Extensions and covers for semigroups whose idempotents form a left regular band

Mario J. J. Branco; Gracinda M. S. Gomes; Victoria Gould

doi:10.1007/s00233-010-9239-9

Extensions and covers for semigroups whose idempotents form a left regular band

Mario J. J. Branco, Gracinda M. S. Gomes, Victoria Gould

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Proper extensions that are "injective on a"'-related idempotents" of a">-unipotent semigroups, and much more generally of the class of generalised left restriction semigroups possessing the ample and congruence conditions, referred to here as glrac semigroups, are described as certain subalgebras of a lambda-semidirect product of a left regular band by an a">-unipotent or by a glrac semigroup, respectively. An example of such is the generalized Szendrei expansion.

As a consequence of our embedding, we are able to give a structure theorem for proper left restriction semigroups. Further, we show that any glrac semigroup S has a proper cover that is a semidirect product of a left regular band by a monoid, and if S is left restriction, the left regular band may be taken to be a semilattice.

Original language	English
Pages (from-to)	51-70
Number of pages	20
Journal	Semigroup forum
Volume	81
Issue number	1
DOIs	https://doi.org/10.1007/s00233-010-9239-9
Publication status	Published - Aug 2010

Keywords

Expansions
R-unipotent semigroups
Generalised restriction
LEFT AMPLE SEMIGROUPS
UNIPOTENT SEMIGROUP
INVERSE-SEMIGROUPS
EXPANSION
MONOIDS

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10.1007/s00233-010-9239-9

Cite this

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title = "Extensions and covers for semigroups whose idempotents form a left regular band",

abstract = "Proper extensions that are {"}injective on a{"}'-related idempotents{"} of a{"}>-unipotent semigroups, and much more generally of the class of generalised left restriction semigroups possessing the ample and congruence conditions, referred to here as glrac semigroups, are described as certain subalgebras of a lambda-semidirect product of a left regular band by an a{"}>-unipotent or by a glrac semigroup, respectively. An example of such is the generalized Szendrei expansion.As a consequence of our embedding, we are able to give a structure theorem for proper left restriction semigroups. Further, we show that any glrac semigroup S has a proper cover that is a semidirect product of a left regular band by a monoid, and if S is left restriction, the left regular band may be taken to be a semilattice.",

keywords = "Expansions, R-unipotent semigroups, Generalised restriction, LEFT AMPLE SEMIGROUPS, UNIPOTENT SEMIGROUP, INVERSE-SEMIGROUPS, EXPANSION, MONOIDS",

author = "Branco, {Mario J. J.} and Gomes, {Gracinda M. S.} and Victoria Gould",

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T1 - Extensions and covers for semigroups whose idempotents form a left regular band

AU - Branco, Mario J. J.

AU - Gomes, Gracinda M. S.

AU - Gould, Victoria

PY - 2010/8

Y1 - 2010/8

N2 - Proper extensions that are "injective on a"'-related idempotents" of a">-unipotent semigroups, and much more generally of the class of generalised left restriction semigroups possessing the ample and congruence conditions, referred to here as glrac semigroups, are described as certain subalgebras of a lambda-semidirect product of a left regular band by an a">-unipotent or by a glrac semigroup, respectively. An example of such is the generalized Szendrei expansion.As a consequence of our embedding, we are able to give a structure theorem for proper left restriction semigroups. Further, we show that any glrac semigroup S has a proper cover that is a semidirect product of a left regular band by a monoid, and if S is left restriction, the left regular band may be taken to be a semilattice.

AB - Proper extensions that are "injective on a"'-related idempotents" of a">-unipotent semigroups, and much more generally of the class of generalised left restriction semigroups possessing the ample and congruence conditions, referred to here as glrac semigroups, are described as certain subalgebras of a lambda-semidirect product of a left regular band by an a">-unipotent or by a glrac semigroup, respectively. An example of such is the generalized Szendrei expansion.As a consequence of our embedding, we are able to give a structure theorem for proper left restriction semigroups. Further, we show that any glrac semigroup S has a proper cover that is a semidirect product of a left regular band by a monoid, and if S is left restriction, the left regular band may be taken to be a semilattice.

KW - Expansions

KW - R-unipotent semigroups

KW - Generalised restriction

KW - LEFT AMPLE SEMIGROUPS

KW - UNIPOTENT SEMIGROUP

KW - INVERSE-SEMIGROUPS

KW - EXPANSION

KW - MONOIDS

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DO - 10.1007/s00233-010-9239-9

M3 - Article

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VL - 81

SP - 51

EP - 70

JO - Semigroup forum

JF - Semigroup forum

IS - 1

ER -

Extensions and covers for semigroups whose idempotents form a left regular band

Abstract

Keywords

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