Representable pseudo-BCK-algebras and integral residuated lattices

Kühr, Jan

doi:10.1016/j.jalgebra.2007.07.003

Cited by 28 publications

(27 citation statements)

References 5 publications

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“…We say that a nonempty subset F of a pseudo-BCK algebra A is a filter (or a deductive system, [26,27]) if (i) 1 ∈ F , and (ii) if a ∈ F and a → b ∈ F, then b ∈ F. It is easy to verify that a set F containing 1 is a filter if and only if (ii)' if a ∈ F and a b ∈ F, then b ∈ F. A filter F is (i) maximal if it is a proper subset of A and not properly contained in another proper filter of A, (ii) normal if a → b ∈ F if and only if a b ∈ F. Given a normal filter F , the relation Θ F on A given by…”

Section: Remarks 22 ([20]) (1) a Pseudo-bck Algebramentioning

confidence: 99%

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Measures, states and de Finetti maps on pseudo-BCK algebras

Ciungu

Dvurečenskij

2010

Fuzzy Sets and Systems

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Abstract. In this paper, we extend the notions of states and measures presented in [12] to the case of pseudo-BCK algebras and study similar properties. We prove that, under some conditions, the notion of a state in the sense of [12] coincides with the Bosbach state, and we extend to the case of pseudo-BCK algebras some results proved by J. Kühr only for pseudo-BCK semilattices. We characterize extremal states, and show that the quotient pseudo-BCK algebra over the kernel of a measure can be embedded into the negative cone of an archimedean ℓ-group. Additionally, we introduce a Borel state and using results by J. Kühr and D. Mundici from [28], we prove a relationship between de Finetti maps, Bosbach states and Borel states.

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Section: Remarks 22 ([20]) (1) a Pseudo-bck Algebramentioning

confidence: 99%

“…Then F = [1]Θ F and the quotient class A/F defined as A/Θ F is again a pseudo-BCK algebra, and we write a/F = [a] Θ F for every a ∈ A, see [26,27].…”

Section: Remarks 22 ([20]) (1) a Pseudo-bck Algebramentioning

confidence: 99%

Measures, states and de Finetti maps on pseudo-BCK algebras

Ciungu

Dvurečenskij

2010

Fuzzy Sets and Systems

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“…Pseudo BCK-algebras and pseudo BCK-join-semilattices are strongly related to residuated lattices (see [12], [13]). Actually, every pseudo BCKalgebra is isomorphic to a {-l}-subreduct of some (bounded integral) residuated lattice, where also existing finite joins are preserved, and hence every pseudo BCK-join-semilattice arises as a {V, ->,~>, l}-subreduct of a residuated lattice.…”

Section: (I) X -• Y < (Y -Y Z) ~> {X -> Z) X Y < (Y Z) -> (X Z) (Iimentioning

confidence: 99%

Pseudo BCK-Semilattices

Kühr

2007

Demonstratio Mathematica

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Abstract. Pseudo BCK-algebras are algebras {A, -1) of type (2,2,0) which generalize BCK-algebras in such a way that if the operations -• and coincide then (A, -1) is a BCK-algebra. They can be also viewed as {-l}-subreducts of non-commutative integral residuated lattices. In the paper, we study pseudo BCK-algebras whose underlying posets are semilattices or lattices; we call them pseudo BCK-join-semilattices, pseudo BCK-meet-semilattices and pseudo BCK-lattices, respectively. After describing their congruence properties we deal mainly with prime deductive systems of pseudo BCKjoin-semilattices.

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“…In view of strongly connections with a BIK + -logic, BCC -algebra also is called BIK + -algebra (see [19,26]). Now, such algebras are studied by many authors in many directions (see for example [4,7,8,10,15,20,23,[25][26][27]). …”

Section: Introductionmentioning

confidence: 99%

BCC-algebras and residuated partially-ordered groupoid

Zhang

2013

Mathematica Slovaca

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ABSTRACT. The aim of the paper is to investigate the relationship between BCC -algebras and residuated partially-ordered groupoids. We prove that an integral residuated partially-ordered groupoid is an integral residuated pomonoid if and only if it is a double BCC -algebra. Moreover, we introduce the notion of weakly integral residuated pomonoid, and give a characterization by the notion of pseudo-BCI algebra. Finally, we give a method to construct a weakly integral residuated pomonoid (pseudo-BCI algebra) from any bounded pseudo-BCK algebra with pseudo product and any group.

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Representable pseudo-BCK-algebras and integral residuated lattices

Abstract: Pseudo-BCK-algebras arise as the {\, /, 1}-subreducts of integral residuated lattices. In this note we characterize pseudo-BCK-algebras that are subdirect products of linearly ordered pseudo-BCK-algebras.

Cited by 28 publications

References 5 publications

Measures, states and de Finetti maps on pseudo-BCK algebras

Measures, states and de Finetti maps on pseudo-BCK algebras

Pseudo BCK-Semilattices

BCC-algebras and residuated partially-ordered groupoid

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