 , isalso homogeneous:(s3) s ( r · x ) = rs ( x ) for any r ∈ [0 , , x ∈ R .Proof. We can safely assume that R = [0 , u ] V for some unital Riesz space ( V, u ).By [28, Theorem 2.4], there is a state s ′ : V → R such that s ′ ( x ) = s ( x ) for any x ∈ [0 , u ]. 21f r ∈ [0 , ∩ Q then r = mn and rx = y in [0 , u ] implies mx = ny in V .It follows that s ′ ( mx ) = s ′ ( ny ), so ms ( x ) = ns ( y ) and we get s ( rx ) = s ( y ) = rs ( x ).If r ∈ (0 ,
1) there are rational sequences ( r n ) n and ( q n ) n such that r n ↑ r and q n ↓ r . Hence r n s ( x ) = s ( r n x ) ≤ s ( rx ) ≤ s ( q n x ) = q n s ( x )for any n and x ∈ [0 , Definition 8.2. If R is a Riesz MV-algebra, a state on R is a function s : R → [0 , which satisfies the conditions (s1), (s2) and (s3) (additivity, normalizationand homogeneity).The previous lemma asserts that the states of a Riesz MV-algebra R coincidewith the states of its MV-algebra reduct U( R ) . The following definition generalizes de Finetti’s notion of coherence and pro-vides an algebraic approach within Lukasiewicz logic [19].
Definition 8.3. [19] If A is an MV-algebra and x , . . . , x k are in A then a map β : { x , . . . , x n } → [0 , is coherent if for any c , · · · , c k ∈ R there exists amorphism of MV-algebras e : A → [0 , such that P ki =1 c i ( β ( x i ) − e ( x i )) ≥ . Theorem 8.4. [19, Theorem 3.2] If A is an MV-algebra, x , . . . , x k ∈ A and β : { x , . . . , x n } → [0 , then the following are equivalent:(i) the map β is coherent,(ii) there exists a state s : A → [0 , such that s ( x i ) = β ( x i ) for any i ∈ { , . . . , k } ,(iii) there exists e , . . . , e m : A → [0 , morphisms of MV-algebras such that m ≤ k + 1 and β is the restriction of a convex combination of { e , . . . , e m } . By Remark 3.12, any morphsim of Riesz MV-algebras is just a morphismbetween the MV-algebra reducts of its domain and codomain. Therefore, thenotion of coherent map remains unchanged on Riesz MV-algebras and an ana-logue of Theorem 8.4 can be proved for Riesz MV-algebras as well.
Corollary 8.5. If R is a Riesz MV-algebra, x , . . . , x k ∈ R and β : { x , . . . , x k } → [0 , then the following are equivalent:(i) the map β is coherent,(ii) there exists a state s : R → [0 , such that s ( x i ) = β ( x i ) for any i ∈ { , . . . , k } , iii) there exists e , . . . , e m : R → [0 , morphisms such that m ≤ k + 1 and β is the restriction of a convex combination of { e , . . . , e m } .Proof. (i) ⇔ (iii) and (ii) ⇒ (i) follow by Theorem 8.4 applied to the MV-algebrareduct of R and by Remark 3.12.(iii) ⇒ (ii) There are α , . . . , α m ∈ [0 ,
1] such that α + · · · + α m = 1 and β ( x i ) = α e ( x i ) + · · · + α m e m ( x i )for any i ∈ { , . . . , k } . We set s = α e + · · · + α m e m satisfies (s1), (s2) and(s3), so s : R → [0 ,
1] is the required state.The above result is an algebraic version of de Finetti’s coherence criterion.In the sequel we provide a logical approach within R L .We firstly recall de Finetti’s coherence criterion for Lukasiewicz logic L ∞ . Theorem 8.6. [30] If ϕ , . . . , ϕ k are formulas of L ∞ with variables v , . . . , v n and r , . . . , r k ∈ [0 , then the following are equivalent:(i) the book { ( ϕ i , r i ) | i ∈ { , . . . , k }} is coherent,(ii) there exists a state s : L n → [0 , such that s ([ ϕ i ]) = r i for any i ∈ { , . . . , k } ,where L n is the Lindenbaum-Tarski algebra of the formulas in n variables and [ ϕ ] is the equivalence class of the formula ϕ in L n . When we consider R L instead of L ∞ , we get the following. Definition 8.7. If ϕ , . . . , ϕ k are formulas of R L and r , . . . , r k ∈ [0 , then the book { ( ϕ i , r i ) | i ∈ { , . . . , k }} is coherent if for any c , · · · , c k ∈ R there exists an evaluation e : F orm ( R L ) → [0 , such that P ki =1 c i ( r i − e ( ϕ i )) ≥ . In order to characterize coherence in logical terms within R L we introducethe notion of quasi-linear combination of piecewise linear functions.In the sequel we assume that n is a natural number and all the formulasfrom R L have variables from the set { v , . . . , v n } . As in the previous section,we use the function ̺ : R → [0 , ̺ ( x ) = ( x ∨ ∧ x ∈ R . Remark 8.8. If f , . . . , f k : [0 , n → R are continuous piecewise linear func-tions and c , . . . , c k ∈ R then P ki =1 c i f i is also a continuous piecewise linear function, so ̺ ◦ ( P ki =1 c i f i ) isin P L n . By Theorem 7.7, there exists a formula ϕ of R L such that e ϕ = ̺ ◦ ( P ki =1 c i f i ) . Therefore, we introduce the following definition. Definition 8.9.
