# Article

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Keywords:
semigroup quasivariety; full embedding; \$f\!f\$-alg-universality; \$Q\$-universality
Summary:
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.
References:
[1] Adámek J., Rosický J.: Locally Presentable and Accessible Categories. Cambridge University Press Cambridge (1994). MR 1294136
[2] Adams M.E., Adaricheva K.V., Dziobiak W., Kravchenko A.V.: Some open questions related to the problem of Birkhoff and Maltsev. Studia Logica 78 (2004), 357-378. MR 2108035
[3] Adams M.E., Dziobiak W.: \$Q\$-universal quasivarieties of algebras. Proc. Amer. Math. Soc. 120 (1994), 1053-1059. MR 1172942 | Zbl 0810.08007
[4] Adams M.E., Dziobiak W.: Lattices of quasivarieties of \$3\$-element algebras. J. of Algebra 166 (1994),181-210. MR 1276823 | Zbl 0806.08005
[5] Adams M.E., Dziobiak W.: Finite-to-finite universal quasivarieties are \$Q\$-universal. Algebra Universalis 46 (2001), 253-283. MR 1835799 | Zbl 1059.08002
[6] Adams M.E., Dziobiak W.: The lattice of quasivarieties of undirected graphs. Algebra Universalis 47 (2002), 7-11. MR 1901728 | Zbl 1059.08003
[7] Adams M.E., Dziobiak W.: Quasivarieties of idempotent semigroups. Internat. J. Algebra Comput. 13 (2003), 733-752. MR 2028101 | Zbl 1042.08002
[8] Demlová M., Koubek V.: Endomorphism monoids in varieties of bands. Acta Sci. Math. (Szeged) 66 (2000), 477-516. MR 1804205
[9] Demlová M., Koubek V.: A weak version of universality in semigroup varieties. Novi Sad J. Math. 34 (2004), 37-86. MR 2136462
[10] Gorbunov V.A.: Algebraic Theory of Quasivarieties. Plenum Publishing Co. New York (1998). MR 1654844 | Zbl 0986.08001
[11] Hedrlín Z., Lambek J.: How comprehensive is the category of semigroups?. J. Algebra 11 (1969), 195-212. MR 0237611
[12] Koubek V., Sichler J.: Universal varieties of semigroups. J. Austral. Math. Soc. Ser. A 36 (1984), 143-152. MR 0725742 | Zbl 0549.20038
[13] Koubek V., Sichler J.: On relative universality and \$Q\$-universality. Studia Logica 78 (2004), 279-291. MR 2108030 | Zbl 1079.08009
[14] Koubek V., Sichler J.: Almost \$ff\$-universal and \$Q\$-universal varieties of modular \$0\$-lattices. Colloq. Math. 101 (2004), 161-182. MR 2110722 | Zbl 1066.06004
[15] Kravchenko A.V.: \$Q\$-universal quasivarieties of graphs. Algebra and Logic 41 (2002), 173-181. MR 1934538 | Zbl 1062.08013
[16] Mendelsohn E.: On a technique for representing semigroups as endomorphism semigroups of graphs with given properties. Semigroup Forum 4 (1972), 283-294. MR 0304533 | Zbl 0262.20083
[17] Pultr A., Trnková V.: Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories. North-Holland Amsterdam (1980). MR 0563525
[18] Sapir M.V.: The lattice of quasivarieties of semigroups. Algebra Universalis 21 (1985), 172-180. MR 0855737 | Zbl 0599.08014
[19] Sizyi S.V.: Quasivarieties of graphs. Siberian Math. J. 35 (1994), 783-794. MR 1302441

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