Abstract
The set of all cancellable elements of the lattice of semigroup varieties has recently been shown to be countably infinite. But the description of all cancellable elements of the lattice \(\mathbb{MON}\) of monoid varieties remains unknown. This problem is addressed in the present article. The first example of a monoid variety with modular but non-distributive subvariety lattice is first exhibited. Then a necessary condition of the modularity of an element in \(\mathbb{MON}\) is established. These results play a crucial role in the complete description of all cancellable elements of the lattice \(\mathbb{MON}\). It turns out that there are precisely five such elements.
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References
J. Almeida, Finite Semigroups and Universal Algebra, World Scientific (Singapore, 1994).
S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Springer Verlag (New York, 1981).
G. Grätzer, Lattice Theory: Foundation, Springer Basel (Basel, 2011).
S. V. Gusev, On the lattice of overcommutative varieties of monoids, Izv. VUZ. Matem., 62 (2018), 28-32 (in Russian); translation in Russ. Math. Izv. VUZ, 62 (2018), 23-26.
S. V. Gusev, Special elements of the lattice of monoid varieties,Algebra Univ., 79 (2018), Article 29, 12 pp.
S. V. Gusev, Standard elements of the lattice of monoid varieties, Algebra Logika, 59 (2020), 615-626 (in Russian); translation in Algebra Logic, 59 (2021), 415-422.
S. V. Gusev and B. M. Vernikov, Chain varieties of monoids, Dissert. Math., 534 (2018), 1-73.
S. V. Gusev and B. M. Vernikov, Two weaker variants of congruence permutability for monoid varieties, Semigroup Forum, 103 (2021), 106-152.
M. Jackson, Small Semigroup Related Structures with Infinite Properties, Ph.D. thesis, University of Tasmania (Hobart, 1999).
M. Jackson, Finiteness properties of varieties and the restriction to finite algebras, Semigroup Forum, 70 (2005), 154-187.
J. Ježek, The lattice of equational theories. Part I: modular elements, Czechoslovak Math. J., 31 (1981), 127-152.
E. W. H. Lee, Maximal Specht varieties of monoids, Mosc. Math. J., 12 (2012), 787- 802.
E. W. H. Lee, Varieties generated by 2-testable monoids, Studia Sci. Math. Hungar., 49 (2012), 366-389.
O. Sapir, Finitely based words, Int. J. Algebra Comput., 10 (2000), 457-480.
V. Yu. Shaprynskiǐ, D. V. Skokov and B. M. Vernikov, Cancellable elements of the lattices of varieties of semigroups and epigroups, Comm. Algebra, 47 (2019), 4697-4712.
B. M. Vernikov, Special elements in lattices of semigroup varieties, Acta Sci. Math. (Szeged), 81 (2015), 79-109.
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The first author is supported by the Ministry of Science and Higher Education of the Russian Federation (project FEUZ-2020-0016).
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Gusev, S.V., Lee, E.W.H. Cancellable elements of the lattice of monoid varieties. Acta Math. Hungar. 165, 156–168 (2021). https://doi.org/10.1007/s10474-021-01177-z
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DOI: https://doi.org/10.1007/s10474-021-01177-z