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

2016 ◽  
Vol 14 (1) ◽  
pp. 531-542
Author(s):  
Ninghua Gao ◽  
Qingguo Li ◽  
Zhaowen Li
Keyword(s):  
Distributive Lattice ◽  
Heyting Algebra ◽  
Lattice Structures ◽  
Boolean Ring ◽  
Fuzzy Subset ◽  
The Family ◽  
Essential Properties ◽  
Fuzzy Ideal

AbstractThe 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.

The semigroup of endomorphisms of a Boolean ring

1970 ◽  
Vol 11 (4) ◽  
pp. 411-416 ◽  
Author(s):  
Kenneth D. Magill
Keyword(s):  
Rational Numbers ◽  
Boolean Ring ◽  
Ring Of Integers ◽  
The Family ◽  
Identity Automorphism

The family (R) of all endomorphisms of a ring R is a semigroup under composition. It follows easily that if R and T are isomorphic rings, then (R) and (T) are isomorphic semigroups. We devote ourselves here to the converse question: ‘If (R) and (T) are isomorphic, must R and T be isomorphic?’ As one might expect, the answer is, in general, negative. For example, the ring of integers has precisely two endomorphisms – the zero endomorphism and the identity automorphism. Since the same is true of the ring of rational numbers, the two endomorphism semigroups are isomorphic while the rings themselves are certainly not.

scholarly journals Fuzzy ideals of ordered semigroups with fuzzy orderings

2016 ◽  
Vol 14 (1) ◽  
pp. 841-856
Author(s):  
Xiaokun Huang ◽  
Qingguo Li
Keyword(s):  
Fuzzy Relation ◽  
Lattice Structures ◽  
Ordered Semigroup ◽  
Fuzzy Subset ◽  
Ordered Semigroups ◽  
Fuzzy Orderings

AbstractThe purpose of this paper is to introduce the notions of ∈, ∈ ∨qk-fuzzy ideals of a fuzzy ordered semigroup with the ordering being a fuzzy relation. Several characterizations of ∈, ∈ ∨qk-fuzzy left (resp. right) ideals and ∈, ∈ ∨qk-fuzzy interior ideals are derived. The lattice structures of all ∈, ∈ ∨qk-fuzzy (interior) ideals on such fuzzy ordered semigroup are studied and some methods are given to construct an ∈, ∈ ∨qk-fuzzy (interior) ideals from an arbitrary fuzzy subset. Finally, the characterizations of generalized semisimple fuzzy ordered semigroups in terms of ∈, ∈ ∨qk-fuzzy ideals (resp. ∈, ∈ ∨qk-fuzzy interior ideals) are developed.

scholarly journals IDEAL PRIMA FUZZY DI SEMIGRUP S

2017 ◽  
Vol 2 (2) ◽  
pp. 46
Author(s):  
Abdurahim Abdurahim
Keyword(s):  
Prime Ideal ◽  
Membership Function ◽  
Closed Interval ◽  
Fuzzy Membership ◽  
Fuzzy Membership Function ◽  
Fuzzy Subset ◽  
Fuzzy Ideal ◽  
Function Domain

A fuzzy membership function is a function that maps the nonempty set  to a closed interval . Furthermore, if the function domain is replaced with a semigroup, then the function is called fuzzy subset. A fuzzy subset mapping  to  is denoted by . A fuzzy subset  is called fuzzy ideal if it satisfies both  and . Moreover,  is called a fuzzy prime ideal if for any fuzzy ideal  and , with  implies  or . In this paper be investigated about some characteristics of prime fuzzy ideals and some example of them.

scholarly journals Soft Set Theory Applied to Hoops

2020 ◽  
Vol 28 (1) ◽  
pp. 61-79
Author(s):  
R. A. Borzooei ◽  
E. Babaei ◽  
Y. B. Jun ◽  
M. Aaly Kologani ◽  
M. Mohseni Takallo
Keyword(s):  
Set Theory ◽  
Congruence Relation ◽  
Heyting Algebra ◽  
Soft Set ◽  
The Family ◽  
Different Types

AbstractIn this paper, we introduced the concept of a soft hoop and we investigated some of their properties. Then, we established different types of intersections and unions of the family of soft hoops. We defined two operations ⊙ and → on the set of all soft hoops and we proved that with these operations, it is a hoop and also is a Heyting algebra. Finally we introduced a congruence relation on the set of all soft hoops and we investigated the quotient of it.

