Studia Logica

, Volume 53, Issue 2, pp 227–234

# On a property of BCK-identities

• Misao Nagayama
Article

## 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 aBCK-identity an equation valid in all BCK-algebras. In this paper using a syntactic method we show that for two termss andt, if neithers=1 nort=1 is a BCK-identity, ands=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.

## Keywords

Mathematical Logic Computational Linguistic Logical System Introduction Rule Exchange Rule
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

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A. Wroński,BCK-Algebras do not Form a Variety Math. Japonica,28 2 (1983), pp. 211–213.Google Scholar
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A. Wroński,Reflections and Distentions of BCK-Algebras Math. Japonica,28 2 (1983), pp. 215–225.Google Scholar