Least V-quasivarieties of MV-algebras

Gispert, Joan

doi:10.1016/j.fss.2014.07.011

Cited by 9 publications

(12 citation statements)

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“…For the finite valued case it is much simpler, since M V n is a locally finite discriminator variety (cf. [7,25]). Any locally finite quasivariety is generated by its critical algebras (see [22]), where an algebra A is said to be critical iff it is a finite algebra not belonging to the quasivariety generated by all its proper subalgebras.…”

Section: Translations Equivalent Logics and Strong Maximality 51 Prmentioning

confidence: 99%

Maximality in finite-valued Łukasiewicz logics defined by order filters

Coniglio

Esteva

Gispert

et al. 2018

Journal of Logic and Computation

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In this paper we consider the logics L i n obtained from the (n + 1)valued Lukasiewicz logics Ln+1 by taking the order filter generated by i/n as the set of designated elements. In particular, the conditions of maximality and strong maximality among them are analyzed. We present a very general theorem which provides sufficient conditions for maximality between logics. As a consequence of this theorem it is shown that L i n is maximal w.r.t. CPL whenever n is prime. Concerning strong maximality between the logics L i n (that is, maximality w.r.t. rules instead of axioms), we provide algebraic arguments in order to show that the logics L i n are not strongly maximal w.r.t. CPL, even for n prime. Indeed, in such case, we show there is just one extension between L i n and CPL obtained by adding to L i n a kind of graded explosion rule. Finally, using these results, we show that the logics L i n with n prime and i/n < 1/2 are ideal paraconsistent logics. arXiv:1803.09815v2 [math.LO]

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Section: Translations Equivalent Logics and Strong Maximality 51 Prmentioning

confidence: 99%

Maximality in finite-valued Łukasiewicz logics defined by order filters

Coniglio

Esteva

Gispert

et al. 2018

Journal of Logic and Computation

Self Cite

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“…A congruence α on an algebra A is called a Q-congruence provided A/α ∈ Q. The set Con Q (A) of all Q-congruences of A forms an algebraic lattice with the same meets, but possibly different joins, as in the lattice Con(A) of all congruences of A [37,Corollary 1.4.11]. We say that a (quasi)variety Q is (relatively) congruence-distributive if for every A ∈ Q the lattice Con Q (A) is distributive.…”

Section: Concepts From Quasivariety Theorymentioning

confidence: 99%

“…(When Q is a variety we also simplify the notation by dropping the subscript Q.) We say that an algebra is Q-finitely presented if it is isomorphic to F(k)/θ Q (H) for some natural number k and some finite set H [37,Chapter 2]. The class of all Q-finitely presented algebras will be denoted by Q FP .…”

Section: Concepts From Quasivariety Theorymentioning

confidence: 99%

“…Condition (4) in Theorem 3.1 may be formulated with the use of relative antivarieties [37]. Recall that a subclass of a quasivariety Q is a Q-antivariety if it is closed under taking subalgebras, ultraproducts and homomorphic preimages from Q.…”

Section: Corollary 35 Suppose That Every Nontrivial Algebra From Q mentioning

confidence: 99%

“…The reader may consult the monograph [70] and references therein for older results concerning SC property. Let us here recall recent works on SC: [67] on varieties of positive Sugihara monoids, [68] on substructural logics, [63] on fragments of R-mingle, [18] on fuzzy logics, [36] on quasivarieties of MV-algebras, [81] on some fragment of the intuitionistic logic, [52] on BCK logic, [15] on semisimple quasivarieties and discriminator varieties, and [74] which contains general considerations from abstract algebraic logic perspective and results for some non-algebraizable deductive systems. Although the SC property was often investigated algebraically, there are few papers on it for algebras not connected to logic.…”

Section: Introductionmentioning

confidence: 99%

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Almost structural completeness; an algebraic approach

Dzik

Stronkowski

2016

Annals of Pure and Applied Logic

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A deductive system is structurally complete if all of its admissible inference rules are derivable. For several important systems, like the modal logic S5, failure of structural completeness is caused only by the underivability of a passive rule, i.e., a rule whose premise is not unifiable by any substitution. Neglecting passive rules leads to the notion of almost structural completeness, that means, to the derivability of admissible non-passive rules. We investigate almost structural completeness for quasivarieties and varieties of general algebras. The results apply to all algebraizable deductive systems. Firstly, various characterizations of almost structurally complete quasivarieties are presented. Two of them are general: the one expressed with finitely presented algebras, and the one expressed with subdirectly irreducible algebras. The next one is restricted to quasivarieties with the finite model property and equationally definable principal relative congruences, where the condition is verifiable on finite subdirectly irreducible algebras. Some connections with exact and projective unification are included.Secondly, examples of almost structurally complete varieties are provided. Particular emphasis is put on varieties of closure algebras, that are known to constitute adequate semantics for normal extensions of the modal logic S4. A certain infinite family of such almost structurally complete, but not structurally complete, varieties is constructed. Every variety from this family has a finitely presented unifiable algebra which does not embed into any free algebra for this variety. Hence unification is not unitary there. This shows that almost structural completeness is strictly weaker than projective unification for varieties of closure algebras.

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