A study of fuzzy sets and systems with applications to group theory and decision making

In this study we apply the knowledge of fuzzy sets to group structures and also to decision-making implications. We study fuzzy subgroups of finite abelian groups. We set G = Z[subscript p[superscript n]] + Z[subscript q[superscript m]]. The classification of fuzzy subgroups of G using equivalence classes is introduced. First, we present equivalence relations on fuzzy subsets of X, and then extend it to the study of equivalence relations of fuzzy subgroups of a group G. This is then followed by the notion of flags and keychains projected as tools for enumerating fuzzy subgroups of G. In addition to this, we use linear ordering of the lattice of subgroups to characterize the maximal chains of G. Then we narrow the gap between group theory and decision-making using relations. Finally, a theory of the decision-making process in a fuzzy environment leads to a fuzzy version of capital budgeting. We define the goal, constraints and decision and show how they conflict with each other using membership function implications. We establish sets of intervals for projecting decision boundaries in general. We use the knowledge of triangular fuzzy numbers which are restricted field of fuzzy logic to evaluate investment projections

A survey of the classification of fuzzy subgroups of some finite groups

In this lecture we survey the classification of fuzzy subgroups of finite groups as studied byProf. B.B Makamba and V. Murali. We present the impact of the research on our postgraduate students. The classification is focusing on finite abelian p-groups and dihedral groups, giving a mixture of abelian and non-abelian groups. We show some highlights and what still needs to be done in the classification of fuzzy subgroups. We also touch on what other researchers have achieved in the classification of fuzzy subgroups and how our work is related to theirs. We begin with a historical background of fuzzy logic.Inaugural Lecture Address by Prof. Babington Makamba- A survey of the classification of fuzzy subgroups of some finite groups

COMPUTING THE NUMBER OF FUZZY SUBGROUPS BY EXPANSION METHOD

Abstract: The purpose of this paper is to construct the method to find the number of fuzzy subgroups for rectangle groups. The method is formulated by using the pattern of their lattice diagram. We construct the expansion methods, raw and column expansion

The classification of fuzzy groups of finite cyclic groups Zpn Zqm Zr and Zp1 Zp2 Zpn for distinct prime numbers p; q; r; p1; p2; ; pn and n;m 2 Z+

Let G be the cyclic group Zpn _ Zqm _ Zr where p; q; r are distinct primes and n;m 2 Z+. Using the criss-cut method by Murali and Makamba, we determine in general the number of distinct fuzzy subgroups of G. This is achieved by using the maximal chains of subgroups of the respective groups, and the equivalence relation given in their research papers. For cases of m, the number of fuzzy subgroups is _rst given, from which the general pattern for G is achieved. Murali and Makamba discussed the number of fuzzy subgroups of Zpn _ Zqm using the cross-cut method. A brief revisit of the group Zpn _Zqm is done using the criss-cut method. The formulae for _nding the number of distinct fuzzy subgroups in each of the cases is given and proofs provided. Furthermore, we classify the fuzzy subgroups of the group Zp1_Zp2__ _ __Zpn for p1; p2; _ _ _ ; pn distinct primes and n 2 Z+ using the criss-cut method. An algorithm for counting the distinct fuzzy subgroups of this group is developed

The classsification of fuzzy subgroups of some finite Abelian p-groups of rank 3

An important trend in fuzzy group theory in recent years has been the notion of classification of fuzzy subgroups using a suitable equivalence relation. In this dissertation, we have successfully used the natural equivalence relation defined by Murali and Makamba in [81] and a natural fuzzy isomorphism to classify fuzzy subgroups of some finite abelian p-groups of rank three of the form Zpn + Zp + Zp for any fixed prime integer p and any positive integer n. This was achieved through the usage of a suitable technique of enumerating distinct fuzzy subgroups and non-isomorphic fuzzy subgroups of G. We commence by giving a brief discussion on the theory of fuzzy sets and fuzzy subgroups from the perspective of group theory through to the theory of sets, leading us to establish a linkage among these theories. We have also shown in this dissertation that the converse of theorem 3.1 proposed by Das in [24] is incorrect by giving a counter example and restate the theorem. We have then reviewed and enriched the study conducted by Ngcibi in [94] by characterising the non-isomorphic fuzzy subgroups in that study. We have also developed a formula to compute the crisp subgroups of the under-studied group and provide its proof. Furthermore, we have compared the equivalence relation under which the classification problem is based with various versions of equivalence studied in the literature. We managed to use this counting technique to obtain explicit formulae for the number of maximal chains, distinct fuzzy subgroups, non-isomorphic maximal chains and non-isomorphic fuzzy subgroups of these groups and their proofs are provided

The classification of fuzzy subgroups of some finite non-cyclic abelian p- groups of rank 3, with emphasis on the number of distinct fuzzy subgroups

In [6] and [7] we classi_ed fuzzy subgroups of some rank-3 abelian groups of the form G = Zpn + Zp + Zp for any _xed prime integer p and any positive integer n, using the natural equivalence relation de_ned in [40]. In this thesis, we extend our classi_cation of fuzzy subgroups in [6] to the group G = Zpn + Zpm + Zp for any _xed prime integer p; m = 2 and any positive integer n using the same natural equivalence relation studied in [40]. We present and prove explicit polynomial formulae for the number of (i) subgroups, (ii) maximal chains of subgroups of G for any n;m _ 2 and (iii) distinct fuzzy subgroups for m = 2 and n _ 2. We have also developed user-friendly polynomial formulae for the number of (iv) subgroups, (v) maximal chains for the group G = Zpn + Zpm for any n;m _ 2; any _xed prime positive integer p and (vi) distinct fuzzy subgroups of Zpn + Zpm for m equal to 2 and 3, and n _ 2 and provided their proofs.Thesis (PhD) -- Faculty of Science and Agriculture, 202

The classification of fuzzy subgroups of some finite non-cyclic abelian p- groups of rank 3, with emphasis on the number of distinct fuzzy subgroups

In [6] and [7] we classi_ed fuzzy subgroups of some rank-3 abelian groups of the form G = Zpn + Zp + Zp for any _xed prime integer p and any positive integer n, using the natural equivalence relation de_ned in [40]. In this thesis, we extend our classi_cation of fuzzy subgroups in [6] to the group G = Zpn + Zpm + Zp for any _xed prime integer p; m = 2 and any positive integer n using the same natural equivalence relation studied in [40]. We present and prove explicit polynomial formulae for the number of (i) subgroups, (ii) maximal chains of subgroups of G for any n;m _ 2 and (iii) distinct fuzzy subgroups for m = 2 and n _ 2. We have also developed user-friendly polynomial formulae for the number of (iv) subgroups, (v) maximal chains for the group G = Zpn + Zpm for any n;m _ 2; any _xed prime positive integer p and (vi) distinct fuzzy subgroups of Zpn + Zpm for m equal to 2 and 3, and n _ 2 and provided their proofs.Thesis (PhD) -- Faculty of Science and Agriculture, 202