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 Square Integer Matrix with a Single Non-Integer Entry in Its Inverse

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2226
Author(s):  
Arif Mandangan ◽  
Hailiza Kamarulhaili ◽  
Muhammad Asyraf Asbullah
Keyword(s):  
Computer Graphics ◽  
Matrix Inversion ◽  
Floating Point ◽  
Large Integer ◽  
Identity Matrix ◽  
Singular Matrix ◽  
Integer Matrix ◽  
Unimodular Matrix ◽  
The Matrix ◽  
Lattice Based Cryptography

Matrix inversion is one of the most significant operations on a matrix. For any non-singular matrix A∈Zn×n, the inverse of this matrix may contain countless numbers of non-integer entries. These entries could be endless floating-point numbers. Storing, transmitting, or operating such an inverse could be cumbersome, especially when the size n is large. The only square integer matrix that is guaranteed to have an integer matrix as its inverse is a unimodular matrix U∈Zn×n. With the property that det(U)=±1, then U−1∈Zn×n is guaranteed such that UU−1=I, where I∈Zn×n is an identity matrix. In this paper, we propose a new integer matrix G˜∈Zn×n, which is referred to as an almost-unimodular matrix. With det(G˜)≠±1, the inverse of this matrix, G˜−1∈Rn×n, is proven to consist of only a single non-integer entry. The almost-unimodular matrix could be useful in various areas, such as lattice-based cryptography, computer graphics, lattice-based computational problems, or any area where the inversion of a large integer matrix is necessary, especially when the determinant of the matrix is required not to equal ±1. Therefore, the almost-unimodular matrix could be an alternative to the unimodular matrix.

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.

scholarly journals Toeplitz Operators with Lagrangian Invariant Symbols Acting on the Poly-Fock Space of ℂ n

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Jorge Luis Arroyo Neri ◽  
Armando Sánchez-Nungaray ◽  
Mauricio Hernández Marroquin ◽  
Raquiel R. López-Martínez
Keyword(s):  
Toeplitz Operators ◽  
Complex Space ◽  
Fock Space ◽  
Identity Matrix ◽  
The Matrix

We introduce the so-called extended Lagrangian symbols, and we prove that the C ∗ -algebra generated by Toeplitz operators with these kind of symbols acting on the homogeneously poly-Fock space of the complex space ℂ n is isomorphic and isometric to the C ∗ -algebra of matrix-valued functions on a certain compactification of ℝ n obtained by adding a sphere at the infinity; moreover, the matrix values at the infinity points are equal to some scalar multiples of the identity matrix.

scholarly journals Membangkitkan Suatu Matriks Unimodular Dengan Python

2018 ◽  
Vol 5 (2) ◽  
pp. 1-9
Author(s):  
Samsul Arifin ◽  
Indra Bayu Muktyas
Keyword(s):  
Matrix Method ◽  
Coefficient Matrix ◽  
Inverse Matrix ◽  
Solution Vector ◽  
Unimodular Matrix ◽  
All Solutions ◽  
Cramer’S Rule

An SPL can be represented as a multiplication of the coefficient matrix and solution vector of the SPL. Determining the solution of an SPL can use the inverse matrix method and Cramer's rule, where both can use the concept of the determinant of a matrix. If the coefficient matrix is a unimodular matrix, then all solutions of an SPL are integers. In this paper we will present a method of generating a unimodular matrix using Python so that it can be utilized on an SPL. Keywords: SPL, Unimodular Matrix, Python

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.

8. Matrices and groups

2015 ◽  
pp. 101-110
Author(s):  
Peter M. Higgins
Keyword(s):  
Dihedral Group ◽  
Inverse Matrix ◽  
Identity Matrix ◽  
Matrix Products ◽  
Square Matrix

‘Matrices and groups’ continues with the example of geometric matrix products to see what happens when we compose the mappings involved. It explains several features, including the identity matrix, the inverse matrix, the square matrix, and the concept of isomorphism. If a collection of matrices represent the elements of a group, such as the eight matrices that represent the dihedral group D, then each of these matrices A will have an inverse, A −1, such that AA-1 = A-1A =I, the identity matrix. This prompts the twin questions of when the inverse of a square matrix A exists and, if it does, how to find it.

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