The class of weakly algebrizable logics is defined as the class of logics having
monotonic and injective Leibniz operator. We show that \monotonicity" can-
not be discarded on this definition, by presenting an example of a system with
injective and non monotonic Leibniz operator.
We also show that the non injectivity of the non protoalgebraic inf-sup
fragment of the Classic Propositional Calculus, CPC_{inf,sup}, holds only from the fact that the empty set is a CPC_{inf,sup}-filter.FCT via UIM

Descalço, L.

Martins, Manuel A.

Bulletin of the Section of Logic

Repositório Institucional da Universidade de Aveiro

Bulletin of the Section of LogicVolume 34/4 (2005), pp. 203–211L. Descalc¸oManuel A. Martins∗1ON THE INJECTIVITY OF THE LEIBNIZ OPERATORAbstractThe class of weakly algebrizable logics is defined as the class of logics havingmonotonic and injective Leibniz operator. We show that “monotonicity” can-not be discarded on this definition, by presenting an example of a system withinjective and non monotonic Leibniz operator.We also show that the non injectivity of the non protoalgebraic inf-supfragment of the Classic Propositional Calculus, CPC∧∨, holds only from the factthat the empty set is a CPC∧∨-filter.1. IntroductionAn important paradigm in algebraic logic is the Lindenbaum-Tarski processfor building Boolean algebras from the classical propositional logic. Amain aim on abstract algebraic logic is the study of the generalizationof this process for other deductive systems. This study has lead to theestablishment of an algebraic hierarchy defined by properties of the Leibnizoperator. The relevant classes of this hirarchy for the present note are theclass of protoalgebraic logics and the class of weakly algebraizable logics.Relevant references about this subject are the papers by Blok andPigozzi [2] and [3]; Czelakowski’s book, Protoalgebraic Logics [6] and thepaper [5]; Hermann’s papers [14] and [15], the book A General Semantics∗The authors wish to thank Don Pigozzi and Isabel Ferreirim for some fruitful dis-cussions concerning this work.1Research partially supported by project PRODEP III.204 L. Descalc¸o and Manuel A. Martinsfor Sentential Logic by Font and Jansana [10] and the survey [11] by Font,Jansana and Pigozzi.We follow the notation by Blok and Pigozzi in [1]. Some basic defini-tions and proofs of well known results will be omitted. However, referencesare provided.We begin by stating some definitions and results about algebraic logicwe require. Then we study the {∨,∧,¬,>,⊥}-fragment of the IntuitionisticPropositional Calculus, IPC∗, and we show that its Leibniz operator isinjective but not monotonic. Therefore, the class of weakly algebraizablelogics may not be simply defined as the class of logics with injective Leibnizoperator. Finally, we show that the non injectivity of the inf-sup fragmentCPC∧∨ of the Classic Propositional Calculus follows from the fact that theempty set is a CPC∧∨-filter.2. PreliminariesLet Λ = {ωi : i ∈ I} be a (countable algebraic similarity) language, a setof finitary connectives with associated natural numbers called their arity,and let V = {x1, x2, x3, ...