BCIalgebraDistributed 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 BCIalgebras. It introduces the general theoretical basis of BCIalgebras, 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 easytofollow steps into this field.

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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 BCIhomomorphism BCIordering BCKalgebra of order BCKalgebra with condition BCKpart binary operation Boolean algebra called commutative BCKalgebra commutative BCKchain commutative BCKlattice 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 Icongruence ideal of X irreducible ideals isomorphic Lemma lower BCKsemilattice lower semilattice mapping maximal ideal minimal element multiply commutative multiply implicative multiply positive implicative nfold commutative nfold implicative nfold positive implicative natural number nonempty subset nonzero element obtain Obviously psemisimple algebra partially ordered set pomonoid positive implicative BCKalgebra positive implicative ideal proper BCIalgebra proper ideal Proposition quasiassociative quotient algebra X/A rideal satisfies the following semilattice Show subalgebra subdirect product subdirect sum submonoid Suppose Theorem upper semilattice Verify x o y Yliei zero ideal