Let f , . . . , f k : [0 , n → R be continuous piecewise linearfunctions. We say that ϕ is a quasi-linear combination of f , . . . , f k whenever ϕ = ̺ ◦ ( P ki =1 c i f i ) for some c , . . . , c k ∈ R . We define qspan ( f , . . . , f k ) as the subset of F orm ( R L ) that contains all the quasi-linear combinations of f , . . . , f k . If ϕ , . . . , ϕ k areformulas of R L then qspan ( f ϕ , . . . , f ϕ k ) will be denoted by qspan ( ϕ , . . . , ϕ k ) . If ϕ ∈ qspan ( ϕ , . . . , ϕ k ) then we say that ϕ is a quasi-linear combination of theformulas ϕ , . . . , ϕ k in R L . Lemma 8.10. If ϕ , ϕ , . . . , ϕ k are formulas in R L such that e ϕ = ̺ ◦ ( P ki =1 c i e ϕ i ) for some c , . . . , c k ∈ R and e : F orm ( R L ) → [0 , is an evaluation, then e ( ϕ ) = ̺ ( P ki =1 c i e ( ϕ i )) .Proof. We have e ( ϕ ) = e ϕ ( e ( v ) , · · · , e ( v n )) = ̺ ( P ki =1 c i e ϕ i ( e ( v ) , · · · , e ( v n ))) = ̺ ( P ki =1 c i e ( ϕ i )). Lemma 8.11. If ϕ , . . . , ϕ k are formulas of R L and r , · · · , r k ∈ [0 , then ∇ r ϕ ⊕ · · · ⊕ ∇ r k ϕ k ∈ qspan ( ϕ , . . . , ϕ k ) .Proof. Under the above hypothesis, we get ̺ ( P ki =1 r i e ( ϕ i )) = r e ( ϕ ) ⊕ · · · ⊕ r k e ( ϕ k ) = e ( ∇ r ϕ ) ⊕ · · · ⊕ e ( ∇ r k ϕ k )for any evaluation e : F orm ( R L ) → [0 , Definition 8.12.
We say that a formula ϕ of R L is invalid if there exists anevaluation e : F orm ( R L ) → [0 , such that e ( ϕ ) = 0 Theorem 8.13.
Let ϕ , . . . , ϕ k are formulas of R L and r , · · · , r k ∈ [0 , .The following are equivalent:(i) the book { ( ϕ i , r i ) | i ∈ { , . . . , k }} is coherent,(ii) there exists a state s : RL n → [0 , such that s ([ ϕ i ]) = r i for any i ∈{ , . . . , k } ,(iii) qspan ( f ϕ − r , . . . , f ϕ k − r k ) is a set of invalid formulas of R L .Proof. (i) ⇔ (ii) Apply Corollary 8.5 to β ([ ϕ i ]) = r i for any i ∈ { , . . . , k } .(i) ⇔ (iii) The following facts are equivalent:(1) qspan ( f ϕ − r , . . . , f ϕ k − r k ) is a set of invalid formulas,(2) for any Ψ ∈ qspan ( f ϕ − r , . . . , f ϕ k − r k ) there exists an evaluation e : F orm ( R L ) → [0 ,
1] such that e (Ψ) = 0,(3) for any c , . . . , c k ∈ R , if e Ψ = ̺ ◦ ( P ki =1 c i ( e ϕ i − r i )) then there exists anevaluation e : F orm ( R L ) → [0 ,
1] such that e (Ψ) = 0,(4) for any c , . . . , c k ∈ R , there exists an evaluation e : F orm ( R L ) → [0 , ̺ ◦ ( P ki =1 c i ( e ( ϕ i ) − r i ) = 0,(5) for any c , . . . , c k ∈ R , there exists an evaluation e : F orm ( R L ) → [0 , ki =1 c i ( r i − e ( ϕ i )) ≥ ⇔ (4) and Lemma 7.4 (b) is used for(4) ⇔ (5).If r ∈ [0 ,
1] and ϕ a formula then the piecewise linear function r − e ϕ :[0 , n → R may have negative values, therefore it may not correspond to aformula of R L . The next result provides a necessary condition for a book to becoherent using quasi-linear combinations of formulas. We also prove a sufficientcondition in Corollary 8.16, but using different formulas.We set r = ∆ r ( ϕ → ϕ ) and ϕ ⊖ ψ = ϕ ⊙ ¬ ψ whenever r ∈ [0 ,
1] and ϕ , ψ ∈ F orm ( R L ). Note that χ = ϕ ⊖ ψ implies e χ = 0 ∨ ( e ϕ − e ψ ). Proposition 8.14.