} be a countable infinite set of (propositional)variables. An algebra A of type Λ consists of a set A and, for each elementw ∈ Λ, a function from An to A where n is the arity of w.We denote by Fm(Λ) the set of formulas over Λ with variables inV and, defining the operations on Fm(Λ) in the usual way, we obtain analgebra over the language Λ, called the formula algebra, that we denoteby Fm(Λ). We write δ(x0, . . . , xn) to denote a formula whose variablesbelong to the set {x0, . . . , xn}. An equation over Λ is a pair 〈δ, ²〉, withδ, ² ∈ Fm(Λ), which we denote by δ ≈ ².The algebra Fm(Λ) has the universal mapping property over V , i.e.for any algebra A, of type Λ, with domain A, and any mapping h : V → A,there is a unique homomorphism from Fm(Λ) into A, denoted by h. Foreach δ(x0, . . . , xn) ∈ Fm(Λ) we denote by δA(a0, . . . , an) its image by h,for any h such that h(xi) = ai (i = 0, . . . , n).A matrix over Λ is a pair A = 〈A,F〉, where A is an algebra of typeΛ and F is a subset of A. The algebra A is called the algebraic reduct ofA and the set F is called the designated filter of A. A congruence relationθ on A is compatible with F if for any two elements a, a′ ∈ A such thata θ a′, either a, a′ ∈ F or a, a′ /∈ F .On the Injectivity of the Leibniz Operator 205For a matrix A = 〈A,F〉 the Leibniz congruence on A over F isthe largest congruence relation on A compatible with F ; we denote it byΩA(F ) or simply by Ω(F ). For an algebra A the Leibniz operator ΩA isdefined byΩA : P(A) → Cong(A)F 7→ ΩA(F ).The Leibniz operator ΩA is injective if it is an injective map and it is calledmonotonic if∀ F,G ∈ P(A), F ⊆ G⇒ ΩA(F ) ⊆ ΩA(G).A (finitary) deductive system is a pair S = 〈Λ,`S〉, where Λ is alanguage and `S is a structural (or substitution-invariant) finitary conse-quence relation (a finitary deductive system can also be defined by axiomsand inference rules, see [2]). We write ∆ `S δ (or simply ∆ ` δ) to meanthat the pair 〈∆, δ〉 belongs to `S .By a theorem of S we mean a formula ϕ such that ∅ `S ϕ; we denoteby Thm(S) the set of all theorems. A set of formulas Γ is said to be closedunder the consequence relation `S if Γ `S ϕ implies ϕ ∈ Γ and it is called atheory of S or an S-theory. The set of all theories of S is denoted by Th(S).Given any set of formulas Γ, the set of all consequences of Γ, ConS(Γ), isthe smallest theory that contains Γ. Clearly, ConS(Γ) = {ϕ : Γ `S ϕ}.Let Λ and Λ′ ⊆ Λ be two languages and let S be a deductive systemover Λ. The Λ′-fragment S ′ of S is the deductive system with language Λ′and consequence relation `S′ defined by:Γ `S′ ϕ iff Γ `S ϕ (Γ ⊆ Fm(Λ′), ϕ ∈ Fm(Λ′)).Given a matrix A = 〈A,F〉, a formula ϕ is a (semantic) consequence ofa set of formulas Γ, and we write Γ |=A ϕ, if for every mapping h : V → A,h(ϕ) ∈ F whenever h(δ) ∈ F for every δ ∈ Γ. We call A a model of aformula ϕ if ∅ |=A ϕ and we write A |= ϕ.Let K be a class of matrices over a language Λ. A formula ϕ is aconsequence of a set of formulas Γ in K, and we write Γ |=K ϕ, if for everymatrix A ∈ K, Γ |=A ϕ.Let S be a deductive system over a language Λ. A matrix A is anS-matrix (or a model of S) if the consequence in S implies the semantic206 L. Descalc¸o and Manuel A. Martinsconsequence in A, i.e., Γ ` ϕ ⇒ Γ |=A ϕ. The class of all S-matrices isdenoted by Mod(S). An S-matrix A = 〈A,F〉 is reduced if Ω(F ) = idA andwe denote by Mod∗(S) the class of all reduced S-matrices. A set F ⊆ Asuch that 〈A,F〉 ∈ Mod(S) is called an S-filter of A. The set of all S-filtersof A is denoted by FiS(A).3. Injectivity for non protoalgebraic logicsA deductive system S is said to be protoalgebraic if the restriction of ΩFmto the set of theories of S is monotonic. The subclass of the protoalgebraiclogics for