BCK-algebras.

*(English)*Zbl 0906.06015
Korea: Kyung Moon Sa Co. vi, 294 p. (1994).

BCK-algebras were introduced by K. Iséki about thirty years ago as a generalization of some ideas of set theory, and also of some ideas in propositional calculus. Precisely, a set \(X\) with a binary operation \(*\) and a constant 0 is a BCK-algebra if it satisfies the following axioms:
\[
(1)\quad \bigl\{(x*y)* (x*z)\bigr\}* (z*y)=0, \qquad (2)\quad \bigl\{x*(x*y) \bigr\}*y=0, \qquad (3) \quad x*x=0,
\]

\[ (4)\quad x*y=0 \text{ and } y*x=0 \text{ imply that }x=y, \qquad (5) \quad 0*x=0. \] For example, the set of nonnegative integers with operation given by \(m*n=m-n\) if \(n\leq m\), and \(m*n=0\) if \(n>m\), is a BCK-algebra. The subject was studied intensively by Iséki and his Japanese colleagues and they obtained most of the basic results. Other non-Japanese friends soon joined in elaborating the theory (the reviewer among them), and the subject has gained some popularity. If one drops axiom (5) above, one obtains a BCI-algebra. Iséki and his colleagues did this and they again obtained a lot of the basic results. As usual, others followed, the reviewer among them. The authors also participated in both cases. If one imposes further axioms, one obtains a richer theory. For example, by requiring the existence of a largest element and requiring commutativity, that is, the axiom \(x*(x*y)= y*(y*x)\), one obtains the theory of MV-algebras. There is now a relatively rich theory of such algebras and quite a number of people work in this area. Of course, in the ultimate, by imposing further axioms, one obtains the theory of Boolean algebras.

This book deals strictly with BCK-algebras and not much attention is paid to the more general theory of BCI-algebras or the richer theory of MV-algebras. The authors state that the text is designed primarily for the graduate student who wants to learn the basic ideas and techniques of BCK-algebras. The treatment is extremely elementary and only the most basic knowledge of mathematics is assumed. Also only the most basic results of BCK-algebras are considered. They deal with all the usual basic ideas of algebras, stated especially for BCK-algebras. They also include a final chapter on more recent results, for example on free, injective, projective and fuzzy BCK-algebras. However, these topics are only skimmed. This book is suitable if a student wants a quick review of the fundamentals of the subject. It is convenient to have them all collected in one place. The level is elementary and the novice is not greatly taxed. This reviewer has noticed only a few errors.

\[ (4)\quad x*y=0 \text{ and } y*x=0 \text{ imply that }x=y, \qquad (5) \quad 0*x=0. \] For example, the set of nonnegative integers with operation given by \(m*n=m-n\) if \(n\leq m\), and \(m*n=0\) if \(n>m\), is a BCK-algebra. The subject was studied intensively by Iséki and his Japanese colleagues and they obtained most of the basic results. Other non-Japanese friends soon joined in elaborating the theory (the reviewer among them), and the subject has gained some popularity. If one drops axiom (5) above, one obtains a BCI-algebra. Iséki and his colleagues did this and they again obtained a lot of the basic results. As usual, others followed, the reviewer among them. The authors also participated in both cases. If one imposes further axioms, one obtains a richer theory. For example, by requiring the existence of a largest element and requiring commutativity, that is, the axiom \(x*(x*y)= y*(y*x)\), one obtains the theory of MV-algebras. There is now a relatively rich theory of such algebras and quite a number of people work in this area. Of course, in the ultimate, by imposing further axioms, one obtains the theory of Boolean algebras.

This book deals strictly with BCK-algebras and not much attention is paid to the more general theory of BCI-algebras or the richer theory of MV-algebras. The authors state that the text is designed primarily for the graduate student who wants to learn the basic ideas and techniques of BCK-algebras. The treatment is extremely elementary and only the most basic knowledge of mathematics is assumed. Also only the most basic results of BCK-algebras are considered. They deal with all the usual basic ideas of algebras, stated especially for BCK-algebras. They also include a final chapter on more recent results, for example on free, injective, projective and fuzzy BCK-algebras. However, these topics are only skimmed. This book is suitable if a student wants a quick review of the fundamentals of the subject. It is convenient to have them all collected in one place. The level is elementary and the novice is not greatly taxed. This reviewer has noticed only a few errors.

Reviewer: C.S.Hoo (Edmonton)