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On a Property of BCK-Identities

Misao Nagayama
Studia Logica: An International Journal for Symbolic Logic
Vol. 53, No. 2 (May, 1994), pp. 227-234
Published by: Springer
Stable URL: http://www.jstor.org/stable/20015718
Page Count: 8
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On a Property of BCK-Identities
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Abstract

A BCK-algebra is an algebra in which the terms are generated by a set of variables, 1, and an arrow. We mean by a BCK-identity an equation valid in all BCK-algebras. In this paper using a syntactic method we show that for two terms s and t, if neither s = 1 nor t = 1 is a BCK-identity, and s = t is a BCK-identity, then the rightmost variables of the two terms are identical. This theorem was conjectured firstly in [5], and then in [3]. As a corollary of this theorem, we derive that the BCK-algebras do not form a variety, which was originally proved algebraically by Wroński ([4]). To prove the main theorem, we use a Gentzen-type logical system for the BCK-algebras, introduced by Komori, which consists of the identity axiom, the right and the left introduction rules of the implication, the exchange rule, the weakening rule and the cut. As noted in [2], the cut-elimination theorem holds for this system.

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