which this restriction of ΩFm is also injective is called the class ofweakly algebraizable logics. We say that a deductive system S has injectiveLeibniz operator (for simplicity, we say that S is injective) if for any algebraA, the restriction of ΩA to the set FiS(A) is injective. In the context ofprotoalgebraic logics it was shown that the injectivity of the restriction ofΩFm to the set of theories of S implies the deductive system to be injective(see [7], Theorem 4.7).Lemma 1. ([6]). Let S be a deductive system. Then S is protoalgebraic ifand only if for all F ⊆ Th(S), Ω(⋂F) = ⋂{Ω(T ) : T ∈ F}.As a consequence we have:Proposition 2. If S is protoalgebraic then Ω(Th(S)) is closed under(arbitrary) intersections.The converse of this proposition holds in the case where the S-theoriesare definable by a set of equations in the sense of the following definition:Definition 3. Let K be a set of matrices over a language Λ. We saythat the filters of the matrices in K are equationally definable by a set ofequations E ={δi(x) ≈ εi(x) : i ∈ I}if for each matrix A = 〈A,F〉 ∈ Kwe have:F ={a ∈ A : for all δ ≈ ε ∈ E(x), δA(a) ≡ εA(a) (Ω(F ) )}.This notion of equational definability generalizes the concept of ex-plicit definability of the truth predicate introduced by Czelakowski andOn the Injectivity of the Leibniz Operator 207Jansana in [7] (see also [6]). Moreover, if all the matrices in K are reducedthe two notions coincide.It is not difficult to see that equationally definability implies injectivityin the following sense:Proposition 4. Let S be a deductive system over Λ. If the class of allmodels of S has equationally definable filters by a set of equations, then forany algebra A, the restriction of ΩA to the S-filters of A is injective.For the class of all models having the algebraic part being the set offormulas, with filters equationally definable, protoalgebraicity (monotonic-ity) can be characterized by the condition of Ω(Th(S)) being closed underfinite intersections.Theorem 5. Let S be a deductive system over Λ. Assume that the class ofall models of S, of the form 〈Fm(Λ),T〉 has equationally definable filtersby a set of equations E. Then, S is protoalgebraic if and only if Ω(Th(S))is closed under finite intersections.Proof. By Proposition 2, for any protoalgebraic system S, Ω(Th(S)) isclosed under finite intersections.Assume now that Ω(Th(S)) is closed under finite intersections. LetT,G ∈ Th(S) such that T ⊆ G and F = {F,G}. Since Ω(Th(S)) isclosed under finite intersections,⋂Ω(F) = Ω(T )∩Ω(G) = Ω(H), for someH ∈ Th(S). Moreover, a ∈ H if and only if for all δ ≈ ε ∈ E, δA(a) ≡εA(a)(Ω(H)). That is, for all δ ≈ ε ∈ E, δA(a) ≡ εA(a) (Ω(T ) ∩ Ω(G)).This implies that for all δ ≈ ε ∈ E, δA(a) ≡ εA(a) (Ω(T )). Hence a ∈ Tand so H ⊆ T . Let now a ∈ T . Since T ⊆ G, a ∈ G. Hence, for all δ ≈ ε ∈E, δA(a) ≡ εA(a) (Ω(T )) and for all δ ≈ ε ∈ E, δA(a) ≡ εA(a) (Ω(G)).Then for all δ ≈ ε ∈ E, δA(a) ≡ εA(a) (Ω(T ) ∩ Ω(G)) which means thatδA(a) ≡ εA(a) (Ω(H)) and so T ⊆ H.Therefore T = H and thus Ω(T ) = Ω(H) = Ω(T ) ∩ Ω(G) ⊆ Ω(G).Hence, S is protoalgebraic. 