Weak alg-universality and Q -universality of semigroup quasivarieties

Marie Demlová; Václav Koubek

Commentationes Mathematicae Universitatis Carolinae (2005)

  • Volume: 46, Issue: 2, page 257-279
  • ISSN: 0010-2628

Abstract

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In an earlier paper, the authors showed that standard semigroups 𝐌 1 , 𝐌 2 and 𝐌 3 play an important role in the classification of weaker versions of alg-universality of semigroup varieties. This paper shows that quasivarieties generated by 𝐌 2 and 𝐌 3 are neither relatively alg-universal nor Q -universal, while there do exist finite semigroups 𝐒 2 and 𝐒 3 generating the same semigroup variety as 𝐌 2 and 𝐌 3 respectively and the quasivarieties generated by 𝐒 2 and/or 𝐒 3 are quasivar-relatively f f -alg-universal and Q -universal (meaning that their respective lattices of subquasivarieties are quite rich). An analogous result on Q -universality of the variety generated by 𝐌 2 was obtained by M.V. Sapir; the size of our semigroup is substantially smaller than that of Sapir’s semigroup.

How to cite

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Demlová, Marie, and Koubek, Václav. "Weak alg-universality and $Q$-universality of semigroup quasivarieties." Commentationes Mathematicae Universitatis Carolinae 46.2 (2005): 257-279. <http://eudml.org/doc/249558>.

@article{Demlová2005,
abstract = {In an earlier paper, the authors showed that standard semigroups $\mathbf \{M\}_1$, $\mathbf \{M\}_2$ and $\mathbf \{M\}_3$ play an important role in the classification of weaker versions of alg-universality of semigroup varieties. This paper shows that quasivarieties generated by $\mathbf \{M\}_2$ and $\mathbf \{M\}_3$ are neither relatively alg-universal nor $Q$-universal, while there do exist finite semigroups $\mathbf \{S\}_2$ and $\mathbf \{S\}_3$ generating the same semigroup variety as $\mathbf \{M\}_2$ and $\mathbf \{M\}_3$ respectively and the quasivarieties generated by $\mathbf \{S\}_2$ and/or $\mathbf \{S\}_3$ are quasivar-relatively $f\!f$-alg-universal and $Q$-universal (meaning that their respective lattices of subquasivarieties are quite rich). An analogous result on $Q$-universality of the variety generated by $\mathbf \{M\}_2$ was obtained by M.V. Sapir; the size of our semigroup is substantially smaller than that of Sapir’s semigroup.},
author = {Demlová, Marie, Koubek, Václav},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {semigroup quasivariety; full embedding; $f\!f$-alg-universality; $Q$-universality; semigroup quasivarieties; full embeddings; semigroup varieties; -alg-universality; -universality; finite semigroups; lattices of quasivarieties},
language = {eng},
number = {2},
pages = {257-279},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Weak alg-universality and $Q$-universality of semigroup quasivarieties},
url = {http://eudml.org/doc/249558},
volume = {46},
year = {2005},
}

TY - JOUR
AU - Demlová, Marie
AU - Koubek, Václav
TI - Weak alg-universality and $Q$-universality of semigroup quasivarieties
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2005
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 46
IS - 2
SP - 257
EP - 279
AB - In an earlier paper, the authors showed that standard semigroups $\mathbf {M}_1$, $\mathbf {M}_2$ and $\mathbf {M}_3$ play an important role in the classification of weaker versions of alg-universality of semigroup varieties. This paper shows that quasivarieties generated by $\mathbf {M}_2$ and $\mathbf {M}_3$ are neither relatively alg-universal nor $Q$-universal, while there do exist finite semigroups $\mathbf {S}_2$ and $\mathbf {S}_3$ generating the same semigroup variety as $\mathbf {M}_2$ and $\mathbf {M}_3$ respectively and the quasivarieties generated by $\mathbf {S}_2$ and/or $\mathbf {S}_3$ are quasivar-relatively $f\!f$-alg-universal and $Q$-universal (meaning that their respective lattices of subquasivarieties are quite rich). An analogous result on $Q$-universality of the variety generated by $\mathbf {M}_2$ was obtained by M.V. Sapir; the size of our semigroup is substantially smaller than that of Sapir’s semigroup.
LA - eng
KW - semigroup quasivariety; full embedding; $f\!f$-alg-universality; $Q$-universality; semigroup quasivarieties; full embeddings; semigroup varieties; -alg-universality; -universality; finite semigroups; lattices of quasivarieties
UR - http://eudml.org/doc/249558
ER -

References

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