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Idempotents and one-sided units: lattice invariants and a semigroup of functors on the category of monoids. (English) Zbl 1456.20074

Author’s abstract: For a monoid \(M\), we denote by \(\mathbb{G}(M)\) the group of units, \(\mathbb{E}(M)\) the submonoid generated by the idempotents, and \(\mathbb{G}_L(M)\) and \(\mathbb{G}_R(M)\) the submonoids consisting of all left or right units. Writing \(\mathcal{M}\) for the (monoidal) category of monoids, \(\mathbb{G}\), \(\mathbb{E}\), \(\mathbb{G}_L\) and \(\mathbb{G}_R\) are all (monoidal) functors \(\mathcal{M}\to \mathcal{M}\). There are other natural functors associated to submonoids generated by combinations of idempotents and one- or two-sided units. The above functors generate a monoid with composition as its operation. We show that this monoid has size 15, and describe its algebraic structure. We also show how to associate certain lattice invariants to a monoid, and classify the lattices that arise in this fashion. A number of examples are discussed throughout, some of which are essential for the proofs of the main theoretical results.

MSC:

20M50 Connections of semigroups with homological algebra and category theory
20M10 General structure theory for semigroups
20M15 Mappings of semigroups
20M20 Semigroups of transformations, relations, partitions, etc.
18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories)

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References:

[1] Araújo, J.; Mitchell, J. D., Relative ranks in the monoid of endomorphisms of an independence algebra, Monatshefte Math., 151, 1, 1-10 (2007) · Zbl 1126.20045
[2] Banach, S., Sur un thèoréme de M. Sierpiński, Fundam. Math., 25, 5-6 (1935) · JFM 61.0230.01
[3] Bergman, G. M., Generating infinite symmetric groups, Bull. Lond. Math. Soc., 38, 3, 429-440 (2006) · Zbl 1103.20003
[4] Bulman-Fleming, S., Regularity and products of idempotents in endomorphism monoids of projective acts, Mathematika, 42, 2, 354-367 (1995) · Zbl 0836.20088
[5] Bulman-Fleming, S.; Fountain, J., Products of idempotent endomorphisms of free acts of infinite rank, Monatshefte Math., 124, 1, 1-16 (1997) · Zbl 0888.20041
[6] Chen, S. Y.; Hsieh, S. C., Factorizable inverse semigroups, Semigroup Forum, 8, 4, 283-297 (1974) · Zbl 0298.20055
[7] Clifford, A. H.; Preston, G. B., The Algebraic Theory of Semigroups. Vol. I, Mathematical Surveys, vol. 7 (1961), American Mathematical Society: American Mathematical Society Providence, R.I. · Zbl 0111.03403
[8] Dolinka, I.; East, J., Idempotent generation in the endomorphism monoid of a uniform partition, Commun. Algebra, 44, 12, 5179-5198 (2016) · Zbl 1349.20056
[9] Dolinka, I.; East, J., The idempotent-generated subsemigroup of the Kauffman monoid, Glasg. Math. J., 59, 3, 673-683 (2017) · Zbl 1472.20136
[10] Dolinka, I.; East, J., Twisted Brauer monoids, Proc. R. Soc. Edinb., Sect. A, 148, 4, 731-750 (2018) · Zbl 1489.20025
[11] Dolinka, I.; East, J.; Gray, R. D., Motzkin monoids and partial Brauer monoids, J. Algebra, 471, 251-298 (2017) · Zbl 1406.20065
[12] Dolinka, I.; East, J.; Mitchell, J. D., Idempotent rank in the endomorphism monoid of a nonuniform partition, Bull. Aust. Math. Soc., 93, 1, 73-91 (2016) · Zbl 1342.20056
[13] Dolinka, I.; Gray, R. D.; Ruškuc, N., On regularity and the word problem for free idempotent generated semigroups, Proc. Lond. Math. Soc. (3), 114, 3, 401-432 (2017) · Zbl 1434.20039
[14] Dolinka, I.; Ruškuc, N., Every group is a maximal subgroup of the free idempotent generated semigroup over a band, Int. J. Algebra Comput., 23, 3, 573-581 (2013) · Zbl 1281.20068
[15] Droste, M.; Göbel, R., The normal subsemigroups of the monoid of injective maps, Semigroup Forum, 87, 2, 298-312 (2013) · Zbl 1288.20087
[16] Easdown, D., Biordered sets come from semigroups, J. Algebra, 96, 2, 581-591 (1985) · Zbl 0602.20055
[17] East, J., Generators and relations for partition monoids and algebras, J. Algebra, 339, 1-26 (2011) · Zbl 1277.20069
[18] East, J., On the singular part of the partition monoid, Int. J. Algebra Comput., 21, 1-2, 147-178 (2011) · Zbl 1229.20066
[19] East, J., Generation of infinite factorizable inverse monoids, Semigroup Forum, 84, 2, 267-283 (2012) · Zbl 1254.20045
[20] East, J., Infinite partition monoids, Int. J. Algebra Comput., 24, 4, 429-460 (2014) · Zbl 1305.20070
[21] East, J., Infinite dual symmetric inverse monoids, Period. Math. Hung., 75, 2, 273-285 (2017) · Zbl 1399.20075
[22] East, J., Idempotents and one-sided units in infinite partial Brauer monoids, J. Algebra, 534, 427-482 (2019) · Zbl 1444.20038
[23] East, J.; FitzGerald, D. G., The semigroup generated by the idempotents of a partition monoid, J. Algebra, 372, 108-133 (2012) · Zbl 1281.20077
[24] East, J.; Gray, R. D., Diagram monoids and Graham-Houghton graphs: idempotents and generating sets of ideals, J. Comb. Theory, Ser. A, 146, 63-128 (2017) · Zbl 1351.05227
[25] East, J.; Higgins, P. M., Green’s relations and stability for subsemigroups, Semigroup Forum (2020), in press · Zbl 1508.20082
[26] Erdos, J. A., On products of idempotent matrices, Glasg. Math. J., 8, 118-122 (1967) · Zbl 0157.07101
[27] Everitt, B.; Fountain, J., Partial symmetry, reflection monoids and Coxeter groups, Adv. Math., 223, 5, 1782-1814 (2010) · Zbl 1238.20073
[28] Everitt, B.; Fountain, J., Partial mirror symmetry, lattice presentations and algebraic monoids, Proc. Lond. Math. Soc. (3), 107, 2, 414-450 (2013) · Zbl 1297.20068
[29] FitzGerald, D. G., A presentation for the monoid of uniform block permutations, Bull. Aust. Math. Soc., 68, 2, 317-324 (2003) · Zbl 1043.20037
[30] FitzGerald, D. G., Factorizable inverse monoids, Semigroup Forum, 80, 3, 484-509 (2010) · Zbl 1202.20067
[31] FitzGerald, D. G.; Leech, J., Dual symmetric inverse monoids and representation theory, J. Aust. Math. Soc. Ser. A, 64, 3, 345-367 (1998) · Zbl 0927.20040
[32] Fountain, J., Products of idempotent integer matrices, Math. Proc. Camb. Philos. Soc., 110, 3, 431-441 (1991) · Zbl 0751.20049
[33] Fountain, J.; Gould, V., Products of idempotent endomorphisms of relatively free algebras with weak exchange properties, Proc. Edinb. Math. Soc. (2), 50, 2, 343-362 (2007) · Zbl 1132.20040
[34] Fountain, J.; Lewin, A., Products of idempotent endomorphisms of an independence algebra of finite rank, Proc. Edinb. Math. Soc. (2), 35, 3, 493-500 (1992) · Zbl 0794.20066
[35] Fountain, J.; Lewin, A., Products of idempotent endomorphisms of an independence algebra of infinite rank, Math. Proc. Camb. Philos. Soc., 114, 2, 303-319 (1993) · Zbl 0819.20070
