Two weaker variants of congruence permutability for monoid varieties. (English) Zbl 1484.20100
Here, varieties of monoids are studied as semigroups with an additional 0-ary operation fixing the identity element and a description of fi-permutable and almost fi-permutable monoid varieties is presented. A variety of algebras \(V\) is called fi-permutable if every two fully invariant congruences \(\alpha\), \(\beta \) (congruences preserved by endomorphisms) on any \(V\)-free object \(F\) permute (\(\alpha \beta = \beta \alpha \)); \(V\) is called almost fi-permutable, if instead of all fully invariant congruences permuting is required only of those of them that are contained in the least semilattice congruence on \(F\).
Reviewer: Jaak Henno (Tallinn)
Keywords:
monoid; variety; free object of a variety; fully invariant congruence; permutative congruencesReferences:
| [1] | Almeida, J., Finite Semigroups and Universal Algebra (1994), Singapore: World Scientific, Singapore · Zbl 0844.20039 |
| [2] | Evans, T., The lattice of semigroup varieties, Semigroup Forum, 2, 1-43 (1971) · Zbl 0225.20043 · doi:10.1007/BF02572269 |
| [3] | Freese, R.; Nation, JB, Congruence lattices of semilattices, Pacif. J. Math., 49, 51-58 (1973) · Zbl 0287.06002 · doi:10.2140/pjm.1973.49.51 |
| [4] | Grätzer, G., Lattice Theory: Foundation (2011), Basel: Springer, Basel · Zbl 1233.06001 · doi:10.1007/978-3-0348-0018-1 |
| [5] | Gusev, SV, Special elements of the lattice of monoid varieties, Algebra Universalis, 79, 29, 1-12 (2018) · Zbl 1498.20138 |
| [6] | Gusev, SV, On the ascending and descending chain conditions in the lattice of monoid varieties, Siberian Electron. Math. Rep., 16, 983-997 (2019) · Zbl 1444.20035 |
| [7] | 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)] · Zbl 1515.08006 |
| [8] | Gusev, SV; Vernikov, BM, Chain varieties of monoids, Dissert. Math., 534, 1-73 (2018) · Zbl 1434.20042 · doi:10.4064/dm772-2-2018 |
| [9] | Head, TJ, The varieties of commutative monoids, Nieuw Arch. Wiskunde. III Ser., 16, 203-206 (1968) · Zbl 0165.33601 |
| [10] | Hobby, D., McKenzie, R.N.: The Structure of Finite Algebras, vol. 76. American Mathematical Society, Providence Rhode Island (1988) · Zbl 0721.08001 |
| [11] | Jackson, M., Finiteness properties of varieties and the restriction to finite algebras, Semigroup Forum, 70, 154-187 (2005) · Zbl 1073.20052 · doi:10.1007/s00233-004-0161-x |
| [12] | Jackson, M.; Lee, EWH, Monoid varieties with extreme properties, Trans. Am. Math. Soc., 370, 4785-4812 (2018) · Zbl 1401.20067 · doi:10.1090/tran/7091 |
| [13] | Jones, PR, Congruence semimodular varieties of semigroups, Lect. Notes Math., 1320, 162-171 (1988) · Zbl 0643.20038 · doi:10.1007/BFb0083429 |
| [14] | Jónsson, B., On the representation of lattices, Math. Scand., 1, 193-206 (1953) · Zbl 0053.21304 · doi:10.7146/math.scand.a-10377 |
| [15] | Jónsson, B., The class of Arguesian lattices is self-dual, Algebra Universalis, 2, 396 (1972) · Zbl 0261.06008 · doi:10.1007/BF02945054 |
| [16] | Lee, EWH, Varieties generated by 2-testable monoids, Studia Sci. Math. Hungar., 49, 366-389 (2012) · Zbl 1274.20057 |
| [17] | Lee, EWH, Almost Cross varieties of aperiodic monoids with central idempotents, Beitr. Algebra Geom., 54, 121-129 (2013) · Zbl 1267.20083 · doi:10.1007/s13366-012-0094-6 |
| [18] | 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) · Zbl 1334.20048 |
| [19] | Lipparini, P., \(n\)-permutable varieties satisfy non trivial congruence identities, Algebra Universalis, 33, 159-168 (1995) · Zbl 0821.08007 · doi:10.1007/BF01190927 |
| [20] | Pastijn, F., Commuting fully invariant congruences on free completely regular semigroups, Trans. Am. Math. Soc., 323, 79-92 (1990) · Zbl 0716.20040 · doi:10.1090/S0002-9947-1991-1016809-9 |
| [21] | Petrich, M.; Reilly, NR, The modularity of the lattice of varieties of completely regular semigroups and related representations, Glasgow Math. J., 32, 137-152 (1990) · Zbl 0715.20036 · doi:10.1017/S0017089500009162 |
| [22] | Tully, EJ, The equivalence, for semigroup varieties, of two properties concerning congruence relations, Bull. Am. Math. Soc., 70, 399-400 (1964) · Zbl 0115.25101 · doi:10.1090/S0002-9904-1964-11111-3 |
| [23] | 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)] · Zbl 1115.20048 |
| [24] | 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)] · Zbl 1079.20076 |
| [25] | Vernikov, BM, Completely regular semigroup varieties whose free objects have weakly permutable fully invariant congruences, Semigroup Forum, 68, 154-158 (2004) · Zbl 1055.20047 · doi:10.1007/s00233-003-0009-9 |
| [26] | Vernikov, BM; Shaprynskiǐ, VY, Three weaker variants of congruence permutability for semigroup varieties, Siberian Electron. Math. Rep., 11, 567-604 (2014) · Zbl 1330.20085 |
| [27] | Vernikov, BM; Volkov, MV, Permutability of fully invariant congruences on relatively free semigroups, Acta Sci. Math. (Szeged), 63, 437-461 (1997) · Zbl 0889.20031 |
| [28] | Vernikov, BM; Volkov, MV, Commuting fully invariant congruences on free semigroups, Contrib. General Algebra, 12, 391-417 (2000) · Zbl 0978.20028 |
| [29] | Volkov, MV, Modular elements of the lattice of semigroup varieties, Contrib. General Algebra, 16, 275-288 (2005) · Zbl 1090.20030 |
| [30] | Wismath, SL, The lattice of varieties and pseudovarieties of band monoids, Semigroup Forum, 33, 187-198 (1986) · Zbl 0591.20060 · doi:10.1007/BF02573192 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