23.1. Injective deductive systemsOne important question related to the class of weakly algebraizable logicscan be formulated: “Is injectivity enough to guarantee monotonicity?” Wewill show below that the answer is “no” by giving an example of a deductivesystem that is injective but not protoalgebraic.208 L. Descalc¸o and Manuel A. MartinsVery few examples of non protoalgebraic deductive systems have beeninvestigated. Among them we have: the inf-sup fragment of the ClassicPropositional Calculus (CPC∧∨) (see [9], [10] and [13]); Belnap’s Logic (see[8] and [10]), the {∨,∧,¬,>,⊥}-fragment of the Intuitionistic PropositionalCalculus IPC, denoted by IPC∗ (see [2], [10] and [17]). Recently, PositiveModal Logics and some Subintuitionistic Logics have also been investigated(see [4], [16]).If a deductive system does not have theorems, then each algebraA hasnon injective Leibniz operator, since the Leibniz congruence for both theempty set and the universal set is the universal congruence. An immediateconsequence is that an injective deductive system must have theorems. Itis well known that the deductive system IPC∗ has theorems. We will showthat it is injective.Proposition 6. Let S be a deductive system such that for every algebraA there is at most one reduced S-matrix with algebraic reduct A. Then Sis injective.Proof. Suppose that S is not injective. That is, there exists an algebraA such that ΩA is not injective. Hence, there are F1, F2 ∈ FiS(A) suchthat F1 6= F2 and ΩA(F1) = ΩA(F2). Let A∞ = 〈A,F1〉 and A∈ =〈A,F2〉. Clearly, A∞ and A∞ are S-matrices. Moreover, the matrices〈A/ΩA(F1),F1/ΩA(F1)〉 and 〈A/ΩA(F2),F2/ΩA(F2)〉 are reduced S-matrices with the same algebraic reduct A/ΩA(F1) (= A/ΩA(F2) ) andsuch that F1/ΩA(F1) 6= F2/ΩA(F2), a contradiction. 2The following result, recalled by Rebagliato and Verdu´ in [17], charac-terizes the reduced IPC∗-matrices (PCDL denotes the variety of the pseu-docomplemented distributive lattices).Theorem 7. ([17]) The following are equivalent:(i) A = 〈A,F〉 is a reduced IPC∗-matrix;(ii) (a) A ∈ PCDL;(b) F = {1} and(c) ∀ a, b ∈ A, if a < b then there exists c ∈ A, c 6= 1, that satisfiesc ∧ a = a and (c ∨ b) ∧ ¬(¬a ∧ b) = ¬(¬a ∧ b).From this theorem we conclude that for every algebra A there is atmost one reduced IPC∗-matrix with algebraic reduct A. Moreover, whenOn the Injectivity of the Leibniz Operator 209such reduced matrix exists, it is equal to 〈A, {1}〉. Hence, by Proposition6, IPC∗ is injective. Therefore, IPC∗ is an example of a deductive systemwhich is non protoalgebraic but injective.3.2. Non protoalgebraic deductive systems without the-oremsAs we pointed out above, there are non protoalgebraic systems withouttheorems, which obviously are non injective. Since the non injectivity isshown by using that Ω(∅) = Ω(Fm(Λ)), it is natural to investigate theinjectivity of the restriction of the Leibniz operator to the set of non emptyS-filters.We say that a deductive system is quasi-injective if for every algebraA, the restriction of ΩA to FiS(A) \ {∅}, ΩA : FiS(A) \ {∅} → Cong(A),is injective.We will need the following result, proved by Font and Jansana in [10],that characterizes the algebraic reducts of the CPC∧∨-matrices:Theorem 8. ([9]) The class of algebraic reducts of the reduced CPC∧∨-matrices is the class of distributive lattices with maximum 1 such that forevery a, b ∈ A, if a < b then there is c ∈ A, with a ∨ c 6= 1 and b ∨ c = 1.On the other hand, if A is not a trivial algebra then A = 〈A, ∅〉 cannot be a reduced CPC∧∨-matrix, since ΩA(∅) = A2 6= ∆A.Next proposition is a weaker version of a result in [9] that characterizesCPC∧∨-matrices:Proposition 9. ([9]) Let A be an algebra and A = 〈A,F〉 a reducedCPC∧∨-matrix. If A is non trivial then F = {1}, otherwise F = ∅.Now, we are able to state our final result about the injectivity of theLeibniz operator on CPC∧∨.Theorem 10. CPC∧∨ is quasi-injective.Proof. LetA be an algebra. IfA is not trivial the statement follows fromProposition 9 and Proposition 6. If A = {a} then FiCPC∧∨(A)\{∅} = ∅and, obviously, ΩA : FiCPC∧∨(A) \ {∅} → Cong(A) is injective. 