Bounded BCK‐algebras and their generated variety

Gispert, Joan; Torrens, Antoni

doi:10.1002/malq.200610040

Cited by 11 publications

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“…If we represent the set of all ⊆-minimal prime i-filters of a bounded BCK-algebra A by m Prim(A), then we have: Proof Since (i) ⇔ (ii) follows from Lemma 1.2 and since every finitely subdirectly irreducible bBCK-algebra is directly indecomposable, it suffices to see that (ii) implies (iii), which follows from the fact that for any P ∈ Prim(A), Prim(B F (A)). To analyze the case in which the stalks are simple algebras, we recall that, as the authors show in Gispert and Torrens (2007), in bounded BCK-algebras maximal proper congruences are BCK-congruences, and so any bBCKsimple algebra is simple. In what follows, for a given bounded BCK-algebra A, the set of its maximal proper i-filters is denoted by Max (A).…”

Section: Pierce Representations With Finitely Subdirectly Irreduciblementioning

confidence: 99%

Boolean representation of bounded BCK-algebras

Gispert

Torrens

2007

Soft Comput

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We define the Boolean center and the Boolean skeleton of a bounded BCK-algebra, and we use the Boolean skeleton to obtain a representation of bounded BCKalgebras, called (weak) Pierce bBCK-representation, as (weak) Boolean products of bounded BCK-algebras. We analyze the cases in which the stalks in these representations are directly indecomposable, finitely subdirectly irreducible or simple algebras. We give some examples of algebras and relative subvarieties of bounded BCK-algebras to illustrate the results.

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Section: Pierce Representations With Finitely Subdirectly Irreduciblementioning

confidence: 99%

Boolean representation of bounded BCK-algebras

Gispert

Torrens

2007

Soft Comput

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“…We denote by Max (A) the set of maximal (proper) i-filters of A. The following two lemmas are well known (see for instance [9]). …”

Section: Implicative Filters and Congruences Given A Bounded Bck-algmentioning

confidence: 99%

“…It is clear that in semisimple quasivarieties, subdirectly irreducible algebras coincide with simple algebras. In particular, bBCK is not semisimple, but for any set X, its |X|-free algebra F bBCK (X) is semisimple (see [9]); hence, its generated variety is also generated by simple bounded BCK-algebras.…”

Section: Semisimple Relative Subvarieties Of Bbckmentioning

confidence: 99%

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Semisimplicity and the discriminator in bounded BCK-algebras

Torrens

2010

Algebra Univers.

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By means of an adaptation of a proof given by T. Kowalski in [16], we show that the members of a relative subvariety V of bounded BCK-algebras are all semisimple if and only if V is a variety and satisfies the identityMoreover, we show that discriminator varieties of bounded BCK-algebras are just the semisimple and involutive relative subvarieties. We also explain how these results can be extended to "strong expansions'" of the quasivariety of bounded BCK-algebras.

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“…The {?,1}-reducts 1 of good EQ-algebras are BCKalgebras (for the definitions and basic properties of BCK-algebras see (Gispert and Torrens 2007;Iseki 1978;Pałasinski 1980Pałasinski , 1981Raftery 1987). Actually, we have the following theorem from El-Zekey et al…”

Section: Properties Of Special Eq-algebrasmentioning

confidence: 99%

Representable good EQ-algebras

El-Zekey

2009

Soft Comput

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Recently, a special algebra called EQ-algebra (we call it here commutative EQ-algebra since its multiplication is assumed to be commutative) has been introduced by Novák (Proceedings of the Czech-Japan seminar, ninth meeting, Kitakyushu and Nagasaki, 18-22 August, 2006), which aims at becoming the algebra of truth values for fuzzy type theory. Its implication and multiplication are no more closely tied by the adjunction and so, this algebra generalizes commutative residuated lattice. One of the outcomes is the possibility to relax the commutativity of the multiplication. This has been elaborated by El-Zekey et al. (Fuzzy Sets Syst 2009, submitted). We continue in this paper the study of EQ-algebras (i.e., those with multiplication not necessarily commutative). We introduce prelinear EQ-algebras, in which the join-semilattice structure is not assumed. We show that every prelinear and good EQ-algebra is a lattice EQ-algebra. Moreover, the f^; _; !; 1g-reduct of a prelinear and separated lattice EQ-algebra inherits several lattice-related properties from product of linearly ordered and separated EQ-algebras. We show that prelinearity alone does not characterize the representable class of all good (commutative) EQ-algebras. One of the main results of this paper is to characterize the representable good EQ-algebras. This is mainly based on the fact that f!; 1g-reducts of good EQ-algebras are BCK-algebras and run on lines of van Alten's (J Algebra 247:672-691, 2002) characterization of representable integral residuated lattices. We also supply a number of potentially useful results, leading to this characterization.

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Bounded BCK‐algebras and their generated variety

Cited by 11 publications

References 10 publications

Boolean representation of bounded BCK-algebras

Boolean representation of bounded BCK-algebras

Semisimplicity and the discriminator in bounded BCK-algebras

Representable good EQ-algebras

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