×

zbMATH — the first resource for mathematics

Cancellable elements of the lattices of varieties of semigroups and epigroups. (English) Zbl 1444.20036
An element \(x\) of a lattice \(\langle L; \vee , \wedge \rangle \) is called cancellable iff \((\forall x,y \in L)(x \vee y = x \vee z \, \& \, x \wedge y = x \wedge z \to y = z)\). An epigroup is a semigroup in which some power of any element lies in a subgroup. Here, a description of all semigroup (epigroup) varieties which are cancellable elements of the lattice of all semigroup (epigroup) varieties is presented.
MSC:
20M07 Varieties and pseudovarieties of semigroups
08B15 Lattices of varieties
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aǐzenštat, A. Y.; Boguta, B. K.; Lyapin, E. S., Semigroup Varieties and Endomorphism Semigroups, On the lattice of semigroup varieties, 3-46, (1979), Leningrad: Leningrad State Pedagogic Institute, Leningrad
[2] Burris, S.; Nelson, E., Embedding the dual of \(##?##\) in the lattice of equational classes of semigroups, Algebra Univ., 1, 2, 248-254, (1971) · Zbl 0227.08006
[3] Evans, T., The lattice of semigroup varieties, Semigroup Forum, 2, 1, 1-43, (1971) · Zbl 0225.20043
[4] Grätzer, G., Lattice Theory: Foundation, (2011), Basel: Birkhäuser, Springer Basel, Basel · Zbl 1233.06001
[5] Grätzer, G.; Schmidt, E. T., Standard ideals in lattices, Acta Math. Acad. Sci. Hungar., 12, 1-2, 17-86, (1964) · Zbl 0115.01901
[6] Gusev, S. V.; Skokov, D. V.; Vernikov, B. M., Cancellable elements of the lattice of semigroup varieties, Algebra Discr. Math., 26, 1, 34-46, (2018) · Zbl 07083030
[7] Gusev, S. V.; Vernikov, B. M., Endomorphisms of the lattice of epigroup varieties, Semigroup Forum, 93, 3, 554-574, (2016) · Zbl 1371.08004
[8] Ježek, J., Intervals in lattices of varieties, Algebra Univ., 6, 1, 147-158, (1976) · Zbl 0354.08007
[9] Ježek, J., The lattice of equational theories. Part I: Modular elements, Czechosl. Math. J., 31, 1, 127-152, (1981) · Zbl 0477.08006
[10] Ježek, J.; Mckenzie, R. N., Definability in the lattice of equational theories of semigroups, Semigroup Forum, 46, 1, 199-245, (1993) · Zbl 0782.20051
[11] Kalicki, J.; Scott, D., Equationally completeness in abstract algebras, Proc. Konikl. Nederl. Akad. Wetensch. Ser. A, 58, 17, 650-659, (1955) · Zbl 0073.24501
[12] Mitsch, H., Semigroups and their lattice of congruences, Semigroup Forum, 26, 1, 1-63, (1983) · Zbl 0513.20047
[13] Petrich, M.; Reilly, N. R., Completely Regular Semigroups, (1999), New York: John Wiley & Sons, New York · Zbl 0967.20034
[14] Sapir, M. V.; Sukhanov, E. V., Varieties of periodic semigroups, Izv. VUZ. Matematika., 4, 48-55, (1981) · Zbl 0471.20042
[15] Shaprynskiǐ, V. Y., Modular and lower-modular elements of lattices of semigroup varieties, Semigroup Forum, 85, 1, 97-110, (2012) · Zbl 1261.20069
[16] Shaprynskiǐ, V. Y.; Skokov, D. V.; Vernikov, B. M., Special elements of the lattice of epigroup varieties, Algebra Univ., 76, 1, 1-30, (2016) · Zbl 1356.20035
[17] Shevrin, L. N., On the theory of epigroups. I, II, Matem. Sborn, 185, 8, 129-160, (1994) · Zbl 0841.20056
[18] Shevrin, L. N.; Kudryavtsev, V. B.; Rosenberg, I. G., Structural Theory of Automata, Semigroups, and Universal Algebra, Epigroups, 331-380, (2005), Dordrecht: Springer, Dordrecht
[19] Shevrin, L. N.; Vernikov, B. M.; Volkov, M. V., Lattices of semigroup varieties, Izv. Vuz. Matem., 3, 3-36, (2009) · Zbl 1185.08002
[20] Skokov, D. V., Distributive elements of the lattice of epigroup varieties, Siberian Electron. Math. Rep., 12, 723-731, (2015) · Zbl 1345.20072
[21] Skokov, D. V., Special elements of certain types in the lattice of epigroup varieties, Proc. Inst. Math. Mechan. Ural Branch Russ. Acad. Sci., 22, 3, 244-250, (2016)
[22] Skokov, D. V., Cancellable elements of the lattice of epigroup varieties, Izv. VUZ. Matem., 9, 59-67, (2018) · Zbl 07000199
[23] Skokov, D. V.; Vernikov, B. M., On modular and cancellable elements of the lattice of semigroup varieties, Siberian Electron. Math. Rep., 16, 175-186, (2019) · Zbl 1436.20111
[24] Vernikov, B. M., On modular elements of the lattice of semigroup varieties, Comment. Math. Univ. Carol., 48, 4, 595-606, (2007) · Zbl 1174.20324
[25] Vernikov, B. M., Special elements in lattices of semigroup varieties, Acta Sci. Math. (Szeged), 81, 1-2, 79-109, (2015) · Zbl 1363.20050
[26] Vernikov, B. M.; Shaprynskiǐ, V. Y., Distributive elements of the lattice of semigroup varieties, Algebra Logic, 49, 3, 301-330, (2010) · Zbl 1223.08002
[27] Volkov, M. V., Modular elements of the lattice of semigroup varieties, Contrib. Gen. Algebra, 16, 275-288, (2005) · Zbl 1223.08002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.