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On universality of semigroup varieties

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M. Demlová, V. Koubek
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** Address.**
Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University, Technická 2, 166 27 Praha 6, Czech Republic

Department of Theoretical Computer Science and Mathematical Logic, and Institute of Theoretical Computer Science, The Faculty of Mathematics and Physics, Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic

** E-mail:**
demlova@math.feld.cvut.cz,
koubek@ksi.ms.mff.cuni.cz

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

**AMSclassification.**
20M07, 20M99, 08C15, 18B15.

**Keywords.**
Semigroup variety, band variety, full embedding, $f\!f$-alg-universality, determinacy,
$Q$-universality.