Abstract
We completely determine all varieties of monoids on whose free objects all fully invariant congruences or all fully invariant congruences contained in the least semilattice congruence permute. Along the way, we find several new monoid varieties with the distributive subvariety lattice (only a few examples of varieties with such a property are known so far).
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Almeida, J.: Finite Semigroups and Universal Algebra. World Scientific, Singapore (1994)
Evans, T.: The lattice of semigroup varieties. Semigroup Forum 2, 1–43 (1971)
Freese, R., Nation, J.B.: Congruence lattices of semilattices. Pacif. J. Math. 49, 51–58 (1973)
Grätzer, G.: Lattice Theory: Foundation. Springer, Basel (2011)
Gusev, S.V.: Special elements of the lattice of monoid varieties. Algebra Universalis 79(29), 1–12 (2018)
Gusev, S.V.: On the ascending and descending chain conditions in the lattice of monoid varieties. Siberian Electron. Math. Rep. 16, 983–997 (2019)
Gusev, S.V.: Standard elements of the lattice of monoid varieties. Algebra i Logika 59, 615–626 (2020). [Russian; Engl. translation: Algebra and Logic 59, 415–422 (2021)]
Gusev, S.V., Vernikov, B.M.: Chain varieties of monoids. Dissert. Math. 534, 1–73 (2018)
Head, T.J.: The varieties of commutative monoids. Nieuw Arch. Wiskunde. III Ser. 16, 203–206 (1968)
Hobby, D., McKenzie, R.N.: The Structure of Finite Algebras, vol. 76. American Mathematical Society, Providence Rhode Island (1988)
Jackson, M.: Finiteness properties of varieties and the restriction to finite algebras. Semigroup Forum 70, 154–187 (2005)
Jackson, M., Lee, E.W.H.: Monoid varieties with extreme properties. Trans. Am. Math. Soc. 370, 4785–4812 (2018)
Jones, P.R.: Congruence semimodular varieties of semigroups. Lect. Notes Math. 1320, 162–171 (1988)
Jónsson, B.: On the representation of lattices. Math. Scand. 1, 193–206 (1953)
Jónsson, B.: The class of Arguesian lattices is self-dual. Algebra Universalis 2, 396 (1972)
Lee, E.W.H.: Varieties generated by 2-testable monoids. Studia Sci. Math. Hungar. 49, 366–389 (2012)
Lee, E.W.H.: Almost Cross varieties of aperiodic monoids with central idempotents. Beitr. Algebra Geom. 54, 121–129 (2013)
Lee, E.W.H.: Inherently non-finitely generated varieties of aperiodic monoids with central idempotents. Zapiski Nauchnykh Seminarov POMI (Notes of Sci. Seminars of the St Petersburg Branch of the Math. Institute of the Russ. Acad. Sci.) 423, 166–182 (2014); see also J. Math. Sci., 209, 588–599 (2015)
Lipparini, P.: \(n\)-permutable varieties satisfy non trivial congruence identities. Algebra Universalis 33, 159–168 (1995)
Pastijn, F.: Commuting fully invariant congruences on free completely regular semigroups. Trans. Am. Math. Soc. 323, 79–92 (1990)
Petrich, M., Reilly, N.R.: The modularity of the lattice of varieties of completely regular semigroups and related representations. Glasgow Math. J. 32, 137–152 (1990)
Tully, E.J.: The equivalence, for semigroup varieties, of two properties concerning congruence relations. Bull. Am. Math. Soc. 70, 399–400 (1964)
Vernikov, B.M.: On weaker variant of congruence permutability for semigroup varieties. Algebra i Logika 43, 3–31 (2004). [Russian; Engl. translation: Algebra and Logic 43, 1–16 (2004)]
Vernikov, B.M.: On semigroup varieties on whose free objects almost all fully invariant congruences are weakly permutable. Algebra i Logika 43, 635–649 (2004). [Russian; Engl. translation: Algebra and Logic 43, 357–364 (2004)]
Vernikov, B.M.: Completely regular semigroup varieties whose free objects have weakly permutable fully invariant congruences. Semigroup Forum 68, 154–158 (2004)
Vernikov, B.M., Shaprynskiǐ, V.Y.: Three weaker variants of congruence permutability for semigroup varieties. Siberian Electron. Math. Rep. 11, 567–604 (2014). [Russian]
Vernikov, B.M., Volkov, M.V.: Permutability of fully invariant congruences on relatively free semigroups. Acta Sci. Math. (Szeged) 63, 437–461 (1997)
Vernikov, B.M., Volkov, M.V.: Commuting fully invariant congruences on free semigroups. Contrib. General Algebra 12, 391–417 (2000)
Volkov, M.V.: Modular elements of the lattice of semigroup varieties. Contrib. General Algebra 16, 275–288 (2005)
Wismath, S.L.: The lattice of varieties and pseudovarieties of band monoids. Semigroup Forum 33, 187–198 (1986)
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Communicated by Edmond W. H. Lee.
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The work was supported by the Ministry of Science and Higher Education of the Russian Federation (Project FEUZ-2020-0016).
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Gusev, S.V., Vernikov, B.M. Two weaker variants of congruence permutability for monoid varieties. Semigroup Forum 103, 106–152 (2021). https://doi.org/10.1007/s00233-021-10196-9
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DOI: https://doi.org/10.1007/s00233-021-10196-9