scholarly journals Rings Graded By a Generalized Group

2014 ◽  
Vol 2 (1) ◽  
Author(s):  
Farzad Fatehi ◽  
Mohammad Reza Molaei
Keyword(s):  
Simple Semigroup ◽  
Orthonormal Basis ◽  
Homogeneous Ideal ◽  
Graded Ring ◽  
Noetherian Module ◽  
Completely Simple Semigroup ◽  
Completely Simple Semigroups ◽  
Maximal Ideals ◽  
Completely Simple

AbstractThe aim of this paper is to consider the ringswhich can be graded by completely simple semigroups. We show that each G-graded ring has an orthonormal basis, where G is a completely simple semigroup. We prove that if I is a complete homogeneous ideal of a G-graded ring R, then R/I is a G-graded ring.We deduce a characterization of the maximal ideals of a G-graded ring which are homogeneous. We also prove that if R is a Noetherian graded ring, then each summand of it is also a Noetherian module..

Residuated Completely Simple Semigroups

2014 ◽  
Vol 21 (02) ◽  
pp. 181-194
Author(s):  
T. S. Blyth ◽  
G. A. Pinto
Keyword(s):  
Cyclic Group ◽  
Simple Semigroup ◽  
Lexicographic Order ◽  
Ordered Semigroup ◽  
Completely Simple Semigroup ◽  
Completely Simple Semigroups ◽  
Completely Simple

We consider particular compatible orders on a given completely simple semigroup Sx=M(〈x〉; I ,Λ;P) where 〈x〉 is an ordered cyclic group with x > 1 and P11=x-1. Of these, only the lexicographic and bootlace orders yield residuated semigroups. With the lexicographic order, Sx is orthodox and has a biggest idempotent. With the bootlace order, the maximal idempotents of Sx are identified by specific locations in the sandwich matrix. In the orthodox case there is also a biggest idempotent and, for sandwich matrices of a given size, uniqueness up to ordered semigroup isomorphism is established.

Free Involutorial Completely Simple Semigroups

1985 ◽  
Vol 37 (2) ◽  
pp. 271-295 ◽  
Author(s):  
J. A. Gerhard ◽  
Mario Petrich
Keyword(s):  
Maximal Subgroup ◽  
Universal Algebra ◽  
Simple Semigroup ◽  
Binary Operation ◽  
Unary Operation ◽  
Completely Simple Semigroup ◽  
Completely Simple Semigroups ◽  
Image Position ◽  
Completely Simple

An involution x → x* of a semigroup S is an antiautomorphism of S of order at most 2, that is (xy)* = y*x* and x** = x for all x, y ∊ S. In such a case, S is called an involutorial semigroup if regarded as a universal algebra with the binary operation of multiplication and the unary operation *. If S is also a completely simple semigroup, regarded as an algebra with multiplication and the unary operation x → x−1 of inversion (x−1 is the inverse of x in the maximal subgroup of S containing x), then (S, −1, *), or simply S, is an involutorial completely simple semigroup. All such S form a variety determined by the identities above concerning * andwhere x0 = xx−1.

Expansions of completely simple semigroups

2004 ◽  
Vol 41 (1) ◽  
pp. 39-58
Author(s):  
B. Billhardt
Keyword(s):  
Semidirect Product ◽  
Simple Semigroup ◽  
Completely Simple Semigroup ◽  
Completely Simple Semigroups ◽  
Group 6 ◽  
The Relationship ◽  
Completely Simple

For any completely simple semigroup C a regular expansion S(C) is constructed which is the Birget-Rhodes prefix expansion CPr if C is a group [6]. We show that our construction generalizes two important features of CPr. Moreover we embed S (C) into a restricted semidirect product of a semilattice by C and investigate the relationship to the expansion P(C), introduced by Meakin [14].

Generalized Inflations of Completely Simple Semigroups

2007 ◽  
Vol 14 (01) ◽  
pp. 103-116
Author(s):  
Qiang Wang ◽  
Shelly L. Wismath
Keyword(s):  
Simple Semigroup ◽  
Completely Simple Semigroup ◽  
Completely Simple Semigroups ◽  
Completely Simple

Clarke and Monzo defined a construction called a generalized inflation of a semigroup, and proved that for unions of groups, all null extensions are generalized inflations. We characterize when a groupoid containing a completely simple semigroup B is an associative generalized inflation of B, and thus provide a way to construct all associative generalized inflations (null extensions) of a given completely simple semigroup. This answers a question posed by Clarke and Monzo.

