Studia Logica

, Volume 53, Issue 2, pp 227–234

On a property of BCK-identities

  • Misao Nagayama
Article

DOI: 10.1007/BF01054710

Cite this article as:
Nagayama, M. Stud Logica (1994) 53: 227. doi:10.1007/BF01054710

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.

Copyright information

© Kluwer Academic Publishers 1994

Authors and Affiliations

  • Misao Nagayama
    • 1
  1. 1.Department of MathematicsTokyo Woman's Christian UniversityTokyoJapan