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.

Regular Semigroups Which Are Amalgamation Bases for Finite Semigroups

2007 ◽  
Vol 14 (02) ◽  
pp. 245-254 ◽  
Author(s):  
Kunitaka Shoji
Keyword(s):  
Regular Semigroup ◽  
Simple Semigroup ◽  
Finite Semigroup ◽  
Principal Factor ◽  
Regular Semigroups ◽  
Completely Simple Semigroup ◽  
Finite Semigroups ◽  
Completely Simple ◽  
Finite Regular Semigroup

In this paper, we prove that a completely 0-simple (or completely simple) semigroup is an amalgamation base for finite semigroups if and only if it is an amalgamation base for semigroups. By adopting the same method as used in a previous paper, we prove that a finite regular semigroup is an amalgamation base for finite semigroups if its [Formula: see text]-classes are linearly ordered and all of its principal factor semigroups are amalgamation bases for finite semigroups. Finally, we give an example of a finite semigroup U which is an amalgamation base for semigroups, but not all of its principal factor semigroups are amalgamation bases either for semigroups or for finite semigroups.

Completely simple and inverse semigroups

1961 ◽  
Vol 57 (2) ◽  
pp. 234-236 ◽  
Author(s):  
R. McFadden ◽  
Hans Schneider
Keyword(s):  
Simple Semigroup ◽  
Zero Element ◽  
Inverse Semigroups ◽  
Completely Simple Semigroup ◽  
Relative Inverse ◽  
Matrix Semigroup ◽  
Completely Simple

The purpose of this paper is to investigate the structure of certain types of semigroups. Rees(6),(7) has determined the structure of a completely simple semigroup, and has shown that such a system may be realized as a type of matrix semigroup. Clifford (2) and Schwarz (8) have found conditions, namely, the existence of minimal left and minimal right ideals, under which a simple semigroup is completely simple, and have made a more detailed study of such semigroups. Preston (4), (5) has studied inverse semigroups, in which each non-zero element has a unique relative inverse, and has also considered inverse semigroups which contain minimal right or left ideals. In the present paper we obtain a set of conditions on a simple semigroup, each of which is equivalent to the semigroup being both completely simple and inverse. Section 2 defines the terms used and gives a brief resume of the main results which have already been proved. Section 3 is devoted to our present considerations.

Completely 0-simple semigroup with the basis property

2020 ◽  
pp. 2150141
Author(s):  
Ibrahim Al-Dayel ◽  
Ahmad Al Khalaf
Keyword(s):  
Simple Semigroup ◽  
Basis Property ◽  
Addition Formula ◽  
Completely Simple Semigroup ◽  
Matrix Formula ◽  
Completely Simple

A semigroup [Formula: see text] is said to have the Basis Property if for any subsemigroup [Formula: see text] of a semigroup [Formula: see text], any two bases for [Formula: see text] have the same cardinality. The structure of completely [Formula: see text]-simple semigroup with the Basis Property is described. In particular, we proved that each completely [Formula: see text]-simple semigroup [Formula: see text] has the Basis Property if and only if [Formula: see text] satisfies one of the following conditions: (1) [Formula: see text] is produced from a completely simple semigroup with adjoint zero. (2) [Formula: see text] is an isomorphic to Rees’s semigroup [Formula: see text] over a group [Formula: see text] with sandwich matrix [Formula: see text] such that [Formula: see text], [Formula: see text], in addition [Formula: see text] has a zero in every row and column.

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.

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.

PROFINITE SEMIGROUPS, VARIETIES, EXPANSIONS AND THE STRUCTURE OF RELATIVELY FREE PROFINITE SEMIGROUPS

2001 ◽  
Vol 11 (06) ◽  
pp. 627-672 ◽  
Author(s):  
JOHN RHODES ◽  
BENJAMIN STEINBERG
Keyword(s):  
Fixed Points ◽  
Structural Properties ◽  
Simple Semigroup ◽  
Cayley Graphs ◽  
Minimal Ideal ◽  
Completely Simple Semigroup ◽  
Finite Semigroups ◽  
Finite Set ◽  
Profinite Semigroup ◽  
Completely Simple

Building on the now generally accepted thesis that profinite semigroups are important to the study of finite semigroups, this paper proposes to apply various of the techniques, already used in studying algebraic semigroups, to profinite semigroups. The goal in mind is to understand free profinite semigroups on a finite set. To do this we define profinite varieties. We then introduce expansions of profinite semigroups, giving examples of several classes of such expansions. These expansions will then be useful in studying various structural properties of relatively free profinite semigroups, since these semigroups will be fixed points of certain expansions. This study also requires a look at profinite categories, semigroupoids, and Cayley graphs, all of which we handle in turn. We also study the structure of the minimal ideal of relatively free profinite semigroups showing, in particular, that the minimal ideal of the free profinite semigroup on a finite set with more than two generators is not a relatively free profinite completely simple semigroup, as well as some generalizations to related pseudovarieties.

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