On hyper BCC-algebras

نویسندگان

Rajab Ali Borzooei

Wieslaw A. Dudek

N. Koohestani

چکیده

The study of BCK-algebras was initiated by Iséki [7] as a generalization of the concept of set-theoretic difference and propositional calculus. Iséki posed an interesting problem; whether the class of BCK-algebras is a variety. In connection with this problem Komori introduced in [9] a notion of BCC-algebra which is a generalization of a BCK-algebra and proved that the class of all BCC-algebras is not a variety. Dudek [3–5] followed this theory and has got a lot of related results. BCC-algebras are algebraic models of BCClogic, implicational logic whose axiom schemes are the principal-type schemes of the combinators B, I , andK , and whose inference rules aremodus ponens andmodus ponens 2. So, in fact, such algebras ought to have been named BIK-algebras. In this convention, a BCK-algebra is a BCC-algebra satisfying the identity y → (x→ z)= x→ (y → z). The hyperstructure theory (called also multialgebras) was introduced by Marty [10] at the eighth congress of Scandinavian mathematicians. Now hyperstructures of different types have many important applications (see, e.g., [1, 2]). In this paper, we study hyper BCC-algebras which are a common generalization of BCC-algebras and hyper BCK-algebras investigated by many authors. In particular, we investigate different types of hyper BCC-ideals and describe the relationship among them. Next, we calculate all nonisomorphic 22 hyper BCC-algebras of order 3 of which only three are not hyper BCK-algebras.

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