Assume ϕ , . . . , ϕ k are formulas of R L and r , · · · , r k ∈ [0 , such that the book { ( ϕ i , r i ) | i ∈ { , . . . , k }} is coherent. Hence qspan (( r ⊖ ϕ ) , . . . , ( r k ⊖ ϕ k )) is a set of invalid formulas.Proof. Let χ ∈ qspan (( r ⊖ ϕ ) , . . . , ( r k ⊖ ϕ k )) and c , · · · , c k ∈ R such that e χ = ̺ ◦ ( P ki =1 c i (0 ∨ ( r i − e ϕ i ))).Since the book { ( ϕ i , r i ) | i ∈ { , . . . , k }} is coherent there is an evaluation e such that P ki =1 ( − c i )( r i − e ( ϕ i )) ≥
0, which implies that P ki =1 ( − c i )(( r i − e ( ϕ i )) ∨ ≥ P ki =1 ( − c i ) e ( r i ⊖ ϕ i ) ≥
0. By Lemma 7.4 (b), ̺ ( P ki =1 c i e ( r i ⊖ ϕ i )) =0. Using Lemma 8.10 it follows that e ( χ ) = 0, so χ is an invalid formula.If ϕ ∈ F orm ( R L ), r ∈ [0 ,
1] and c ∈ R we denote ψ ( ϕ, r, c ) = (cid:26) ϕ ⊖ r , if c ≥ r ⊖ ϕ, if c < . Proposition 8.15.
Assume ϕ , . . . , ϕ k ∈ F orm ( R L ) and r , · · · , r k ∈ [0 , such that for any c , . . . , c k ∈ R the formula Φ of R L is invalid whenever e Φ = ̺ ◦ ( P ki =1 | c i | e ψ i ) , where | c i | is the module of c i and ψ i = ψ ( ϕ i , r i , c i ) for any i ∈ { , . . . , k } . Then the book { ( ϕ i , r i ) | i ∈ { , . . . , k }} is coherent.Proof. If c , · · · , c k ∈ R and Φ in F orm ( R L ) is an invalid formula such that e Φ = ̺ ◦ ( P ki =1 | c i | e ψ i ), then there exists an evaluation e such that e (Φ) = 0 and,by Lemma 8.10, we get ̺ ( P ki =1 | c i | e ( ψ i )) = 0, so ̺ ( P c i ≥ c i (( e ( ϕ i ) − r i ) ∨ P c i < ( − c i )(( r i − e ( ϕ i )) ∨
0) ) = 025y Lemma 7.4(a), ̺ (( P c i ≥ c i ( e ( ϕ i ) − r i ) + P c i < ( − c i )( r i − e ( ϕ i ))) ∨
0) = 0and, by Lemma 7.4(c), ̺ ( P c i ≥ c i ( e ( ϕ i ) − r i ) + P c i < ( − c i )( r i − e ( ϕ i ))) = 0.Using Lemma 7.4 (b), we get − ( P c i ≥ c i ( e ( ϕ i ) − r i ) + P c i < ( − c i )( r i − e ( ϕ i ))) ≥ P ki =1 c i ( r i − e ( ϕ i )) ≥ Corollary 8.16.
Assume ϕ , . . . , ϕ k ∈ F orm ( R L ) and r , · · · , r k ∈ [0 , such that qspan ( α , . . . , α k , β , . . . , β k ) is a set of invalid formulas, where α i = r i ⊖ ϕ i and β i = ϕ i ⊖ r i for any i ∈ { , . . . , k } .Then the book { ( ϕ i , r i ) | i ∈ { , . . . , k }} is coherent.Proof. For any c , · · · , c k ∈ R , if Φ is a formula of R L such that e Φ = ̺ ◦ P ki =1 | c i | ψ i as in Proposition 8.15, then Φ ∈ qspan ( α , . . . , α k , β , . . . , β k ). Inconsequence, we can apply Proposition 8.15. Remark 8.17.
We initiate the theory of quasi-linear combinations in R L andwe relate it with de Finetti’s notion of coherence, which can be expressed byan invalidity condition. The linear combinations of formulas from Lukasiewiczlogic were approached in [5] and [1], as representations for a particular class ofneural networks. The composition between the function ̺ and a linear combina-tion of formulas Lukasiewicz logic can be naturally represented by a formula inour logic R L and, therefore, the theory of linear combinations can be approachedwithin a simple defined logical system. Note that R L is a conservative exten-sion of Lukasiewicz logic. It has standard completeness theorem with respect to [0 , and it is supported by the algebraic theory of Riesz MV-algebras which arecategorically equivalent with unital Riesz spaces. Hence, our hope for the futureis that the system R L is enough expressive for representing classes of neuralnetworks in a pure logical frame, having in mind the role of classical logic in thesynthesis and analysis of boolean circuits. Acknowledgement.
I. Leu¸stean was supported by the strategic grant POSDRU/89/1.5/S/58852,cofinanced by ESF within SOP HRD 2007-2013. Part of the research has beencarried out while visiting the University of Salerno. The investigations fromSection 5 were initiated in 2009, when V. Marra (University of Milan) pointedout Kakutani’s theorem and its relevance to the theory of Riesz MV-algebras.26 eferenceseferences