2210 L. Descalc¸o and Manuel A. MartinsIn the previous subsection we saw that the non injectivity of the Leib-niz operator of the deductive system CPC∧∨ follows only from the fact thatthe empty set is a CPC∧∨-filter. It is not expected that it happens for anydeductive systems without theorems, since CPC∧∨ is a very particular caseof a non protoalgebraic fragment of an algebraizable logic. In fact, Belnap’slogic, [8], is an example of a deductive system, without theorems, which isnot quasi-injective. Moreover, not all fragments of an algebraizable logicare quasi-injective. For example, the trivial fragment of CPC, CPC∅, isnot quasi-injective. It would be interesting to find a characterization ofquasi-injective deductive systems.References[1] W. Blok and D. Pigozzi, Algebraic semantics for universal Hornlogic without equality,Universal algebra and quasigroup theory, Lect.Conf., Jadwisin/Pol. 1989, Res. Expo. Math. 19 (1992), pp. 1–56.[2] W. Blok and D. Pigozzi, Algebraizable logics,Mem. Am. Math.Soc., 396 (1989).[3] W. Blok and D. Pigozzi, Protoalgebraic logics, Studia Logica, 45(1986), pp. 337–369.[4] S. Celani and R. Jansana, A closer look at some subintuitionisticlogics, Notre Dame J. Formal Logic, 42 (2001), pp. 225–255.[5] J. Czelakowski, Equivalential logics (I,II), Studia Logica, 40(1981), pp. 227–236 and 355–372[6] J. Czelakowski, Protoalgebraic logics., Trends in Logic–StudiaLogica Library. 10. Dordrecht: Kluwer Academic Publishers (2001).[7] J. Czelakowski and R. Jansana, Weakly algebraizable logics,J. Symb. Log., 65-2 (2000), pp. 641–668.[8] J. Font, Belnap’s Four-Valued Logic and De Morgan Lattice,L. J. of the IGPL, 5-3 (1997), pp. 1–29.[9] J. Font, F. Guzma´n and V. Verdu´, Characterization of the reducedmatrices for the {∧,∨}-fragment of classical logic, Bull. Sect. Logic, 20(1991), pp. 124–128.[10] J. Font and R. Jansana, A general algebraic semantics for sen-tential logics, Lecture Notes in Logic, 7 (1996).[11] J. Font, R. Jansana and D. Pigozzi, A survey of abstract algebraiclogic, Studia Logica, 74 (2003), pp. 13–97.On the Injectivity of the Leibniz Operator 211[12] J. Font and G. Rodr´ıguez, Algebraic Study of two DeductiveSystems of Relevance Logic, Notre Dame J. Formal Logic, 35-3 (1994),pp. 369–397.[13] J. Font and V. Verdu´, Algebraic Logic for Classical Conjunctionand Disjunction, Studia Logica, L (1991), pp. 391–418.[14] B. Hermann, Equivalential and Algebraizable Logics, StudiaLogica, 57-2 (1996), pp. 419–436.[15] B. Hermann, Characterizating equivalential and algebraizable log-ics, Studia Logica, 58-2 (1997), pp. 305–323.[16] R. Jansana, Full models for positive modal logic, MLQ Math.Log. Q., 48 (2002), pp. 427–445.[17] J. Rebagliato and V. Verdu´, On the Algebraization of someGentzen Systems, Fundamenta Informaticae, 18-2 (1993), pp. 319–338.Universidade de AveiroDepartamento de Matema´tica3810-193 Aveiro, Portugale-mails: luisd@mat.ua.pt, martins@mat.ua.pt.

On the injectivity of the Leibniz operator

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On the injectivity of the Leibniz operator

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