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

Fifty years of similarity relations: a survey of foundations and applications

On the occasion of the 50th anniversary of the publication of Zadeh's significant paper Similarity Relations and Fuzzy Orderings, an account of the development of similarity relations during this time will be given. Moreover, the main topics related to these fuzzy relations will be reviewed.Peer ReviewedPostprint (author's final draft

Classification of Distinct Fuzzy Subgroups of the Dihedral Group Dp nq for p and q distinct primes and n ∈ N

In this dissertation, we classify distinct fuzzy subgroups of the dihedral group Dpnq, for p and q distinct primes and n ∈ N, under a natural equivalence relation of fuzzy subgroups and a fuzzy isomorphism. We aim to present formulae for the number of maximal chains and the number of distinct fuzzy subgroups of this group. Our study will include some theory on non-abelian groups since the classification of distinct fuzzy subgroups of this group relies on the crisp characterization of maximal chains. We give the definition of a natural equivalence relation introduced by Murali and Makamba in [67] which we will use in this study. Based on this definition, we introduce two counting techniques that we will use to compute the number of distinct fuzzy subgroups of Dpnq. In this dissertation, we use the criss-cut counting technique as our primary method of enumeration, and the cross-cut method serves as a means of verifying results we obtain from our primary method. To classify distinct fuzzy subgroups of this group, we begin by investigating the dihedral groups Dpnq, for p and q distinct primes and specific values of n = 2 and 3 to observe a trend. We classify the flags of these groups using the characterization of flags introduced in [93]. From this characterization, we then present formulae for the number of distinct fuzzy subgroups attributed to the flags of Dp 2q and Dp 3q . To generalise results for Dpnq, for p and q distinct primes and n ∈ N, we characterize the flags of this group and classify them as either cyclic, mdcyclic for 1 ≤ m ≤ n, or b-cyclic. Finally, we establish a general formula for the number of distinct fuzzy subgroups obtainable from these flags. We conclude by comparing results obtained from using our general formula to those obtained by other researchers for the same group. Based on the results from this study, we give an outline of future research wor

On the number of distinct fuzzy subgroups of dihedral group of order 60

In this paper, we compute the number of distinct fuzzy subgroups of dihedral group of order 60 with respect to a new equivalence relation existing in literature. Our computation shows that the number of distinct fuzzy subgroups of dihedral group of order 60 is 150

Deformed matrix models, supersymmetric lattice twists and N=1/4 supersymmetry

A manifestly supersymmetric nonperturbative matrix regularization for a twisted version of N=(8,8) theory on a curved background (a two-sphere) is constructed. Both continuum and the matrix regularization respect four exact scalar supersymmetries under a twisted version of the supersymmetry algebra. We then discuss a succinct Q=1 deformed matrix model regularization of N=4 SYM in d=4, which is equivalent to a non-commutative $A_4^*$ orbifold lattice formulation. Motivated by recent progress in supersymmetric lattices, we also propose a N=1/4 supersymmetry preserving deformation of N=4 SYM theory on $\R^4$. In this class of N=1/4 theories, both the regularized and continuum theory respect the same set of (scalar) supersymmetry. By using the equivalence of the deformed matrix models with the lattice formulations, we give a very simple physical argument on why the exact lattice supersymmetry must be a subset of scalar subalgebra. This argument disagrees with the recent claims of the link approach, for which we give a new interpretation.Comment: 47 pages, 3 figure

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