Study of Properties f BCK and BCI-Agebras and Thei Categorical Aspects

Abstract

BCK and BCI-algebras were introduced by Japanese mathematicians K. Iseki and Y. Imai in 1966 as generalization of certain properties of propositional calculi. The major part of the research done on these algebraic structures is due to Japanese, Polish, Chinese, Canadian, Pakistani and Indian mathematicians namely, K. Iseki, S. Tanaka, H. Yutani, M. Palan ski, Q.P. Hu, M. Daoji, C.S. Hoo, J. Ahsan, A.B. Taheem, M.A. Chaudhry, B. Ahmad, S.K. Goel, R. Murti and many others. They discovered the first order theory of these algebraic structures and studied various aspects including ideal theory. But the categorial aspects of BCI-algebras remained un-investigated. The classification of BCK-algebras like positive implicative, Implicated, commutative and bounded BCK-algebras are found in the literature but their generalization regarding BCI-algebras are yet to be investigated. In this project we have studied he categorial aspects of BCI-algebras. We have also investigated classification of BCI-algebras. The report is divided into two parts. In part I, we have investigated the category of BCI-algebras and have proved that: (1) The category BCI of BCI-algebras and BCI homomorphisms has products and equalizers. (2) Let f ϵ BCI (x,y) be onto, then f is a coequalizer. (3) The category BCI has Kernel pairs. (4) In BCI, coequalizers and onto homomorphisms coincide. (5) The restriction of BCI-monomorphism f: x → y to the centres fG: XG→YG is a monomorphism as well as one-one. (6) Epimorphisms in MBCI, the category of medical BCI- algebras and BCI homomorphisms, are onto. (7) The category BCI has co-equalizers. (8) (coequalizers, monomorphisms) form an image factorization system in the category BCI. (9) The category MBCI is reflexive subcategory of BCI. In part-II we have introduced two new classes of BCI-algebras namely the class of weakly positive implicative BCI-algebras and he class of weaky implicative BCI-algebras. The class of weakly positive implicative BCI-algebras contains the class of weakly implicative BCI-algebras, the class of medical BCI-algebras, the class of associative BCI-algebras and the class of positive implicative BCK-algebras, where as the class of weakly implicative BCI-algebras contains the class of medical BCI-algebras, the class of associative BCI-algebras and the class of implicative BCK-algebras. We have further proved that: (10) Let X be a BCI-algebra. X is weakly positive implicative if an only if X*y = ((x*y)*y)*(o*y). (11) Every weakly positive implicative BCI-algebra is quasi commutative of type (I,j;j,i+1) for all I ≥ o, j ≥ 1. (12) Every weakly implicative BCI-algebra is weakly positive implicative. (13) A weakly implicative BCI-algebra x with weak unit is weakly commutative. (14) A weakly implicative BCI-algebra with weak nit is weakly positive implicative and weakly commutative. The investigation of categorical properties will help in establishing the relationship of BCI-algebras with other algebraic structures, whereas introduction of two new classes of BCI-algebras and their classification will help in investigating the structure of the branches of BCI-algebras. Further the results regarding ideal theory of BCK-algebras may be extended now to BCI-algebras.