Definability of principal congruences in equivalential algebras

Idziak, Paweł M.; Wronski, Andrzej

doi:10.4064/cm-74-2-225-238

Cited by 6 publications

(6 citation statements)

References 12 publications

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“…, y t } ⊆ Y , where t ∈ ω. But if G is a finite subset of Y , then B(G) is finitely generated, so by (33), the sentence ∀x∀y D(x, y) is true in S B (G) , and therefore in f G [S B (G) ]. Now ∀x∀y D(x, y) is logically equivalent to a universal-existential sentence, and all such sentences persist in directed unions of algebras, so ∀x∀y D(x, y) is true in S F r , as required.…”

Section: Quasivarietiesmentioning

confidence: 98%

“…2.7]. The intuitionistic equivalential calculus has no local DDT, however, as it lacks the filter extension property [33]. Of course, it is protoalgebraic, with Δ = {v 1 ↔ v 2 } to witness Definition 3.9.…”

Section: Theorem 72 a Local Cdd-sequence For A Finitary Deductive Smentioning

confidence: 98%

“…Consequently, by the assumption and the remark preceding Definition 8.3, S B |= ∀x∀y D(x, y) for all finitely generated algebras B of K's type. (33) Let A ∈ K be an infinitely generated algebra. Then there is a surjective homomorphism h : F r → A, where F r is an absolutely free algebra on some infinite set Y of free generators.…”

Section: Quasivarietiesmentioning

confidence: 99%

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Contextual Deduction Theorems

Raftery

2011

Stud Logica

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Logics that do not have a deduction-detachment theorem (briefly, a DDT) may still possess a contextual DDT -a syntactic notion introduced here for arbitrary deductive systems, along with a local variant. Substructural logics without sentential constants are natural witnesses to these phenomena. In the presence of a contextual DDT, we can still upgrade many weak completeness results to strong ones, e.g., the finite model property implies the strong finite model property. It turns out that a finitary system has a contextual DDT iff it is protoalgebraic and gives rise to a dually Brouwerian semilattice of compact deductive filters in every finitely generated algebra of the corresponding type. Any such system is filter distributive, although it may lack the filter extension property. More generally, filter distributivity and modularity are characterized for all finitary systems with a local contextual DDT, and several examples are discussed. For algebraizable logics, the well-known correspondence between the DDT and the equational definability of principal congruences is adapted to the contextual case.

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Section: Quasivarietiesmentioning

confidence: 98%

Section: Theorem 72 a Local Cdd-sequence For A Finitary Deductive Smentioning

confidence: 98%

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Contextual Deduction Theorems

Raftery

2011

Stud Logica

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“…In this note we will consider Hilbert algebras where the order define a join-semilattice. We note that Hilbert algebras are also known as positive implicative BCK-algebras, and Hilbert algebras with lattice operations are a particular case of the BCK-algebras with lattice operations studied by Idziak in [16] and [17].…”

Section: Introductionmentioning

confidence: 99%

Notes on bounded Hilbert algebras with supremum

Celani¹

2014

Acta Sci. Math.

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In this note we will investigate some particular classes of ideals in Hilbert algebras with supremum. We shall study the relation between αideals and annihilator ideals in bounded Hilbert algebras with supremum. We shall introduce the class of σ-ideals and we will see that this class is strongly connected with the deductive systems. We will also characterize the bounded Hilbert algebras with supremum satisfying the Stone identity.

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“…All these results can be partially generalized to subtractive Fregean varieties (Proposition 5.1, Theorem 5.2) and so cover such different algebraic structures as Brouwerian semilattices, Hilbert algebras or equivalential algebras. However, while in the first two cases one can use a term defining principal filters to get a projective unifier, in the general case it is impossible: for example, there is no non-trivial subvariety of equivalential algebras which has equationally definable principle filters and the only one with definable principle filters is the variety of Boolean groups [14].…”

Section: Introductionmentioning

confidence: 99%

Unification and projectivity in Fregean varieties

Słomczyńska

2011

Logic Journal of IGPL

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In some varieties of algebras one can reduce the question of finding most general unifiers (mgus) to the problem of the existence of unifiers that fulfill the additional condition called projectivity. In this paper we study this problem for Fregean (1-regular and orderable) varieties that arise from the algebraization of fragments of intuitionistic or intermediate logics. We investigate properties of Fregean varieties, guaranteeing either for a given unifiable term or for all unifiable terms, that projective unifiers exist. We indicate the identities which fully characterize congruence permutable Fregean varieties having projective unifiers. In particular, we show that for such a variety there exists the largest subvariety that have projective unifiers.

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Definability of principal congruences in equivalential algebras

Cited by 6 publications

References 12 publications

Contextual Deduction Theorems

Contextual Deduction Theorems

Notes on bounded Hilbert algebras with supremum

Unification and projectivity in Fregean varieties

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