scholarly journals When does a generalized Boolean quasiring become a Boolean ring?

2017 ◽  
Vol 22 (20) ◽  
pp. 6877-6879
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Boolean Ring

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.

scholarly journals Green's relations for regular elements of sandwich semigroups, II; semigroups of continuous functions

1978 ◽  
Vol 25 (1) ◽  
pp. 45-65 ◽  
Author(s):  
K. D. Magill ◽  
S. Subbiah
Keyword(s):  
Continuous Functions ◽  
Maximal Subgroups ◽  
Green’S Relations ◽  
Boolean Ring ◽  
Regular Elements ◽  
Ring Homomorphisms ◽  
Special Cases ◽  
Green's Relations ◽  
Sandwich Semigroup ◽  
Sandwich Semigroups

AbstractA sandwich semigroup of continuous functions consists of continuous functions with domains all in some space X and ranges all in some space Y with multiplication defined by fg = foαog where α is a fixed continuous function from a subspace of Y into X. These semigroups include, as special cases, a number of semigroups previously studied by various people. In this paper, we characterize the regular elements of such semigroups and we completely determine Green's relations for the regular elements. We also determine the maximal subgroups and, finally, we apply some of these results to semigroups of Boolean ring homomorphisms.

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.

Boolean powers of groups

1982 ◽  
Vol 91 (3) ◽  
pp. 375-396 ◽  
Author(s):  
A. B. Apps
Keyword(s):  
Finite Groups ◽  
Algebraic Structure ◽  
Boolean Ring ◽  
Direct Power ◽  
Boolean Power ◽  
Boolean Extension

If M is any algebraic structure, and R is any Boolean ring, then a structure called the (bounded) Boolean power of M by R, denoted MR, can be defined. This construction, which is also called a bounded Boolean extension, is a sort of generalized direct power, and was introduced by Foster in the 1950's (as a refinement of his previous notion of a Boolean extension). In this paper we shall study isomorphism types and automorphisms of Boolean powers of groups, and obtain information about their characteristic subgroups: we shall be chiefly concerned with Boolean powers of finite groups.

Quasi-clean rings and strongly quasi-clean rings

2021 ◽  
pp. 2150079
Author(s):  
Gaohua Tang ◽  
Huadong Su ◽  
Pingzhi Yuan
Keyword(s):  
Positive Integer ◽  
Natural Generalization ◽  
Extensive Study ◽  
Central Unit ◽  
Boolean Ring ◽  
Semilocal Ring ◽  
Clean Rings ◽  
Maximal Ideals ◽  
Open Questions ◽  
Unit Formula

An element [Formula: see text] of a ring [Formula: see text] is called a quasi-idempotent if [Formula: see text] for some central unit [Formula: see text] of [Formula: see text], or equivalently, [Formula: see text], where [Formula: see text] is a central unit and [Formula: see text] is an idempotent of [Formula: see text]. A ring [Formula: see text] is called a quasi-Boolean ring if every element of [Formula: see text] is quasi-idempotent. A ring [Formula: see text] is called (strongly) quasi-clean if each of its elements is a sum of a quasi-idempotent and a unit (that commute). These rings are shown to be a natural generalization of the clean rings and strongly clean rings. An extensive study of (strongly) quasi-clean rings is conducted. The abundant examples of (strongly) quasi-clean rings state that the class of (strongly) quasi-clean rings is very larger than the class of (strongly) clean rings. We prove that an indecomposable commutative semilocal ring is quasi-clean if and only if it is local or [Formula: see text] has no image isomorphic to [Formula: see text]; For an indecomposable commutative semilocal ring [Formula: see text] with at least two maximal ideals, [Formula: see text]([Formula: see text]) is strongly quasi-clean if and only if [Formula: see text] is quasi-clean if and only if [Formula: see text], [Formula: see text] is a maximal ideal of [Formula: see text]. For a prime [Formula: see text] and a positive integer [Formula: see text], [Formula: see text] is strongly quasi-clean if and only if [Formula: see text]. Some open questions are also posed.

On The Extension Of Measure By The Method Of Borel

1956 ◽  
Vol 8 ◽  
pp. 516-523 ◽  
Author(s):  
L. LeBlanc ◽  
G. E. Fox
Keyword(s):  
Outer Measure ◽  
Transfinite Induction ◽  
Boolean Ring

Introduction. This paper concerns the problem of extending a given measure defined on a Boolean ring to a measure on the generated σ-ring. Two general methods are familiar to the literature, that of Lebesgue (outer measure) and a method proposed by Borel using transfinite induction (4, 49-134; 2, 228-238).

scholarly journals Matrices with Elements in a Boolean Ring

1957 ◽  
Vol 9 ◽  
pp. 47-59
Author(s):  
A. T. Butson
Keyword(s):  
Identity Matrix ◽  
Boolean Ring ◽  
Unimodular Matrix

1. Introduction. Let be a Boolean ring of at least two elements containing a unit 1. Form the set of matrices A, B, … of order n having entries aiJ, bij, … (i, j = 1, 2, …, n), which are members of . A matrix U of is called unimodular if there exists a matrix V of such that VU= I, the identity matrix. Two matrices A and B are said to be left-associates if there exists a unimodular matrix U satisfying UA = B.

scholarly journals Injective and projective Boolean-like rings

1982 ◽  
Vol 33 (1) ◽  
pp. 40-49 ◽  
Author(s):  
V. Swaminathan
Keyword(s):  
Commutative Ring ◽  
Boolean Ring

AbstractA Boolean-like ring R is a commutative ring with unity in which 2x = 0 and xy(1 + x)(1 + y) = 0 hold for all elements x, y of the ring R. It is shown in this paper that in the category of Boolean-like rings, R is injective if and only if R is a complete Boolean ring and R is projective if and only if R = {0, 1}.

Boolean Near-Rings

1969 ◽  
Vol 12 (3) ◽  
pp. 265-273 ◽  
Author(s):  
James R. Clay ◽  
Donald A. Lawver
Keyword(s):  
Vector Space ◽  
Maximal Ideal ◽  
Boolean Ring ◽  
Affine Transformations ◽  
Isomorphism Classes ◽  
Boolean Rings ◽  
Lattice Of Ideals ◽  
Unique Maximal Ideal

In this paper we introduce the concept of Boolean near-rings. Using any Boolean ring with identity, we construct a class of Boolean near-rings, called special, and determine left ideals, ideals, factor near-rings which are Boolean rings, isomorphism classes, and ideals which are near-ring direct summands for these special Boolean near-rings.Blackett [6] discusses the near-ring of affine transformations on a vector space where the near-ring has a unique maximal ideal. Gonshor [10] defines abstract affine near-rings and completely determines the lattice of ideals for these near-rings. The near-ring of differentiable transformations is seen to be simple in [7], For near-rings with geometric interpretations, see [1] or [2].

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