Pakistan Research Repository 







Title
of Thesis 

SELFMAPS AND CATEGORICAL ASPECTS OF BCK/BCLALGEBRAS 

Author(s) 

Muhammad Anwar Chaudhary 

Institute/University/Department
Details 

Bahauddin Zakariya University, Multan / Centre For Advanced Studies In Pure & Applied Mathematics 

Session 

1991 

Subject 

Education 

Number
of Pages 

97 



Keywords
(Extracted from title, table of contents and abstract of thesis) 

BCIAlgebras, Homomorphisms, CoEqualizers, S3–Algebras, PSemisimple Algebras, SelfMaps 



Abstract 

During the past twenty years, various aspects of BCIalgebras were studied by a number of researchers, but so far, the notions like selfmaps of BCIalgebras and Categorical aspects of BCIalgebras 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 BCIalgebras which are used in the subsequent chapters. In chapter 2, we study the configurational structure of BCIalgebras and classify them into different classes. M. Daoje [18] solved the problem of configurational structure of psemisimple BCIalgebras and proved that every psemisimple algebra is an abelian group. We define the centre of a BCIalgebra X and show that it is a psemisimple subalgebra of X, which consequently implies that every BCIalgebra contains a psemisimple BCIalgebra. Noting this, we investigate configurational structure of BCIalgebras and classify them into Sialgebras, i = 1, 2, 3, 4 and study some properties of S3algebra and S4algebra. Certain results of BCIalgebra are also established which are used in the subsequent work reported in this thesis. In chapter 3, we classify ideals of BCIalgebras. K. Iseki and S. Tanaka [33], introduced the concept of an ideal in a BCKalgebra and proved that every ideal in BCK algebra is a subalgebra. K. Iseki [30], extended the notion to BCIalgebras. It has been shown that in a BCIalgebra, every ideal is not necessarily a subalgebra. Thus a question arises what type of ideals are subalgebras? 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 subalgebras, 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 psemisimple subalgebra 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 BCKalgebra X by an ideal A. L. Tiande and X. Changchang [39], extended the concept to BCIalgebras and considered the quotient algebra X/M, where M is thc BCKpart 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 subalgcbra) 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 BCIalgebra remained unexplored. We show that if A is an arbitrary ideal in a BCIalgcbra X, then X/A is also a BCIalgebra. We also show that every arbitrary ideal A in a BCIalgebra 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 BCIalgebras. The homological study of BCIalgebras was initiated by K. Iseki and A. B. Thaheem [36]. They posed the problem whether Hom(X), the set of all BCIhomomorphisms 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 BCIalgebra. However, in [18], it was established that Hom(X) is a BCIalgebra, if X is an associative BCIalgebra. But an associative BCIalgebra is again a psemisimple algebra. Thus homological study of BCIalgebras did not develop for BCIalgebras in general. In [13], [16] and [17], the researchers diverted their attention to weaken the notions of BCKhomomorphisms so that such weak selfmaps could form a BCIalgebra. The notion of leftregular selfmaps was defined and shown that LR(X), the set of all leftregular self maps of a positive implicative BCKalgebra X, forms a positive implicative BCKalgebra. But no such effort was made for BCIalgebras. In this chapter, we give the concepts of a Weak Right Selfmap, Weak Left Self map and Weak LeftRegular Selfmap, which are generalizations of different, selfmaps 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 BCIalgebras in terms of its Right Selfmaps and Weak Right Selfmaps. Further, some properties of Weak Right Self maps, Weak Left Selfmaps and Weak LeftRegular Selfmaps have been studied. It has also been shown taht WLR(X), the set of all Weak LeftRegular Selfmaps of a weakly positive implicative BCIalgebra X, is a weakly positive implicative BCIalgebra. Thus homological study has been made in the class of weakly positive implicative BCIalgebras a class which contains the class of psemisimple BCIalgebras, the class of associative BCIalgebra, the class of weakly implicative BCKalgebras and weakly positive implicative BCKalgebras. We have also defined the notions of annihilators and coannihilators in BCIalgebras 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 BCIalgebras, a class which contains the class of psemisimple BCIalgebras 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 BCKalgebras as its objects and BCKhomomorphisms as its morphisms [32, 40, 41, and 42]. K. Iseki [31] emphasized the importance of the category theoretic approach to BCKalgebras and proved that BCK has limits. H. Yutani [42] proved that BCK has colimits. C. S. H00 [24], initiated the study of BCI, the category of BCIalgebras as its objects and BCI homomorphisms as its morphisms. He proved that oneone 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, coequalizers and kernel pairs. We prove that onto homomorphisms and coequalizers coincide in BCI. Further, we establish that coequalizers and monomorphisms form an image factorization system in BCI. It is shown that MBCI, the category of medial BCIalgebras, is a reflexive subcategory of BCI. Moreover, a functor from BCI into MBCI has been investigated. 



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