The Arithmetic of the Quasi-Uniserial Semigroups without Zero

1971 ◽  
Vol 23 (3) ◽  
pp. 507-516 ◽  
Author(s):  
Ernst August Behrens
Keyword(s):  
Simple Semigroup ◽  
Disjoint Union ◽  
Maximal Subgroups ◽  
Ordered Semigroup ◽  
Partially Ordered ◽  
Completely Simple Semigroup ◽  
Image Position ◽  
Previous Paragraph ◽  
Completely Simple ◽  
Identity Elements

An element a in a partially ordered semigroup T is called integral ifis valid. The integral elements form a subsemigroup S of T if they exist. Two different integral idempotents e and f in T generate different one-sided ideals, because eT = fT, say, implies e = fe ⊆ f and f = ef ⊆ e.Let M be a completely simple semigroup. M is the disjoint union of its maximal subgroups [4]. Their identity elements generate the minimal one-sided ideals in M. The previous paragraph suggests the introduction of the following hypothesis on M.Hypothesis 1. Every minimal one-sided ideal in M is generated by an integral idempotent.

scholarly journals Cross-connections of completely simple semigroups

2016 ◽  
Vol 09 (03) ◽  
pp. 1650053 ◽  
Author(s):  
P. A. Azeef Muhammed ◽  
A. R. Rajan
Keyword(s):  
Simple Semigroup ◽  
Structure Theory ◽  
Cambridge Philos ◽  
Regular Semigroups ◽  
Local Isomorphism ◽  
Completely Simple Semigroup ◽  
Matrix Formula ◽  
Matrix Semigroup ◽  
Completely Simple ◽  
Cross Connection

A completely simple semigroup [Formula: see text] is a semigroup without zero which has no proper ideals and contains a primitive idempotent. It is known that [Formula: see text] is a regular semigroup and any completely simple semigroup is isomorphic to the Rees matrix semigroup [Formula: see text] (cf. D. Rees, On semigroups, Proc. Cambridge Philos. Soc. 36 (1940) 387–400). In the study of structure theory of regular semigroups, Nambooripad introduced the concept of normal categories to construct the semigroup from its principal left (right) ideals using cross-connections. A normal category [Formula: see text] is a small category with subobjects wherein each object of the category has an associated idempotent normal cone and each morphism admits a normal factorization. A cross-connection between two normal categories [Formula: see text] and [Formula: see text] is a local isomorphism [Formula: see text] where [Formula: see text] is the normal dual of the category [Formula: see text]. In this paper, we identify the normal categories associated with a completely simple semigroup [Formula: see text] and show that the semigroup of normal cones [Formula: see text] is isomorphic to a semi-direct product [Formula: see text]. We characterize the cross-connections in this case and show that each sandwich matrix [Formula: see text] correspond to a cross-connection. Further we use each of these cross-connections to give a representation of the completely simple semigroup as a cross-connection semigroup.

On the rank of completely 0-simple semigroups

1994 ◽  
Vol 116 (2) ◽  
pp. 325-338 ◽  
Author(s):  
N. Ruškuc
Keyword(s):  
Simple Semigroup ◽  
Alternative Proof ◽  
Rees Matrix Semigroup ◽  
Minimal Rank ◽  
Completely Simple Semigroup ◽  
Equivalent Conditions ◽  
Matrix Semigroup ◽  
Completely Simple

AbstractConnected completely 0-simple semigroups are defined by a number of equivalent conditions, and a formula for the rank of these semigroups is proved. As a consequence an alternative proof of the result from [11] is given. In the case of a Rees matrix semigroup M0 [G, I, Λ, P] the rank is expressed in terms of |I|, |Λ|, G and a certain subgroup of G depending on P. At the end the minimal rank of all semigroups M0[G, I, Λ, P] is found for a given group G. Since every completely simple semigroup is connected, every result has a corollary for these semigroups.

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