scholarly journals On the solvability of systems of linear equations on commutative semigroup

2010 ◽  
Vol 43 (4) ◽  
Author(s):  
Nguyen Thi Thu Huyen ◽  
Nguyen Minh Tuan
Keyword(s):  
Linear Operator ◽  
Linear Space ◽  
Commutative Semigroup ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Operator Equations ◽  
Systems Of Linear Equations ◽  
Necessary And Sufficient ◽  
Linear Operator Equations

AbstractThis paper deals with the solvability of systems of linear operator equations in a linear space. Namely, the paper provides necessary and sufficient conditions for the operators under which certain kinds of systems of operator equations are solvable.

Regular, Commutative, Maximal Semigroups of Quotients

1975 ◽  
Vol 18 (1) ◽  
pp. 99-104 ◽  
Author(s):  
Jurgen Rompke
Keyword(s):  
Commutative Semigroup ◽  
Regular Ring ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Necessary And Sufficient Conditions ◽  
Natural Question ◽  
Commutative Case ◽  
Ring Of Quotients ◽  
Singular Ideal ◽  
Necessary And Sufficient

A well-known theorem which goes back to R. E. Johnson [4], asserts that if R is a ring then Q(R), its maximal ring of quotients is regular (in the sense of v. Neumann) if and only if the singular ideal of R vanishes. In the theory of semigroups a natural question is therefore the following: Do there exist properties which characterize those semigroups whose maximal semigroups of quotients are regular? Partial answers to this question have been given in [3], [7] and [8]. In this paper we completely solve the commutative case, i.e. we give necessary and sufficient conditions for a commutative semigroup S in order that Q(S), the maximal semigroup of quotients, is regular. These conditions reflect very closely the property of being semiprime, which in the theory of commutative rings characterizes those rings which have a regular ring of quotients.

scholarly journals Measure of noncompactness of operators and matrices on the spacescandc0

2006 ◽  
Vol 2006 ◽  
pp. 1-5 ◽  
Author(s):  
Bruno De Malafosse ◽  
Eberhard Malkowsky ◽  
Vladimir Rakocevic
Keyword(s):  
Linear Operator ◽  
Hausdorff Measure ◽  
Measure Of Noncompactness ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Hausdorff Measure Of Noncompactness ◽  
Necessary And Sufficient

In this note, using the Hausdorff measure of noncompactness, necessary and sufficient conditions are formulated for a linear operator and matrices between the spacescandc0to be compact. Among other things, some results of Cohen and Dunford are recovered.

A general canonical expression

1933 ◽  
Vol 29 (4) ◽  
pp. 465-469 ◽  
Author(s):  
J. Bronowski
Keyword(s):  
Linear Space ◽  
Dimensional Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
General Point ◽  
Linear Forms ◽  
Powers Of Linear Forms ◽  
Necessary And Sufficient

1. In a recent paper I established new conditions for a form φ of order n, homogeneous in r + 1 variables, to be expressible as the sum of nth powers of linear forms in these variables; and for this expression, if it exists, to be unique. These conditions, I further showed, may be stated as general theorems regarding the secant spaces of manifolds Mr in higher space, namely:Necessary and sufficient conditions that through a general point of a space N, of h (r + 1) − 1 dimensions, there passes (i) no, (ii) a unique (h − 1)-dimensional space containing h points of a manifold Mr lying in N are that(i) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr meets Mr in a curve, so that Mr cannot be so projected upon a linear space of r dimensions;(ii) the space projecting a general point of Mr from the join of h − 1 general r-dimensional tangent spaces of Mr does not meet Mr again, so that Mr can be so projected, birationally, upon a linear space of r dimensions..