[36] Gray, R., Idempotent rank in endomorphism monoids of finite independence algebras, Proc. R. Soc. Edinb., Sect. A, 137, 2, 303-331 (2007) · Zbl 1197.08001
[37] Gray, R.; Ruškuc, N., Maximal subgroups of free idempotent-generated semigroups over the full transformation monoid, Proc. Lond. Math. Soc. (3), 104, 5, 997-1018 (2012) · Zbl 1254.20054
[38] Gray, R.; Ruskuc, N., On maximal subgroups of free idempotent generated semigroups, Isr. J. Math., 189, 147-176 (2012) · Zbl 1276.20063
[39] Green, J. A., On the structure of semigroups, Ann. Math. (2), 54, 163-172 (1951) · Zbl 0043.25601
[40] Higgins, P. M., Techniques of Semigroup Theory, Oxford Science Publications (1992), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York · Zbl 0744.20046
[41] Higgins, P. M.; Howie, J. M.; Mitchell, J. D.; Ruškuc, N., Countable versus uncountable ranks in infinite semigroups of transformations and relations, Proc. Edinb. Math. Soc. (2), 46, 3, 531-544 (2003) · Zbl 1043.20040
[42] Higgins, P. M.; Howie, J. M.; Ruškuc, N., Generators and factorisations of transformation semigroups, Proc. R. Soc. Edinb., Sect. A, 128, 6, 1355-1369 (1998) · Zbl 0920.20056
[43] Howie, J. M., The subsemigroup generated by the idempotents of a full transformation semigroup, J. Lond. Math. Soc., 41, 707-716 (1966) · Zbl 0146.02903
[44] Howie, J. M., Idempotent generators in finite full transformation semigroups, Proc. R. Soc. Edinb., Sect. A, 81, 3-4, 317-323 (1978) · Zbl 0403.20038
[45] Howie, J. M., Fundamentals of Semigroup Theory, London Mathematical Society Monographs. New Series, vol. 12 (1995), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York, Oxford Science Publications · Zbl 0835.20077
[46] Howie, J. M.; McFadden, R. B., Idempotent rank in finite full transformation semigroups, Proc. R. Soc. Edinb., Sect. A, 114, 3-4, 161-167 (1990) · Zbl 0704.20050
[47] Howie, J. M.; Ruškuc, N.; Higgins, P. M., On relative ranks of full transformation semigroups, Commun. Algebra, 26, 3, 733-748 (1998) · Zbl 0902.20027
[48] Jónsson, B.; Rival, I., Lattice varieties covering the smallest nonmodular variety, Pac. J. Math., 82, 2, 463-478 (1979) · Zbl 0424.06004
[49] Joyal, A.; Street, R., Braided tensor categories, Adv. Math., 102, 1, 20-78 (1993) · Zbl 0817.18007
[50] Kudryavtseva, G.; Mazorchuk, V., On presentations of Brauer-type monoids, Cent. Eur. J. Math., 4, 3, 413-434 (2006), (electronic) · Zbl 1130.20041
[51] Lallement, G., Semigroups and Combinatorial Applications, Pure and Applied Mathematics (1979), John Wiley & Sons: John Wiley & Sons New York-Chichester-Brisbane, A Wiley-Interscience Publication · Zbl 0421.20025
[52] Mac Lane, S., Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5 (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0906.18001
[53] Maltcev, V.; Mazorchuk, V., Presentation of the singular part of the Brauer monoid, Math. Bohem., 132, 3, 297-323 (2007) · Zbl 1163.20035
[54] Maltcev, V.; Mitchell, J. D.; Ruškuc, N., The Bergman property for semigroups, J. Lond. Math. Soc. (2), 80, 1, 212-232 (2009) · Zbl 1232.20060
[55] Mazorchuk, V., On the structure of Brauer semigroup and its partial analogue, Probl. Algebra, 13, 29-45 (1998)
[56] McAlister, D. B., Semigroups generated by a group and an idempotent, Commun. Algebra, 26, 2, 515-547 (1998) · Zbl 0893.20045
[57] McDougall, R. G.; Thornton, L. K., On base radical operators for classes of finite associative rings, Bull. Aust. Math. Soc., 98, 2, 239-250 (2018) · Zbl 1434.16009
[58] McKenzie, R., Equational bases and nonmodular lattice varieties, Trans. Am. Math. Soc., 174, 1-43 (1972) · Zbl 0265.08006
[59] Mesyan, Z., Monoids of injective maps closed under conjugation by permutations, Isr. J. Math., 189, 287-305 (2012) · Zbl 1263.20060
[60] Mitchell, J. D., Semigroups - GAP Package, Version 3.1.3 (June 2019)
[61] Mitchell, J. D.; Péresse, Y., Generating countable sets of surjective functions, Fundam. Math., 213, 1, 67-93 (2011) · Zbl 1238.20070
[62] Mitchell, J. D.; Péresse, Y., Sierpiński rank for groups and semigroups, Wiad. Mat., 48, 2, 209-215 (2012) · Zbl 1277.22003
[63] Nambooripad, K. S.S., Structure of regular semigroups. I, Mem. Am. Math. Soc., 22, 224 (1979), vii+119 · Zbl 0457.20051
[64] Nation, J. B.; Nishida, J., A refinement of the equaclosure operator, Algebra Univers., 79, 2 (2018), Paper No. 46 · Zbl 1400.08005
[65] Putcha, M. S., Algebraic monoids whose nonunits are products of idempotents, Proc. Am. Math. Soc., 103, 1, 38-40 (1988) · Zbl 0658.20036
[66] Putcha, M. S., Products of idempotents in algebraic monoids, J. Aust. Math. Soc., 80, 2, 193-203 (2006) · Zbl 1102.20043
[67] Rhodes, J.; Steinberg, B., The q-Theory of Finite Semigroups, Springer Monographs in Mathematics (2009), Springer: Springer New York · Zbl 1186.20043
[68] Sierpiński, W., Sur les suites infinies de fonctions définies dans les ensembles quelconques, Fundam. Math., 24, 209-212 (1935) · JFM 61.0229.03
[69] Šutov, È. G., Semigroups of one-to-one transformations, Dokl. Akad. Nauk SSSR, 140, 1026-1028 (1961) · Zbl 0112.25603
[70] Thornton, L. K., On base radical and semisimple operators for a class of finite algebras, Beitr. Algebra Geom., 59, 2, 361-374 (2018) · Zbl 1428.16019
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