Enlargements, semiabundancy and unipotent monoids

J Fountain; G M S Gomes; V Gould

Enlargements, semiabundancy and unipotent monoids

J Fountain, G M S Gomes, V Gould

Mathematics

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

Abstract

The relation (R) over tilde on a monoid S provides a natural generalisation of Green's relation R. If every (R) over tilde-class of S contains an idempotent, S is left semiabundant; if (R) over tilde is a left congruence then S satisfies (CL). Regular monoids, indeed left abundant monoids, are left semiabundant and satisfy (CL). However, the class of left semiabundant monoids is much larger, as we illustrate with a number of examples.

This is the first of three related papers exploring the relationship between unipotent monoids and left semiabundancy. We consider the situations where the power enlargement or the Szendrei expansion of a monoid yields a left semiabundant monoid with (CL). Using the Szendrei expansion and the notion of the least unipotent monoid congruence sigma on a monoid S, we construct functors (<(circle)over tilde>)(SR) : U --> F and F-sigma : F --> U such that (<(circle)over tilde>)(SR) is a left adjoint of F-sigma. Here U is the category of unipotent monoids and F is a category of left semiabundant monoids with properties echoing those of F-inverse monoids.

Original language	English
Pages (from-to)	595-614
Number of pages	20
Journal	Communications in Algebra
Volume	27
Issue number	2
Publication status	Published - 1999

Keywords

enlargement
expansion
semiabundant
unipotent
SEMIGROUPS

Access to Document

http://www-users.york.ac.uk/~jbf1/unipot.pdf

Cite this

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title = "Enlargements, semiabundancy and unipotent monoids",

abstract = "The relation (R) over tilde on a monoid S provides a natural generalisation of Green's relation R. If every (R) over tilde-class of S contains an idempotent, S is left semiabundant; if (R) over tilde is a left congruence then S satisfies (CL). Regular monoids, indeed left abundant monoids, are left semiabundant and satisfy (CL). However, the class of left semiabundant monoids is much larger, as we illustrate with a number of examples.This is the first of three related papers exploring the relationship between unipotent monoids and left semiabundancy. We consider the situations where the power enlargement or the Szendrei expansion of a monoid yields a left semiabundant monoid with (CL). Using the Szendrei expansion and the notion of the least unipotent monoid congruence sigma on a monoid S, we construct functors (<(circle)over tilde>)(SR) : U --> F and F-sigma : F --> U such that (<(circle)over tilde>)(SR) is a left adjoint of F-sigma. Here U is the category of unipotent monoids and F is a category of left semiabundant monoids with properties echoing those of F-inverse monoids.",

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author = "J Fountain and Gomes, {G M S} and V Gould",

year = "1999",

language = "English",

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pages = "595--614",

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publisher = "Taylor & Francis",

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TY - JOUR

T1 - Enlargements, semiabundancy and unipotent monoids

AU - Fountain, J

AU - Gomes, G M S

AU - Gould, V

PY - 1999

Y1 - 1999

N2 - The relation (R) over tilde on a monoid S provides a natural generalisation of Green's relation R. If every (R) over tilde-class of S contains an idempotent, S is left semiabundant; if (R) over tilde is a left congruence then S satisfies (CL). Regular monoids, indeed left abundant monoids, are left semiabundant and satisfy (CL). However, the class of left semiabundant monoids is much larger, as we illustrate with a number of examples.This is the first of three related papers exploring the relationship between unipotent monoids and left semiabundancy. We consider the situations where the power enlargement or the Szendrei expansion of a monoid yields a left semiabundant monoid with (CL). Using the Szendrei expansion and the notion of the least unipotent monoid congruence sigma on a monoid S, we construct functors (<(circle)over tilde>)(SR) : U --> F and F-sigma : F --> U such that (<(circle)over tilde>)(SR) is a left adjoint of F-sigma. Here U is the category of unipotent monoids and F is a category of left semiabundant monoids with properties echoing those of F-inverse monoids.

AB - The relation (R) over tilde on a monoid S provides a natural generalisation of Green's relation R. If every (R) over tilde-class of S contains an idempotent, S is left semiabundant; if (R) over tilde is a left congruence then S satisfies (CL). Regular monoids, indeed left abundant monoids, are left semiabundant and satisfy (CL). However, the class of left semiabundant monoids is much larger, as we illustrate with a number of examples.This is the first of three related papers exploring the relationship between unipotent monoids and left semiabundancy. We consider the situations where the power enlargement or the Szendrei expansion of a monoid yields a left semiabundant monoid with (CL). Using the Szendrei expansion and the notion of the least unipotent monoid congruence sigma on a monoid S, we construct functors (<(circle)over tilde>)(SR) : U --> F and F-sigma : F --> U such that (<(circle)over tilde>)(SR) is a left adjoint of F-sigma. Here U is the category of unipotent monoids and F is a category of left semiabundant monoids with properties echoing those of F-inverse monoids.

KW - enlargement

KW - expansion

KW - semiabundant

KW - unipotent

KW - SEMIGROUPS

M3 - Article

VL - 27

SP - 595

EP - 614

JO - Communications in Algebra

JF - Communications in Algebra

IS - 2

ER -