Fregean Varieties

Idziak, Paweł M.; Słomczyńska, Katarzyna; Wronski, Andrzej

doi:10.1142/s0218196709005251

Cited by 22 publications

(30 citation statements)

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“…This allows us to apply modular commutator theory as described in [2]. In particular, in [6] we have shown that Fregean varieties satisfy the condition (SC1) introduced and discussed in [5]: Proposition 2.1. [6, Theorem 2.3] In a subdirectly irreducible algebra A from a Fregean variety, the centralizer (0 : μ) does not exceed the monolith μ of A.…”

Section: Preliminariesmentioning

confidence: 99%

“…It is easy to show that the term 0 := xx is constant and that, with 0 being distinguished, the variety E is congruence permutable and Fregean, where (xyz)(xzzx) serves as the Mal'cev term. In fact, the equivalential algebras form a paradigm of congruence permutable Fregean varieties, as the following result, taken from [6], shows. Theorem 1.1.…”

Section: Introductionmentioning

confidence: 94%

“…Following our earlier paper [6], a variety V with a distinguished constant 0 is called Fregean if every algebra A ∈ V is…”

Section: Introductionmentioning

confidence: 99%

“…We refer the reader to [6], where a discussion of the name Fregean is given. Here we only recall that it comes from Frege's idea that sentences should denote their logical values.…”

Section: Introductionmentioning

confidence: 99%

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The commutator in equivalential algebras and Fregean varieties

2011

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Abstract. A class K of algebras with a distinguished constant term 0 is called Fregean if congruences of algebras in K are uniquely determined by their 0-cosets and Θ A (0, a) = Θ A (0, b) implies a = b for all a, b ∈ A ∈ K. The structure of Fregean varieties was investigated in a paper by P. Idziak, K. S lomczyńska, and A. Wroński. In particular, it was shown there that every congruence permutable Fregean variety consists of algebras that are expansions of equivalential algebras, i.e., algebras that form an algebraization of the purely equivalential fragment of the intuitionistic propositional logic. In this paper we give a full characterization of the commutator for equivalential algebras and solvable Fregean varieties. In particular, we show that in a solvable algebra from a Fregean variety, the commutator coincides with the commutator of its purely equivalential reduct. Moreover, an intrinsic characterization of the commutator in this setting is given.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 94%

“…Following our earlier paper [6], a variety V with a distinguished constant 0 is called Fregean if every algebra A ∈ V is…”

Section: Introductionmentioning

confidence: 99%

“…We refer the reader to [6], where a discussion of the name Fregean is given. Here we only recall that it comes from Frege's idea that sentences should denote their logical values.…”

Section: Introductionmentioning

confidence: 99%

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The commutator in equivalential algebras and Fregean varieties

2011

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“…The author is grateful to prof. K. Słomczyńska for discussing some points related to congruence orderability of the variety of semi-Heyting algebras described in [34] and providing a copy of the paper [22], and to the anonymous referee for various suggestions-in particular, for the improved present version of Theorem 21 and the suggestion to gather in a separate subsection facts concerning relations between known subvarieties of EWR ∧ .…”

mentioning

confidence: 99%

Weak relative pseudocomplements in semilattices

Cı̄rulis

2011

Demonstratio Mathematica

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Abstract. Weak relative pseudocomplementation on a meet semilattice S is a partial operation * which associates with every pair (x, y) of elements, where x ≥ y, an element z (the weak pseudocomplement of x relative to y) which is the greatest among elements u such that y = u ∧ x. The element z coincides with the pseudocomplement of x in the upper section [y) and, if S is modular, with the pseudocomplement of x relative to y. A weakly relatively pseudomented semilattice is said to be extended, if it is equipped with a total binary operation extending * . We study congruence properties of the variety of such semilattices and review some of its subvarieties already described in the literature. IntroductionA meet semilattice is said to be weakly relatively pseudocomplemented if, whenever y ≤ x, there is a greatest element u such that y = u ∧ x. The concept goes back to [28], where the congruence lattice of a semilattice was shown to possess this property; the term, however, was introduced later in [35]. Weak relative pseudocomplements in congruence lattices (of various structures) are discussed also, for example, in [2,16,19,41]; they are uncovered as well in lattices of closure operators [18,31], certain subalgebra lattices of semigroups and groups [37,38,42], and in algebraic structures of constraint programming [5,6]. Every meet-semidistributive algebraic (in particular, finite) lattice is weakly relatively pseudocomplemented [12,17,37]. Weak relative pseudocomplementation has been studied also in posets [15]. Sectionally pseudocomplemented semilattices introduced in [8,10] (i.e., semilattices with pseudocomplemented principial filters) are just 1991 Mathematics Subject Classification: 03G25, 06A12, 08A30, 08B12.

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