2014
DOI: 10.1016/j.fss.2013.12.007
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State BCK-algebras and state-morphism BCK-algebras

Abstract: In the paper, we define the notion of a state BCK-algebra and a state-morphism BCKalgebra extending the language of BCK-algebras by adding a unary operator which models probabilistic reasoning. We present a relation between state operators and state-morphism operators and measures and states on BCK-algebras, respectively. We study subdirectly irreducible state (morphism) BCKalgebras. We introduce the concept of an adjoint pair in BCK-algebras and show that there is a oneto-one correspondence between adjoint pa… Show more

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Cited by 29 publications
(15 citation statements)
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“…Then for all x, y, z ∈ H and for all nonempty subsets A and B of H the following hold (see [4] and [5, Proposition 3.6]):…”
Section: Preliminariesmentioning
confidence: 99%
See 1 more Smart Citation
“…Then for all x, y, z ∈ H and for all nonempty subsets A and B of H the following hold (see [4] and [5, Proposition 3.6]):…”
Section: Preliminariesmentioning
confidence: 99%
“…Jun et al [22] introduced and studied hyperBCK-algebra which is a generalization of a BCK-algebra. Borzooei et al constructed the hyper Kalgebras, and studied (weak) implicative hyper K-ideals in hyper K-algebras (see [3,4,5]). Fuzzy sets and hyperstructures introduced by Zadeh and Marty, respectively, are now used in the real world, both from a theoretical point of view and for their many applications.…”
Section: Introductionmentioning
confidence: 99%
“…A more powerful type of logic can be given by algebraic structures with internal states, and they also constitute the varieties of universal algebras. Consequently, the internal states have been extended and intensively studied in other algebraic structures [17][18][19], etc. Recently, the notions of internal states have been applied to algebraic structures of higher order fuzzy logic, for example, equality algebras [20] and pseudo equality algebra [21] where one of the main results is about the relevance with the corresponding state BCK/pseudo-BCK meet-semilatices.…”
Section: Introductionmentioning
confidence: 99%
“…The latter built a different n-valued logical system in 1921 [17]. Then Lukasiewicz and Tarski developed in 1930 [15] a logic for which the truth values are the rationales in [0,1]. In 1940, Gr.C.…”
Section: Introductionmentioning
confidence: 99%
“…Specially, they clarified the relation between the class of hyper M V -algebras and hyper Kalgebras [2]. R. A. Borzooei et al [1] proved that these relations are not true, which unfortunately is used to prove some important results of several hyper M V -algebras paper. L. Torkzadeh et al [18] discussed hyper M V -ideals and define some hyperoperations on it.…”
Section: Introductionmentioning
confidence: 99%