Aggregation of fuzzy vector spaces

summary:This paper contributes to the ongoing investigation of aggregating algebraic structures, with a particular focus on the aggregation of fuzzy vector spaces. The article is structured into three distinct parts, each addressing a specific aspect of the aggregation process. The first part of the paper explores the self-aggregation of fuzzy vector subspaces. It delves into the intricacies of combining and consolidating fuzzy vector subspaces to obtain a coherent and comprehensive outcome. The second part of the paper centers around the aggregation of similar fuzzy vector subspaces, specifically those belonging to the same equivalence class. This section scrutinizes the challenges and considerations involved in aggregating fuzzy vector subspaces with shared characteristics. The third part of the paper takes a broad perspective, providing an analysis of the aggregation problem of fuzzy vector subspaces from a general standpoint. It examines the fundamental issues, principles, and implications associated with aggregating fuzzy vector subspaces in a comprehensive manner. By elucidating these three key aspects, this paper contributes to the advancement of knowledge in the field of aggregation of algebraic structures, shedding light on the specific domain of fuzzy vector spaces

An L-Point Characterization of Normality and Normalizer of an L-Subgroup of an L-Group

AbstractIn this paper, we study the notion of normal L-subgroup of an L-group and provide its characterization by an L-point. We also provide a construction of the normalizer of an L-subgroup of a given L-group by using L-points. Moreover, we also discuss the product, homomorphic images and homomorphic preimages of normalizers

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

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

Studies in fuzzy groups

In this thesis we first extend the notion of fuzzy normality to the notion of normality of a fuzzy subgroup in another fuzzy group. This leads to the study of normal series of fuzzy subgroups, and this study includes solvable and nilpotent fuzzy groups, and the fuzzy version of the Jordan-Hõlder Theorem. Furthermore we use the notion of normality to study products and direct products of fuzzy subgroups. We present a notion of fuzzy isomorphism which enables us to state and prove the three well-known isomorphism theorems and the fact that the internal direct product of two normal fuzzy subgroups is isomorphic to the external direct product of the same fuzzy subgroups. A brief discussion on fuzzy subgroups generated by fuzzy subsets is also presented, and this leads to the fuzzy version of the Basis Theorem. Finally, the notion of direct product enables us to study decomposable and indecomposable fuzzy subgroups, and this study includes the fuzzy version of the Remak-Krull-Schmidt Theorem

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