# SELF MAPS AND CATEGORICAL ASPECTS OF BCK/BCL ALGEBRAS

Bhatti, Shaban Ali (1991) SELF MAPS AND CATEGORICAL ASPECTS OF BCK/BCL ALGEBRAS. PhD thesis, Bahauddin Zakariya University, Multan.

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## Abstract

During the past twenty years, various aspects of BCI-algebras were studied by a number of researchers, but so far, the notions like self-maps of BCI-algebras and Categorical aspects of BCI-algebras had not been investigated intensively. The purpose of this thesis is to study these notions. We divide the work into five chapters. Chapter 1 gives a brief survey of those results about BCI-algebras which are used in the subsequent chapters. In chapter 2, we study the configurational structure of BCI-algebras and classify them into different classes. M. Daoje [18] solved the problem of configurational structure of p-semisimple BCI-algebras and proved that every p-semisimple algebra is an abelian group. We define the centre of a BCI-algebra X and show that it is a p-semisimple sub-algebra of X, which consequently implies that every BCI-algebra contains a p-semisimple BCI-algebra. Noting this, we investigate configurational structure of BCI-algebras and classify them into Si-algebras, i = 1, 2, 3, 4 and study some properties of S3-algebra and S4-algebra. Certain results of BCI-algebra are also established which are used in the subsequent work reported in this thesis. In chapter 3, we classify ideals of BCI-algebras. K. Iseki and S. Tanaka [33], introduced the concept of an ideal in a BCK-algebra and proved that every ideal in BCK- algebra is a sub-algebra. K. Iseki [30], extended the notion to BCI-algebras. It has been shown that in a BCI-algebra, every ideal is not necessarily a sub-algebra. Thus a question arises -what type of ideals are sub-algebras? In this chapter, we have classified ideals into the following classes: Weak Ideals, (ii) Strong Ideals, (iii) Obstinate Ideals. It is proved that strong ideals and obstinate ideals are always sub-algebras, whereas weak ideals are not necessarily so. In chapter 2, the notion of the centre I of a BCI- algebra X has been defined and shown that I is a p-semisimple sub-algebra of X. We study the relationship of ideals in I and ideals in X. K. Iseki and S. Tanaka [33], studied the quotient algebra X/A of a BCK-algebra X by an ideal A. L. Tiande and X. Changchang [39], extended the concept to BCI-algebras and considered the quotient algebra X/M, where M is thc BCK-part of X and itsclf a sub- algebra. C. S. H00 and P. V. R. Murty [25], gave the idea of a closcd ideal (an ideal which is also a sub-algcbra) and investigated its proper tics in terms of quoticnt algebras along with some others. But the quotient structure with respect to an arbitrary ideal of a BCI-algebra remained un-explored. We show that if A is an arbitrary ideal in a BCI-algcbra X, then X/A is also a BCI-algebra. We also show that every arbitrary ideal A in a BCI-algebra X contains a closed ideal R(A). A. A. Siddiqui and M. A. Chaudhry [46], established some isomorphism theorems constructing the quotient algebras with respect to closed ideals. We study them for arbitrary ideals and prove that the results shown in [46] are particular cases of our results contained in this chapter. In chapter 4, we undertake the homological study of BCI-algebras. The homological study of BCI-algebras was initiated by K. Iseki and A. B. Thaheem [36]. They posed the problem whether Hom(X), the set of all BCI-homomorphisms of X into itself, is a BCI- algcbra or not? In [44], E. Y. Deeba and S. K. Goal proved that Hom(X) is not always a BCI-algebra. However, in [18], it was established that Hom(X) is a BCI-algebra, if X is an associative BCI-algebra. But an associative BCI-algebra is again a p-semisimple algebra. Thus homological study of BCI-algebras did not develop for BCI-algebras in general. In [13], [16] and [17], the researchers diverted their attention to weaken the notions of BCK-homomorphisms so that such weak self-maps could form a BCI-algebra. The notion of left-regular self-maps was defined and shown that LR(X), the set of all left-regular self- maps of a positive implicative BCK-algebra X, forms a positive implicative BCK-algebra. But no such effort was made for BCI-algebras. In this chapter, we give the concepts of a Weak Right Self-map, Weak Left Self- map and Weak Left-Regular Self-map, which are generalizations of different, self-maps introduced in [13], [16] and [17]. Our study is so general that all results proved therein become special cases of our results. We have characterized weakly positive implicative BCI-algebras in terms of its Right Self-maps and Weak Right Self-maps. Further, some properties of Weak Right Self- maps, Weak Left Self-maps and Weak Left-Regular Self-maps have been studied. It has also been shown taht WLR(X), the set of all Weak Left-Regular Self-maps of a weakly positive implicative BCI-algebra X, is a weakly positive implicative BCI-algebra. Thus homological study has been made in the class of weakly positive implicative BCI-algebras a class which contains the class of p-semisimple BCI-algebras, the class of associative BCI-algebra, the class of weakly implicative BCK-algebras and weakly positive implicative BCK-algebras. We have also defined the notions of annihilators and co-annihilators in BCI-algebras and shew that all results of [13] become particular cases of our results. Further, we inintiate duality theory in the class of weakly positive implicative BCI-algebras, a class which contains the class of p-semisimple BCI-algebras chosen by M. Aslam and A. B. Thaheem [43], for the study of duality theory. Chapter 5 is motivated by the notions of BCK, the category of BCK-algebras as its objects and BCK-homomorphisms as its morphisms [32, 40, 41, and 42]. K. Iseki [31] emphasized the importance of the category theoretic approach to BCK-algebras and proved that BCK has limits. H. Yutani [42] proved that BCK has co-limits. C. S. H00 [24], initiated the study of BCI, the category of BCI-algebras as its objects and BCI- homomorphisms as its morphisms. He proved that one-one homomorphisms and monomorphisms coincide in BCI. He also proved that onto homomorphisms are epimorphisms in BCI. But it yet remains to be shown that epimorphisms in BCI are onto homomorphisms. It is an open problem. In this chapter, we partially solve this problem and study some properties of BCI. We show that BCI has limits, co-equalizers and kernel pairs. We prove that onto homomorphisms and co-equalizers coincide in BCI. Further, we establish that co-equalizers and monomorphisms form an image factorization system in BCI. It is shown that MBCI, the category of medial BCI-algebras, is a reflexive sub-category of BCI. Moreover, a functor from BCI into MBCI has been investigated.

Item Type: Thesis (PhD) BCI-Algebras, Homomorphisms, Co-Equalizers, Algebras, P-Semisimple Algebras, Self-Maps Social Sciences(g) > Education(g6) 313 Mr Ghulam Murtaza 14 Dec 2006 04 Oct 2007 21:00

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