On commutative semigroup algebras

1983 ◽  
Vol 93 (2) ◽  
pp. 237-246 ◽  
Author(s):  
W. D. Munn
Keyword(s):  
Commutative Semigroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Semigroup Algebras ◽  
Necessary And Sufficient

This paper is concerned with the problem of finding necessary and sufficient conditions on a commutative semigroup S for the algebra FS of S over a field F to be semiprimitive (Jacobson semisimple).

scholarly journals Minimal sets on tori

1984 ◽  
Vol 4 (4) ◽  
pp. 499-507 ◽  
Author(s):  
Daniel Berend
Keyword(s):  
Hausdorff Dimension ◽  
Commutative Semigroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Dimensional Torus ◽  
Minimal Set ◽  
Circle Group ◽  
Minimal Sets ◽  
Necessary And Sufficient

AbstractLet Σ be a commutative semigroup of continuous endomorphisms of the r-dimensional torus. Generalizing a result of Furstenberg dealing with the circle group, necessary and sufficient conditions are given here for Σ to possess the following property: Any Σ-minimal set consists of torsion elements. Semigroups not having this property are shown to admit minimal sets of positive Hausdorff dimension.

scholarly journals Necessary and Sufficient Conditions for The Solutions of Linear Equation System

2020 ◽  
Vol 17 (1) ◽  
pp. 82-88
Author(s):  
Gregoria Ariyanti
Keyword(s):  
Linear Equation ◽  
Commutative Semigroup ◽  
Algebraic Structure ◽  
Identity Element ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Equation System ◽  
Distributive Property ◽  
Linear Equation System ◽  
Necessary And Sufficient

A Semiring is an algebraic structure (S,+,x) such that (S,+) is a commutative Semigroup with identity element 0, (S,x) is a Semigroup with identity element 1, distributive property of multiplication over addition, and multiplication by 0 as an absorbent element in S. A linear equations system over a Semiring S is a pair (A,b)  where A is a matrix with entries in S  and b is a vector over S. This paper will be described as necessary or sufficient conditions of the solution of linear equations system over Semiring S viewed by matrix X  that satisfies AXA=A, with A in S.  For a matrix X that satisfies AXA=A, a linear equations system Ax=b has solution x=Xb+(I-XA)h with arbitrary h in S if and only if AXb=b.

scholarly journals On linear functional equations modulo $${\mathbb {Z}}$$

2021 ◽  
Author(s):  
Attila Gilányi ◽  
Agata Lewicka
Keyword(s):  
Linear Space ◽  
Functional Equations ◽  
Linear Functional ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Functional Equations ◽  
Necessary And Sufficient

AbstractIn this paper, we consider the condition $$\sum _{i=0}^{n+1}\varphi _i(r_ix+q_iy)\in {\mathbb {Z}}$$ ∑ i = 0 n + 1 φ i ( r i x + q i y ) ∈ Z for real valued functions defined on a linear space V. We derive necessary and sufficient conditions for functions satisfying this condition to be decent in the following sense: there exist functions $$f_i:V\rightarrow {\mathbb {R}}$$ f i : V → R , $$g_i:V\rightarrow {\mathbb {Z}}$$ g i : V → Z such that $$\varphi _i=f_i+g_i$$ φ i = f i + g i , $$(i=0,\dots ,n+1)$$ ( i = 0 , ⋯ , n + 1 ) and $$\sum _{i=0}^{n+1}f_i(r_ix+q_iy)=0$$ ∑ i = 0 n + 1 f i ( r i x + q i y ) = 0 for all $$x, y\in V$$ x , y ∈ V .

scholarly journals Maximal homomorphic images of commutative semigroups

1971 ◽  
Vol 12 (1) ◽  
pp. 12-17 ◽  
Author(s):  
D. B. McAlister ◽  
L. O'Carroll
Keyword(s):  
Homomorphic Image ◽  
Commutative Semigroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Commutative Semigroups ◽  
Necessary And Sufficient

In this paper we give necessary and sufficient conditions on a commutative semigroup in order that it should have a maximal homomorphic image of one of the following types: (1) groups, (2) semigroups which are unions of groups and (3) pseudoinvertible semigroups, i. e. semigroups having the property that some power of each element lies in a subgroup of the semigroup.

scholarly journals Asymptotic Properties of -Expansive Homeomorphisms on a Metric -Space

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Ruchi Das ◽  
Tarun Das
Keyword(s):  
Linear Space ◽  
Metric Space ◽  
Normed Linear Space ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Bounded Subset ◽  
Necessary And Sufficient

We define and study the notions of positively and negatively -asymptotic points for a homeomorphism on a metric -space. We obtain necessary and sufficient conditions for two points to be positively/negatively -asymptotic. Also, we show that the problem of studying -expansive homeomorphisms on a bounded subset of a normed linear -space is equivalent to the problem of studying linear -expansive homeomorphisms on a bounded subset of another normed linear -space.

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