The subjects described in this book are BCC-algebras and an even wider class of weak BCC-algebras. The aim of the book is to summarize the achievements to date in the subject and to present them in the form of a logically created theory. Through appropriate grading and a precise description of the steps of the proofs, this theory is easily assimilated, and it should not take too long for the reader to learn about it.
We begin with the motivation for their creation, many examples, and basic results used later in the book. Then we deal with the constructions of BCC-algebras and calculate the numbers of their subalgebras.
The author describes the so-called solid weak BCC-algebras. They have some properties of BCI-algebras, but this requires completely new, often difficult, proofs. The important subclasses of weak BCC-algebras and the relationships between them are presented with many examples.
BCC-Algebras is intended for researchers dealing with abstract algebra and for logicians working on the border between logic and algebra. The book is also of interest to students interested in the theory of (weak) BCC-algebras or simply in abstract algebra.
The structure of the book makes it possible to discover topics that require further research, which, depending on the degree of difficulty, may be completed with a thesis or dissertation.
Preface
1 BCC-Algebras - Introduction
1.1 Basic Definitions and Facts
1.2 Constructions of BCC-Algebras
1.3 Estimating the Number of Subalgebras
2 Special Objects in Weak BCC-Algebras
2.1 Atoms
2.2 Branches
2.3 BCC-
2.4 Congruences
2.5 Group-Like Weak BCC-Algebras
2.6 Solid Weak BCC-Algebras
2.7 Nilpotent Weak BCC-Algebras
3 Subclasses of BCC-Algebras
3.1 Commutative Solid Weak BCC-Algebras
3.2 Quasi-Commutative Solid Weak BCC-Algebras
3.3 Implicative Solid Weak BCC-Algebras
3.4 Weak BCC-Algebras with Condition (S)
3.5 Initial Segments
3.6 Fuzzy BCC-Subalgebras
3.7 Derivations of Weak BCC-Algebras
3.8 Para-Associative Weak BCC-Algebras
3.9 Hyper (Weak) BCC-Algebras
3.10 Group-Like Hyper Weak BCC-Algebras
3.11 Soft BCC-Algebras
4 Ideal Theory of Weak BCC-Algebras
4.1 Closed Ideals
4.2 T-Ideals of T-Type Weak BCC-Algebras
4.3 Anti-Grouped Ideals
4.4 Associative Ideals
4.5 p-Ideals
4.6 k-Nil Radicals of Solid Weak BCC-Algebras
4.7 Fuzzy BCC-Ideals
4.8 Cubic Bipolar BCC-Ideals
4.9 Soft BCC-Ideals
Appendix
Bibliography
Index
Biography
Janus Thomys received his Master of Sciences in Mathematics and his Dr. Mathematical Sciences degree at University Jan Dlugosz, Czestohowa, Poland.
