scholarly journals Locally adequate semigroup algebras

2016 ◽  
Vol 14 (1) ◽  
pp. 29-48 ◽  
Author(s):  
Yingdan Ji ◽  
Yanfeng Luo
Keyword(s):  
Direct Product ◽  
Simple Semigroup ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Indecomposable Modules ◽  
Direct Sum ◽  
Multiplicative Basis ◽  
Special Cases ◽  
Direct Sum Decomposition ◽  
Adequate Semigroup

AbstractWe build up a multiplicative basis for a locally adequate concordant semigroup algebra by constructing Rukolaĭne idempotents. This allows us to decompose the locally adequate concordant semigroup algebra into a direct product of primitive abundant $0{\rm{ - }}{\cal J}*$-simple semigroup algebras. We also deduce a direct sum decomposition of this semigroup algebra in terms of the ${\cal R}*$-classes of the semigroup obtained from the above multiplicative basis. Finally, for some special cases, we provide a description of the projective indecomposable modules and determine the representation type.

scholarly journals A Note on Locally Inverse Semigroup Algebras

2008 ◽  
Vol 2008 ◽  
pp. 1-5 ◽  
Author(s):  
Xiaojiang Guo
Keyword(s):  
Inverse Semigroup ◽  
Commutative Ring ◽  
Direct Product ◽  
Necessary Conditions ◽  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Locally Inverse Semigroup ◽  
Maximum Subgroup ◽  
Sufficient And Necessary Conditions

LetRbe a commutative ring andSa finite locally inverse semigroup. It is proved that the semigroup algebraR[S]is isomorphic to the direct product of Munn algebrasℳ(R[GJ],mJ,nJ;PJ)withJ∈S/𝒥, wheremJis the number ofℛ-classes inJ,nJthe number ofℒ-classes inJ, andGJa maximum subgroup ofJ. As applications, we obtain the sufficient and necessary conditions for the semigroup algebra of a finite locally inverse semigroup to be semisimple.

scholarly journals Definition and Properties of Direct Sum Decomposition of Groups

2015 ◽  
Vol 23 (1) ◽  
pp. 15-27
Author(s):  
Kazuhisa Nakasho ◽  
Hiroshi Yamazaki ◽  
Hiroyuki Okazaki ◽  
Yasunari Shidama
Keyword(s):  
Direct Product ◽  
Equivalent Form ◽  
Fourth Section ◽  
Product Group ◽  
The Third ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Direct Product Group ◽  
Product Map

Summary In this article, direct sum decomposition of group is mainly discussed. In the second section, support of element of direct product group is defined and its properties are formalized. It is formalized here that an element of direct product group belongs to its direct sum if and only if support of the element is finite. In the third section, product map and sum map are prepared. In the fourth section, internal and external direct sum are defined. In the last section, an equivalent form of internal direct sum is proved. We referred to [23], [22], [8] and [18] in the formalization.

scholarly journals Self-splitting Abelian groups

2001 ◽  
Vol 64 (1) ◽  
pp. 71-79 ◽  
Author(s):  
P. Schultz
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Free Module ◽  
Research Report ◽  
Original Form ◽  
Original Research ◽  
Direct Sum ◽  
The University ◽  
Torsion Free ◽  
Made In

G is reduced torsion-free A belian group such that for every direct sum ⊕G of copies of G, Ext(⊕G, ⊕G) = 0 if and only if G is a free module over a rank 1 ring. For every direct product ΠG of copies of G, Ext(ΠG,ΠG) = 0 if and only if G is cotorsion.This paper began as a Research Report of the Department of Mathematics of the University of Western Australia in 1988, and circulated among members of the Abelian group community. However, it was never submitted for publication. The results have been cited, widely, and since copies of the original research report are no longer available, the paper is presented here in its original form in Sections 1 to 5. In Section 6, I survey the progress that has been made in the topic since 1988.

Regular Hom-algebras admitting a multiplicative basis

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Antonio J. Calderón Martín
Keyword(s):  
Arbitrary Dimension ◽  
Base Field ◽  
Direct Sum ◽  
Multiplicative Basis ◽  
Arbitrary Base ◽  
The Family

AbstractLet {({\mathfrak{H}},\mu,\alpha)} be a regular Hom-algebra of arbitrary dimension and over an arbitrary base field {{\mathbb{F}}}. A basis {{\mathcal{B}}=\{e_{i}\}_{i\in I}} of {{\mathfrak{H}}} is called multiplicative if for any {i,j\in I}, we have that {\mu(e_{i},e_{j})\in{\mathbb{F}}e_{k}} and {\alpha(e_{i})\in{\mathbb{F}}e_{p}} for some {k,p\in I}. We show that if {{\mathfrak{H}}} admits a multiplicative basis, then it decomposes as the direct sum {{\mathfrak{H}}=\bigoplus_{r}{{\mathfrak{I}}}_{r}} of well-described ideals admitting each one a multiplicative basis. Also, the minimality of {{\mathfrak{H}}} is characterized in terms of the multiplicative basis and it is shown that, in case {{\mathcal{B}}}, in addition, it is a basis of division, then the above direct sum is composed by means of the family of its minimal ideals, each one admitting a multiplicative basis of division.