Lattice Structure of D, T, and SR Fuzzy Flip-Flops Under Max-Min Logic

2005 ◽  
Vol 9 (6) ◽  
pp. 661-668 ◽  
Author(s):  
Shinichi Yoshida ◽  
◽  
Kaoru Hirota ◽  
Keyword(s):  
Fuzzy Logic ◽  
Distributive Lattice ◽  
System Design ◽  
Lattice Structure ◽  
Design Method ◽  
Boolean Lattice ◽  
Lattice Structures ◽  
Flip Flop ◽  
Different Types ◽  
Fuzzy Logical

Lattice structures of fuzzy flip-flops are described. A binary flip-flop (e.g. D, T, set-type SR, or reset-type SR flip-flop) can be extended to a fuzzy flip-flop in various ways. Under max-min fuzzy logic, there are 4 types of D fuzzy flip-flops extended from a binary D flip-flop, 136 types of SR fuzzy flip-flops extended from a binary SR flip-flop, and only one T fuzzy flip-flop. There is a lattice structure among different types of fuzzy flip-flops extended from a same binary flip-flop in terms of the order of ambiguity and the order of fuzzy logical value. These results show that fuzzy flip-flops under max-min fuzzy logic construct distributive lattice structures. Moreover D and T fuzzy flip-flops constructs Boolean lattice. And there exists a order monotone between two lattices of same fuzzy flip-flop under the order of ambiguity and the order of fuzzy logical value. Proposed analysis and results have potential to establish a fuzzy sequential system design method.

scholarly journals Maximal subalgebras of Heyting algebras

1986 ◽  
Vol 29 (3) ◽  
pp. 359-365 ◽  
Author(s):  
M. E. Adams
Keyword(s):  
Distributive Lattice ◽  
Binary Operation ◽  
Heyting Algebra ◽  
Heyting Algebras ◽  
Maximal Subalgebras ◽  
Bounded Distributive Lattice ◽  
Proposition 8

AHeyting algebra is an algebra H;∨,∧ →, 0,1) of type (2,2,2,0,0) for which H;∨,∧,0,1) is a bounded distributive lattice and → is the binary operation of relative pseudocomplementation (i.e., for a,b,c∈H,ac ∧≦birr c≦a→b). Associated with every subalgebra of a Heyting algebra is a separating set. Those corresponding to maximal subalgebras are characterized in Proposition 8 and, subsequently, are used in an investigation of Heyting algebras.

The Family of Lodato Proximities Compatible With a Given Topological Space

1974 ◽  
Vol 26 (02) ◽  
pp. 388-404 ◽  
Author(s):  
W. J. Thron ◽  
R. H. Warren
Keyword(s):  
Lower Bound ◽  
Topological Space ◽  
Distributive Lattice ◽  
Structural Properties ◽  
Upper Bound ◽  
Set Inclusion ◽  
The Family

Let (X, ) be a topological space. By we denote the family of all Lodato proximities on X which induce . We show that is a complete distributive lattice under set inclusion as ordering. Greatest lower bound and least upper bound are characterized. A number of techniques for constructing elements of are developed. By means of one of these constructions, all covers of any member of can be obtained. Several examples are given which relate to the lattice of all compatible proximities of Čech and the family of all compatible proximities of Efremovič. The paper concludes with a chart which summarizes many of the structural properties of , and .

scholarly journals On Fuzzy Ideals ofBL-Algebras

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Biao Long Meng ◽  
Xiao Long Xin
Keyword(s):  
Prime Ideal ◽  
Distributive Lattice ◽  
Fuzzy Set ◽  
Representation Theorem ◽  
Prime Ideals ◽  
Converse Implication ◽  
Fuzzy Ideal ◽  
Stone Representation

In this paper we investigate further properties of fuzzy ideals of aBL-algebra. The notions of fuzzy prime ideals, fuzzy irreducible ideals, and fuzzy Gödel ideals of aBL-algebra are introduced and their several properties are investigated. We give a procedure to generate a fuzzy ideal by a fuzzy set. We prove that every fuzzy irreducible ideal is a fuzzy prime ideal but a fuzzy prime ideal may not be a fuzzy irreducible ideal and prove that a fuzzy prime idealωis a fuzzy irreducible ideal if and only ifω0=1and|Im⁡(ω)|=2. We give the Krull-Stone representation theorem of fuzzy ideals inBL-algebras. Furthermore, we prove that the lattice of all fuzzy ideals of aBL-algebra is a complete distributive lattice. Finally, it is proved that every fuzzy Boolean ideal is a fuzzy Gödel ideal, but the converse implication is not true.

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