Studies of equivalent fuzzy subgroups of finite abelian p-Groups of rank two and their subgroup lattices

We determine the number and nature of distinct equivalence classes of fuzzy subgroups of finite Abelian p-group G of rank two under a natural equivalence relation on fuzzy subgroups. Our discussions embrace the necessary theory from groups with special emphasis on finite p-groups as a step towards the classification of crisp subgroups as well as maximal chains of subgroups. Unique naming of subgroup generators as discussed in this work facilitates counting of subgroups and chains of subgroups from subgroup lattices of the groups. We cover aspects of fuzzy theory including fuzzy (homo-) isomorphism together with operations on fuzzy subgroups. The equivalence characterization as discussed here is finer than isomorphism. We introduce the theory of keychains with a view towards the enumeration of maximal chains as well as fuzzy subgroups under the equivalence relation mentioned above. We discuss a strategy to develop subgroup lattices of the groups used in the discussion, and give examples for specific cases of prime p and positive integers n,m. We derive formulas for both the number of maximal chains as well as the number of distinct equivalence classes of fuzzy subgroups. The results are in the form of polynomials in p (known in the literature as Hall polynomials) with combinatorial coefficients. Finally we give a brief investigation of the results from a graph-theoretic point of view. We view the subgroup lattices of these groups as simple, connected, symmetric graphs

Counting of finite fuzzy subsets with applications to fuzzy recognition and selection strategies

The counting of fuzzy subsets of a finite set is of great interest in both practical and theoretical contexts in Mathematics. We have used some counting techniques such as the principle of Inclusion-Exclusion and the Mõbius Inversion to enumerate the fuzzy subsets of a finite set satisfying different conditions. These two techniques are interdependent with the M¨obius inversion generalizing the principle of Inclusion-Exclusion. The enumeration is carried out each time we redefine new conditions on the set. In this study one of our aims is the recognition and identification of fuzzy subsets with same features, characteristics or conditions. To facilitate such a study, we use some ideas such as the Hamming distance, mid-point between two fuzzy subsets and cardinality of fuzzy subsets. Finally we introduce the fuzzy scanner of elements of a finite set. This is used to identify elements and fuzzy subsets of a set. The scanning process of identification and recognition facilitates the choice of entities with specified properties. We develop a procedure of selection under the fuzzy environment. This allows us a framework to resolve conflicting issues in the market place