Pseudo BCK-Semilattices

Kühr, Jan

doi:10.1515/dema-2007-0302

Cited by 12 publications

(10 citation statements)

References 7 publications

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“…Since the partial order ≤ is determined by either of the two "arrows", we can eliminate ≤ from the signature and denote a pseudo BCK-algebra by B = (X, →, , 1). An equivalent definition of a pseudo BCK-algebra is given in [21]. The structure B = (B, →, , 1) of the type (2, 2, 0) is a pseudo BCK-algebra iff it satisfies the following identities and quasi-identity, for all x, y, z ∈ B:…”

Section: Proposition 22 ([12])mentioning

confidence: 99%

On pseudo-equality algebras

Ciungu

2014

Arch. Math. Logic

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Pseudo equality algebras were initially introduced by Jenei and Kóródi as a possible algebraic semantic for fuzzy type theory, and they have been revised by Dvurečenskij and Zahiri under the name of JK-algebras. The aim of this paper is to investigate the internal states and the state-morphisms on pseudo equality algebras. We define and study new classes of pseudo equality algebras, such as commutative, symmetric, pointed and compatible pseudo equality algebras. We prove that any internal state (state-morphism) on a pseudo equality algebra is also an internal state (state-morphism) on its corresponding pseudo BCK(pC)-meet-semilattice, and we prove the converse for the case of linearly ordered symmetric pseudo equality algebras. We also show that any internal state (state-morphism) on a pseudo BCK(pC)-meet-semilattice is also an internal state (state-morphism) on its corresponding pseudo equality algebra. The notion of a Bosbach state on a pointed pseudo equality algebra is introduced and it is proved that any Bosbach state on a pointed pseudo equality algebra is also a Bosbach state on its corresponding pointed pseudo BCK(pC)-meetsemilattice. For the case of an invariant pointed pseudo equality algebra, we show that the Bosbach states on the two structures coincide.

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Section: Proposition 22 ([12])mentioning

confidence: 99%

On pseudo-equality algebras

Ciungu

2014

Arch. Math. Logic

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“…Generalizing the notion of a pseudo-BCK semilattice (see [13]) we define pseudo-BCH join-semilattices.…”

Section: Pseudo-bch Semilatticesmentioning

confidence: 99%

“…In [13], Kühr investigated pseudo-BCK algebras whose underlying posets are semilattices. In this paper we study pseudo-BCH join-semilattices, that is.…”

Section: Introductionmentioning

confidence: 99%

Pseudo-BCH Semilattices

Walendziak

2018

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In this paper we study pseudo-BCH algebras which are semilattices or lattices with respect to the natural relations ≤; we call them pseudo-BCH join-semilattices, pseudo-BCH meet-semilattices and pseudo-BCH lattices, respectively. We prove that the class of all pseudo-BCH join-semilattices is a variety and show that it is weakly regular, arithmetical at 1, and congruence distributive. In addition, we obtain the systems of identities defininig pseudo-BCH meet-semilattices and pseudo-BCH lattices.

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“…In case of pseudo-BCK-semilattices, though ∨ is not a term operation in the operations \, /, it turns out that the congruence kernels are the compatible deductive systems [8]. Thus the congruence lattice and the lattice of all compatible deductive systems of any pseudo-BCK-semilattice are isomorphic (under the above assignments).…”

Section: Integral Residuated Lattices and Pseudo-bck-semilatticesmentioning

confidence: 99%