## BCI-algebraDistributed by Elsevier Science on behalf of Science Press. This book is mainly designed for graduate students who are interested in the theory of BCK and BCI-algebras. It introduces the general theoretical basis of BCI-algebras, omitting difficult proofs and abstract topics which are less necessary for beginners to learn. With abundant examples and exercises arranged after each section, it provides readers with easy-to-follow steps into this field. * Specially designed for graduate students with emphasis on elementary knowledge in this field * Organizes knowledge points systematically and highlights various arguments on vital topics to make them easy to be understand * Gives many examples to clarify important notations and terminologies and abundant of classified exercises after each chapter for revision purposes |

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### Contents

Introduction | 1 |

General Theory | 11 |

Commutative BCKAlgebras | 93 |

Positive Implicative and Implicative | 153 |

BCIAlgebras with Condition S | 209 |

Normal BCIAlgebras | 259 |

Radicals and Ideals | 291 |

Appendix A | 313 |

Appendix B | 336 |

343 | |

353 | |

### Common terms and phrases

abelian group Assume BCI-homomorphism BCI-ordering BCK-algebra of order BCK-algebra with condition BCK-part binary operation Boolean algebra called commutative BCK-algebra commutative BCK-chain commutative BCK-lattice commutative ideal commutative law congruence Corollary Define a binary Definition Denote distributive lattice epimorphic Example Exercise exists finite order following are equivalent following holds greatest element Hasse diagram Hence I-congruence ideal of X irreducible ideals isomorphic Lemma lower BCK-semilattice lower semilattice mapping maximal ideal minimal element multiply commutative multiply implicative multiply positive implicative n-fold commutative n-fold implicative n-fold positive implicative natural number nonempty subset nonzero element obtain Obviously p-semisimple algebra partially ordered set pomonoid positive implicative BCK-algebra positive implicative ideal proper BCI-algebra proper ideal Proposition quasi-associative quotient algebra X/A r-ideal satisfies the following semilattice Show subalgebra subdirect product subdirect sum submonoid Suppose Theorem upper semilattice Verify x o y Yliei zero ideal