Tom Head's join structure of fuzzy subgroups

Jain, Aparna

doi:10.1016/s0165-0114(00)00128-7

Cited by 5 publications

(4 citation statements)

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“…Unfortunately after the emergence of metatheorem, not much attention has been paid on its further development or its application in other areas of algebra such as group theory and ring theory. There are only few papers related to this topic [3][4][5][6]. In [3], a new subdirect product theorem for fuzzy congruences is established.…”

Section: Introductionmentioning

confidence: 99%

“…In the course of this development, in his erratum [7], Tom Head also defined a very important concept of tip extended pair of fuzzy subgroups in order to formulate the join of two fuzzy normal subgroups of a group. Using this technique for constructing the join, A. Jain [5] has provided a much simpler and direct proof of modularity of the lattice of fuzzy normal subgroups. In the similar manner, I. Jahan [4] devised the join of two L-ideals by using the notion of tip extended pair of Lideals and established the modularity of the lattice of L-ideals of a ring which was an open problem.…”

Section: Introductionmentioning

confidence: 99%

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A New Metatheorem and Subdirect Product Theorem for L-Subgroups

Ajmal

Jahan

2018

Fuzzy Information and Engineering

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This paper is a continuation of the work of Tom Head 'Metatheorem for deriving fuzzy theorems from crisp versions'. The concept of natural extension is introduced which is then applied in the development of a new metatheorem in L-setting, where L is a complete chain. The application of this theorem is demonstrated through the notions of generated L-subgroups and commutator L-subgroups. Moreover, a new subdirect product theorem is developed wherein it is demonstrated that for a group G, the L-subgroup lattice can be represented as a subdirect product of copies of its associated lattice of crisp subgroups. The significance of this theorem is exhibited by applying it to a characterization of generalized cyclic groups in terms of the distributivity of the lattice of L-subgroups of a group.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A New Metatheorem and Subdirect Product Theorem for L-Subgroups

Ajmal

Jahan

2018

Fuzzy Information and Engineering

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“…Liu [3] introduced the fuzzy subring, etc. Standing upon these achievements, many researchers explored the lattice theoretical properties of these structures, such as, modularity of the lattice of the fuzzy normal subgroups was established in a systematic and step wise manner in [4][5][6][7][8][9][10]. By supposing the value "t " at the additive identity of a given ring, the fact that the set of fuzzy ideals with sup property forms a sublattice of the lattice of fuzzy ideals was proved by Ajimal and Thomas [11].…”

Section: Introductionmentioning

confidence: 99%

When do L-fuzzy ideals of a ring generate a distributive lattice?

Gao

2016

Open Mathematics

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Abstract:The notion of L-fuzzy extended ideals is introduced in a Boolean ring, and their essential properties are investigated. We also build the relation between an L-fuzzy ideal and the class of its L-fuzzy extended ideals. By defining an operator " " between two arbitrary L-fuzzy ideals in terms of L-fuzzy extended ideals, the result that "the family of all L-fuzzy ideals in a Boolean ring is a complete Heyting algebra" is immediately obtained. Furthermore, the lattice structures of L-fuzzy extended ideals of an L-fuzzy ideal, L-fuzzy extended ideals relative to an L-fuzzy subset, L-fuzzy stable ideals relative to an L-fuzzy subset and their connections are studied in this paper.

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“…One of these is L fn t , the set of all fuzzy normal subgroups of a group G each of which has finite range set and a fixed tip 't'. This set L fn t was shown to be a modular sublattice of the lattice L. Then, it was shown in [6] that L n , the set of all fuzzy normal subgroups of a group is a modular sublattice of L. Later, in [15], the author provided a much simpler and direct proof of modularity of the lattice of fuzzy normal subgroups. In [16], the authors exhibited the method to construct the lattices of the sets of all fuzzy subsets (fuzzy subgroups) of a fuzzy set (fuzzy group) in the newly defined categories S and G of fuzzy sets and fuzzy groups respectively.…”

Section: Introductionmentioning

confidence: 99%