Pseudo equality algebras: revision

Dvurečenskij, Anatolij; Zahiri, Omid

doi:10.1007/s00500-015-1888-x

Cited by 19 publications

(34 citation statements)

References 12 publications

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“…Pseudo equality algebras have been firstly defined by Jenei and Kóródi in [21] as a generalization of equality algebras. Dvurečenskij and Zahiri showed in [15] that every pseudo equality algebra in the sense of [21] is an equality algebra. They also defined and investigated a new concept of pseudo equality algebras (JK-algebras) and established a connection between pseudo equality algebras and a special class of pseudo BCK-meet-semilattices.…”

Section: Preliminaries On Pseudo Equality Algebrasmentioning

confidence: 99%

“…Definition 2.1. ( [15]) A pseudo equality algebra (or a JK-algebra) is an algebra A = (A, ∧, ∼ , ∽, 1) of the type (2, 2, 2, 0) such that the following axioms are fulfilled for all x, y, z ∈ A: (A 1 ) (A, ∧, 1) is a meet-semilattice with top element 1,…”

Section: Preliminaries On Pseudo Equality Algebrasmentioning

confidence: 99%

“…In what follows we recall some notions and results regarding the deductive systems and congruences on a pseudo equality algebra (see [15]). Let (A, ∧, ∼, ∽, 1) be a pseudo equality algebra.…”

Section: Proposition 22 ([15])mentioning

confidence: 99%

“…According to [15,Prop. 4.5], a deductive system D of a pseudo equality algebra A is closed if and only if 1 ∼ x, x ∽ 1 ∈ D for all x ∈ D.…”

Section: Proposition 22 ([15])mentioning

confidence: 99%

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Commutative pseudo-equality algebras

Ciungu

2016

Soft Comput

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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. In this paper we define and study the commutative pseudo equality algebras. We give a characterization of commutative pseudo equality algebras and we prove that an invariant pseudo equality algebra is commutative if and only if its corresponding pseudo BCK(pC)-meet-semilattice is commutative. Other results consist of proving that every commutative pseudo equality algebra is a distributive lattice and every finite invariant commutative pseudo equality algebra is a symmetric pseudo equality algebra. We also introduce and investigate the commutative deductive systems of pseudo equality algebras. As applications of these notions and results we define and study the measures and measure-morphisms on pseudo equality algebras, we prove new properties of state pseudo equality algebras, and we introduce and investigate the pseudo-valuations on pseudo equality algebras.

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Section: Preliminaries On Pseudo Equality Algebrasmentioning

confidence: 99%

Section: Preliminaries On Pseudo Equality Algebrasmentioning

confidence: 99%

Section: Proposition 22 ([15])mentioning

confidence: 99%

“…According to [15,Prop. 4.5], a deductive system D of a pseudo equality algebra A is closed if and only if 1 ∼ x, x ∽ 1 ∈ D for all x ∈ D.…”

Section: Proposition 22 ([15])mentioning

confidence: 99%

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Commutative pseudo-equality algebras

Ciungu

2016

Soft Comput

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“…About BCK/pseudo-BCK algebras and their application, one can see [4][5][6][7]. Recently, Dvurečenskij found the fact that every pseudo equality algebra in the Jenei's version is an equality algebra and so presents the new revision of pseudo equality algebras in [8]. It generalizes equality algebras and seems to be more reasonable as a candidate for a possible algebraic semantics of fuzzy type theory than the Jenei's version.…”

Section: Introductionmentioning

confidence: 99%

Generalized state maps and states on pseudo equality algebras

Cheng

Xin

2018

Open Mathematics

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Abstract:In this paper, we attempt to cope with states in a universal algebraic setting, that is, introduce a notion of generalized state map from a pseudo equality algebra X to an arbitrary pseudo equality algebra Y. We give two types of special generalized state maps, namely, generalized states and generalized internal states. Also, we study two types of states, namely, Bosbach states and Riečan states. Finally, we discuss the relations among generalized state maps, states and internal states (or state operators) on pseudo equality algebras. We verify the results that generalized internal states are the generalization of internal states, and generalized states are the generalization of state-morphisms on pseudo equality algebras. Furthermore, we obtain that generalized states are the generalization of Bosbach states and Riečan states on linearly ordered and involutive pseudo equality algebras, respectively. Hence we can come to the conclusion that, in a sense, generalized state maps can be viewed as a possible united framework of the states and the internal states, the state-morphisms and the internal state-morphisms on pseudo equality algebras.

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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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Pseudo equality algebras: revision

Cited by 19 publications

References 12 publications

Commutative pseudo-equality algebras

Commutative pseudo-equality algebras

Generalized state maps and states on pseudo equality algebras

On pseudo-equality algebras

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