## M. Demlov\'a, V. Koubek

*Weak alg-universality and $Q$-universality of semigroup quasivarieties *

Comment.Math.Univ.Carolinae 46,2 (2005) 257-279. **Abstract:**In an earlier paper, the authors showed that standard semigroups $\bold M_1$, $\bold M_2$ and $\bold M_3$ play an important role in the classification of weaker versions of alg-universality of semigroup varieties. This paper shows that quasivarieties generated by $\bold M_2$ and $\bold M_3$ are neither relatively alg-universal nor $Q$-universal, while there do exist finite semigroups $\bold S_2$ and $\bold S_3$ generating the same semigroup variety as $\bold M_2$ and $\bold M_3$ respectively and the quasivarieties generated by $\bold S_2$ and/or $\bold S_3$ 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 $\bold M_2$ was obtained by M.V. Sapir; the size of our semigroup is substantially smaller than that of Sapir's semigroup.

**Keywords:** semigroup quasivariety, full embedding, $f\!f$-alg-universality, $Q$-universality

**AMS Subject Classification:** 20M99, 20M07, 08C15, 18B15

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