scholarly journals Polyhedral convex cones and the equational theory of the bicyclic semigroup

2006 ◽  
Vol 81 (1) ◽  
pp. 63-96 ◽  
Author(s):  
F. Pastijn
Keyword(s):  
Simple Semigroup ◽  
Equational Theory ◽  
Semigroup Variety ◽  
Convex Cones ◽  
Semigroup Identity ◽  
Bicyclic Semigroup ◽  
Polyhedral Convex ◽  
The Given ◽  
Completely Simple

AbstractTo any given balanced semigroup identity U ≈ W a number of polyhedral convex cones are associated. In this setting an algorithm is proposed which determines whether the given identity is satisfied in the bicylic semigroup or in the semigroup . The semigroups BC and E deserve our attention because a semigroup variety contains a simple semigroup which is not completely simple (respectively, which is idempotent free) if and only if this variety contains BC (respectively, E). Therefore, for a given identity U ≈ W it is decidable whether or not the variety determined by U ≈ W contains a simple semigroup which is not completely simple (respectively, which is idempotent free).

Analogues of the bicyclic semigroup in simple semigroups without idempotents

1987 ◽  
Vol 106 (1-2) ◽  
pp. 11-24 ◽  
Author(s):  
Peter R. Jones
Keyword(s):  
Simple Semigroup ◽  
Finitely Generated ◽  
Sort It ◽  
Similar Role ◽  
Bicyclic Semigroup ◽  
Important Theorem ◽  
Green’S Relation ◽  
Completely Simple

SynopsisBy an important theorem of Andersen, any semigroup, containing idempotents, which is simple but not completely simple contains a copy of the bicyclic semigroup B = 〈a, b | ab = 1〉. In this paper the semigroups A = 〈a, b | a2b = a〉 and C = 〈a, b | a2b = a, ab2 = b〉 are shown to play a similar role in various classes of simple semigroups without idempotents, particularly in those for which Green's relation is nontrivial. For example it is shown that every right simple semigroup without idempotents is a union of copies of A; every finitely generated simple semigroup without idempotents contains either A or C. In a generalisation of a different sort it is shown that the bicyclic semigroup divides every simple semigroup without idempotents.Similar results are obtained for 0-simple semigroups without nonzero idempotents.

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.

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.

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