Locally ample semigroup algebras

2014 ◽  
Vol 07 (04) ◽  
pp. 1450067 ◽  
Author(s):  
Xiaojiang Guo ◽  
K. P. Shum
Keyword(s):  
Semigroup Algebra ◽  
Semigroup Algebras ◽  
Abundant Semigroup ◽  
Ample Semigroup

An idempotent-connected abundant semigroup S is a locally ample semigroup if for any idempotent e of S, the local submonoid eSe of S is an ample subsemigroup of S. Clearly, an ample semigroup is a locally ample semigroup. In this paper, it is proved that the semigroup algebra of a finite locally ample semigroup is isomorphic to the semigroup algebra of an associate primitive abundant semigroup. As an application of this result, we characterize Jacobson radicals of finite locally ample semigroup algebras.

scholarly journals On Direct Sum Decomposition of Integers and Y. Ito's Conjecture

1998 ◽  
Vol 21 (2) ◽  
pp. 433-440 ◽  
Author(s):  
Masahito DATEYAMA ◽  
Teturo KAMAE
Keyword(s):  
Direct Sum ◽  
Direct Sum Decomposition

scholarly journals Implicit Function Theorem. Part II

2019 ◽  
Vol 27 (2) ◽  
pp. 117-131
Author(s):  
Kazuhisa Nakasho ◽  
Yasunari Shidama
Keyword(s):  
Implicit Function Theorem ◽  
Existence And Uniqueness ◽  
Continuous Linear Operator ◽  
Linear Operators ◽  
Implicit Function ◽  
Continuous Linear ◽  
Lipschitz Continuous ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Continuous Linear Operators

Summary In this article, we formalize differentiability of implicit function theorem in the Mizar system [3], [1]. In the first half section, properties of Lipschitz continuous linear operators are discussed. Some norm properties of a direct sum decomposition of Lipschitz continuous linear operator are mentioned here. In the last half section, differentiability of implicit function in implicit function theorem is formalized. The existence and uniqueness of implicit function in [6] is cited. We referred to [10], [11], and [2] in the formalization.

Sylow Theory for a Certain Class of Operator Groups

1961 ◽  
Vol 13 ◽  
pp. 192-200 ◽  
Author(s):  
Christine W. Ayoub
Keyword(s):  
Direct Product ◽  
Finite Groups ◽  
Complete Lattice ◽  
Classical Group ◽  
Additive Group ◽  
Sylow Subgroups ◽  
Direct Sum

In this paper we consider again the group-theoretic configuration studied in (1) and (2). Let G be an additive group (not necessarily abelian), let M be a system of operators for G, and let ϕ be a family of admissible subgroups which form a complete lattice relative to intersection and compositum. Under these circumstances we call G an M — ϕ group. In (1) we studied the normal chains for an M — ϕ group and the relation between certain normal chains. In (2) we considered the possibility of representing an M — ϕ group as the direct sum of certain of its subgroups, and proved that with suitable restrictions on the M — ϕ group the analogue of the following theorem for finite groups holds: A group is the direct product of its Sylow subgroups if and only if it is nilpotent. Here we show that under suitable hypotheses (hypotheses (I), (II), and (III) stated at the beginning of §3) it is possible to generalize to M — ϕ groups many of the Sylow theorems of classical group theorem.

Solution Space Decompositions for nth Order Linear Differential Equations

1975 ◽  
Vol 27 (3) ◽  
pp. 508-512
Author(s):  
G. B. Gustafson ◽  
S. Sedziwy
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Solution Space ◽  
Decomposition Theorem ◽  
Linear Differential Equations ◽  
Direct Sum ◽  
Direct Sum Decomposition ◽  
Linear Differential ◽  
Image Position

Consider the wth order scalar ordinary differential equationwith pr ∈ C([0, ∞) → R ) . The purpose of this paper is to establish the following:DECOMPOSITION THEOREM. The solution space X of (1.1) has a direct sum Decompositionwhere M1 and M2 are subspaces of X such that(1) each solution in M1\﹛0﹜ is nonzero for sufficiently large t ﹛nono sdilatory) ;(2) each solution in M2 has infinitely many zeros ﹛oscillatory).

Sign in / Sign up

Export Citation Format

Share Document