内容简介
This book is mainly designed for the graduate students who are in-terested in the theory of BCK and BCI-algebras. BCI-algebras are a wider class than BCK-algebras， introduced by Kiyoshi Iseki in 1966. BCI-algebras as a class of logical algebras are the algebraic formulations of the set difference together with its pro-perties in set theory and the implicational functor in logical systems.They are closely related to partially ordered commutative monoids as well as various logical algebras. Their names are originated from the combinators B， C， K and I in combinatory logic. The early re-search work was mainly carried out among the Japanese mathemati-cians Kiyoshi Is~ki and Shotaro Tanaka， etc. who did a great deal of foundation work. Since late 1970s， their work has been paid much attention. In particular， the participation in the research of Polish mathematicians Tadeusz Traczyk and Andrzej Wronski as well as Australian mathematician William H. Cornish， etc. is making this branch of algebra develop rapidly. Many interesting and important results are discovered continuously. Now， the theory of BCI-algebras has been widely spread.
目录
Chapter 0 Introduction
0.1 Mappings Abelian Groups Binary Relations
0.2 Lattices Boolean Algebras
Chapter 1 General Theory
1.1 Definition and Elementary Properties
1.2 BCK-Algebras
1.3 p-Semisimple Algebras
1.4 Ideals
1.5 Congruences and Quotient Algebras
1.6 BCI-Homomorphisms
1.7 Direct Sums and Direct Products
Chapter 2 Commutative BCK-Algebras
2.1 Definition and Elementary Properties
2.2 Commutative BCK-Lattices
2.3 Structures of Certain Commutative BCK-Algebras
2.4 Generalized Commutative BCK-Algebras
2.5 Commutative Ideals
2.6 Quasi-Commutative BCI-Algebras
Chapter 3 Positive Implicative and Implicative BCK-Algebras
3.1 Positive Implicative BCK-Algebras
3.2 Implicative BCK-Algebras
3.3 Positive Implicative and Implicative BCI-Algebras
3.4 Positive Implicative Ideals
3.5 Implicative Ideals
Chapter 4 BCI-Algebras with Condition （S）
4.1 Definition and Elementary Properties
4.2 Commutative BCK-Algebras with Condition （S）
4.3 Positive Implicative BCK-Algebras with Condition （S）
4.4 Implicative BCK-Algebras with Condition （S）
4.5 Commutative Residual Pomonoids
Chapter 5 Normal BCI-Algebras
5.1 Simple BCI-Algebras
5.2 Semisimple BCI-Algebras
5.3 J-Semisimple BCI-Algebras
5.4 Normal BCI-Algebras
5.5 Normal Ideals in BCK-Algebras
Chapter 6 Radicals and Ideals
6.1 Radicals
6.2 p-Semisimple Ideals
6.3 Associative and Quasi-Associative Ideals
6.4 Irreducible Ideals
Appendix A
Appendix B
Bibliography
Index
