When Is a BCC-Algebra Equivalent to an Mv-Algebra?

Chajda, Ivan; Halaš, Radomír; Kühr, Jan

doi:10.1515/dema-2007-0403

Cited by 21 publications

(46 citation statements)

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“…The reader is referred to the monographs [7], [9] or [11] for information on general congruence properties of algebras and varieties mentioned in the subseqent theorem. We only remind that an algebra is arithmetical (at 1) if and only if it is congruence distributive and permutable (at 1); permutable at 1 algebras are known also as subtractive.…”

Section: Some Congruence Properties Of Ewr ∧mentioning

confidence: 99%

“…It is known well that every Brouwerian semilattice is arithmetical (see Theorem 4.3.1 in [11]). Since C(A) ⊆ C(A ), the algebra A itself also is arithmetical.…”

Section: Some Congruence Properties Of Ewr ∧mentioning

confidence: 99%

“…Assume that A is an almost Brouwerian semilattice. According to Proposition 9, then A is a Brouwerian semilattice, so that the mapping F → F ♯ is an isomorphism from F (A ) onto C(A ) (see, e.g., Corollary 4.3.4 in [11]). But the algebras A and A have the same filters, and, by Theorem 19(d), also the same congruences.…”

Section: Theorem 21 An Ewr-semilattice Is Almost Brouwerian If and Omentioning

confidence: 99%

“…However, instead of (6), another relation (11) x → y := (x ∨ y) * y (in notation of the present paper) was used in [8] to turn the partial operation * into a total one. In the further papers [12,13], as well as in the monograph [11] sectionally pseudocomplemented lattices are treated as extended by means of (11). The class of such lattices is a subvariety of EWR-lattices, which is defined by one additional axiom…”

Section: Semilattices With Sectional Join-pseudocomplementationmentioning

confidence: 99%

“…Sectionally pseudocomplemented semilattices introduced in [8,10] (i.e., semilattices with pseudocomplemented principial filters) are just weakly relatively pseudocomplemented semilattices. See also [11] and references therein for information on sectionally pseudocomplemented semilattices and lattices.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Some Congruence Properties Of Ewr ∧mentioning

confidence: 99%

“…It is known well that every Brouwerian semilattice is arithmetical (see Theorem 4.3.1 in [11]). Since C(A) ⊆ C(A ), the algebra A itself also is arithmetical.…”

Section: Some Congruence Properties Of Ewr ∧mentioning

confidence: 99%

Section: Theorem 21 An Ewr-semilattice Is Almost Brouwerian If and Omentioning

confidence: 99%