On universality of semigroup varieties

Marie Demlová; Václav Koubek

Archivum Mathematicum (2006)

  • Volume: 042, Issue: 4, page 357-386
  • ISSN: 0044-8753

Abstract

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A category K is called α -determined if every set of non-isomorphic K -objects such that their endomorphism monoids are isomorphic has a cardinality less than α . A quasivariety Q is called Q -universal if the lattice of all subquasivarieties of any quasivariety of finite type is a homomorphic image of a sublattice of the lattice of all subquasivarieties of Q . We say that a variety V is var-relatively alg-universal if there exists a proper subvariety W of V such that homomorphisms of V whose image does not belong to W contains a full subcategory isomorphic to the category of all graphs. A semigroup variety V is nearly J -trivial if for every semigroup S ∈ V any J -class containing a group is a singleton. We prove that for a nearly J -trivial variety V the following are equivalent: V is Q -universal; V is var-relatively alg-universal; V is α -determined for no cardinal α ; V contains at least one of the three specific semigroups. Dually, for a nearly J -trivial variety V the following are equivalent: V is 3 -determined; V is not var-relatively alg-universal; the lattice of all subquasivarieties of V is finite; V is a subvariety of one of two special finitely generated varieties.

How to cite

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Demlová, Marie, and Koubek, Václav. "On universality of semigroup varieties." Archivum Mathematicum 042.4 (2006): 357-386. <http://eudml.org/doc/249818>.

@article{Demlová2006,
abstract = {A category $K$ is called $\alpha $-determined if every set of non-isomorphic $K$-objects such that their endomorphism monoids are isomorphic has a cardinality less than $\alpha $. A quasivariety $Q$ is called $Q$-universal if the lattice of all subquasivarieties of any quasivariety of finite type is a homomorphic image of a sublattice of the lattice of all subquasivarieties of $Q$. We say that a variety $V$ is var-relatively alg-universal if there exists a proper subvariety $W$ of $V$ such that homomorphisms of $V$ whose image does not belong to $W$ contains a full subcategory isomorphic to the category of all graphs. A semigroup variety $V$ is nearly $J$-trivial if for every semigroup $S\in V$ any $ J$-class containing a group is a singleton. We prove that for a nearly $J$-trivial variety $V$ the following are equivalent: $V$ is $Q$-universal; $ V$ is var-relatively alg-universal; $V$ is $\alpha $-determined for no cardinal $\alpha $; $V$ contains at least one of the three specific semigroups. Dually, for a nearly $J$-trivial variety $V$ the following are equivalent: $V$ is $3$-determined; $V$ is not var-relatively alg-universal; the lattice of all subquasivarieties of $V$ is finite; $V$ is a subvariety of one of two special finitely generated varieties.},
author = {Demlová, Marie, Koubek, Václav},
journal = {Archivum Mathematicum},
keywords = {semigroup variety; band variety; full embedding; $f\!f$-alg-universality; determinacy; $Q$-universality; semigroup varieties; band varieties; full embeddings; alg-universality; determinacy},
language = {eng},
number = {4},
pages = {357-386},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On universality of semigroup varieties},
url = {http://eudml.org/doc/249818},
volume = {042},
year = {2006},
}

TY - JOUR
AU - Demlová, Marie
AU - Koubek, Václav
TI - On universality of semigroup varieties
JO - Archivum Mathematicum
PY - 2006
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 042
IS - 4
SP - 357
EP - 386
AB - A category $K$ is called $\alpha $-determined if every set of non-isomorphic $K$-objects such that their endomorphism monoids are isomorphic has a cardinality less than $\alpha $. A quasivariety $Q$ is called $Q$-universal if the lattice of all subquasivarieties of any quasivariety of finite type is a homomorphic image of a sublattice of the lattice of all subquasivarieties of $Q$. We say that a variety $V$ is var-relatively alg-universal if there exists a proper subvariety $W$ of $V$ such that homomorphisms of $V$ whose image does not belong to $W$ contains a full subcategory isomorphic to the category of all graphs. A semigroup variety $V$ is nearly $J$-trivial if for every semigroup $S\in V$ any $ J$-class containing a group is a singleton. We prove that for a nearly $J$-trivial variety $V$ the following are equivalent: $V$ is $Q$-universal; $ V$ is var-relatively alg-universal; $V$ is $\alpha $-determined for no cardinal $\alpha $; $V$ contains at least one of the three specific semigroups. Dually, for a nearly $J$-trivial variety $V$ the following are equivalent: $V$ is $3$-determined; $V$ is not var-relatively alg-universal; the lattice of all subquasivarieties of $V$ is finite; $V$ is a subvariety of one of two special finitely generated varieties.
LA - eng
KW - semigroup variety; band variety; full embedding; $f\!f$-alg-universality; determinacy; $Q$-universality; semigroup varieties; band varieties; full embeddings; alg-universality; determinacy
UR - http://eudml.org/doc/249818
ER -

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