×

Ehresmann theory and partition monoids. (English) Zbl 1470.18005

The paper studies Ehresmann structures on the partition monoid \({\mathcal P}(X)\) and related diagram monoids. Several new diagram monoids that arise naturally as categorical duals of the partial transformation monoids are introduced. The paper is well-written and self-contained. It provides an extensive survey of literature on Ehresmann monoids and on partition and diagram monoids. In addition, many open questions are posed.

MSC:

18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories)
20M20 Semigroups of transformations, relations, partitions, etc.
20M10 General structure theory for semigroups
20M17 Regular semigroups
20M18 Inverse semigroups
20M25 Semigroup rings, multiplicative semigroups of rings
18B05 Categories of sets, characterizations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Anderson, L. W.; Hunter, R. P.; Koch, R. J., Some results on stability in semigroups, Trans. Am. Math. Soc., 117, 521-529 (1965) · Zbl 0133.27803
[2] Aĭzenštat, A. J., Defining relations of finite symmetric semigroups, Mat. Sb. (N.S.), 45, 87, 261-280 (1958) · Zbl 0081.01901
[3] Bouc, S.; Thévenaz, J., The algebra of essential relations on a finite set, J. Reine Angew. Math., 712, 225-250 (2016) · Zbl 1339.20062
[4] Bouc, S.; Thévenaz, J., Correspondence functors and finiteness conditions, J. Algebra, 495, 150-198 (2018) · Zbl 1427.18001
[5] Bouc, S.; Thévenaz, J., Correspondence functors and lattices, J. Algebra, 518, 453-518 (2019) · Zbl 1467.06006
[6] Bouc, S.; Thévenaz, J., The algebra of Boolean matrices, correspondence functors, and simplicity, J. Comb. Algebra, 4, 3, 215-267 (2020) · Zbl 1484.06022
[7] Branco, M. J.J.; Gomes, G. M.S.; Gould, V., Ehresmann monoids, J. Algebra, 443, 349-382 (2015) · Zbl 1336.20057
[8] Brauer, R., On algebras which are connected with the semisimple continuous groups, Ann. Math. (2), 38, 4, 857-872 (1937) · JFM 63.0873.02
[9] Church, T.; Ellenberg, J. S., Homology of FI-modules, Geom. Topol., 21, 4, 2373-2418 (2017) · Zbl 1371.18012
[10] Church, T.; Ellenberg, J. S.; Farb, B., FI-modules and stability for representations of symmetric groups, Duke Math. J., 164, 9, 1833-1910 (2015) · Zbl 1339.55004
[11] 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
[12] Cockett, J. R.B.; Lack, S., Restriction categories. I. Categories of partial maps, Theor. Comput. Sci., 270, 1-2, 223-259 (2002) · Zbl 0988.18003
[13] Dolinka, I.; Gray, R. D., Maximal subgroups of free idempotent generated semigroups over the full linear monoid, Trans. Am. Math. Soc., 366, 1, 419-455 (2014) · Zbl 1297.20059
[14] 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
[15] Dolinka, I.; Đurđev, I.; East, J., Sandwich semigroups in diagram categories (2019), Preprint
[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., Infinite partition monoids, Int. J. Algebra Comput., 24, 4, 429-460 (2014) · Zbl 1305.20070
[20] East, J., Presentations for (singular) partition monoids: a new approach, Math. Proc. Camb. Philos. Soc., 165, 3, 549-562 (2018) · Zbl 1499.20127
[21] East, J., Idempotents and one-sided units in infinite partial Brauer monoids, J. Algebra, 534, 427-482 (2019) · Zbl 1444.20038
[22] East, J., Presentations for rook partition monoids and algebras and their singular ideals, J. Pure Appl. Algebra, 223, 3, 1097-1122 (2019) · Zbl 1499.20126
[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, 101, 1, 77-86 (2020) · Zbl 1508.20082
[26] East, J.; Mitchell, J. D.; Ruškuc, N.; Torpey, M., Congruence lattices of finite diagram monoids, Adv. Math., 333, 931-1003 (2018) · Zbl 1400.20060
[27] East, J.; Ruškuc, N., Congruences on infinite partition and partial Brauer monoids (2018), Preprint
[28] East, J.; Ruškuc, N., Congruence lattices of ideals in categories and (partial) semigroups, Mem. Am. Math. Soc. (2021), in press
[29] FitzGerald, D. G., Mitsch’s order and inclusion for binary relations and partitions, Semigroup Forum, 87, 1, 161-170 (2013) · Zbl 1285.20063
[30] FitzGerald, D. G.; Lau, K. W., On the partition monoid and some related semigroups, Bull. Aust. Math. Soc., 83, 2, 273-288 (2011) · Zbl 1230.20061
[31] FitzGerald, D. G.; Leech, J., Dual symmetric inverse monoids and representation theory, J. Aust. Math. Soc. A, 64, 3, 345-367 (1998) · Zbl 0927.20040
[32] Freyd, P. J.; Scedrov, A., Categories, Allegories, North-Holland Mathematical Library, vol. 39 (1990), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam · Zbl 0698.18002
[33] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.8.10 (2018)
[34] Gomes, G. M.S.; Gould, V., Fundamental Ehresmann semigroups, Semigroup Forum, 63, 1, 11-33 (2001) · Zbl 0998.20051
[35] Gould, V., Notes on restriction semigroups and related structures (2010)
[36] Gould, V., Restriction and Ehresmann semigroups, (Proceedings of the International Conference on Algebra 2010 (2012), World Sci. Publ.: World Sci. Publ. Hackensack, NJ), 265-288 · Zbl 1264.20067
[37] Gray, R., Hall’s condition and idempotent rank of ideals of endomorphism monoids, Proc. Edinb. Math. Soc. (2), 51, 1, 57-72 (2008) · Zbl 1138.20056
[38] Gray, R.; Ruskuc, N., On maximal subgroups of free idempotent generated semigroups, Isr. J. Math., 189, 147-176 (2012) · Zbl 1276.20063
[39] 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
[40] Gray, R. D., The minimal number of generators of a finite semigroup, Semigroup Forum, 89, 1, 135-154 (2014) · Zbl 1321.20048
[41] Green, J. A., On the structure of semigroups, Ann. Math. (2), 54, 163-172 (1951) · Zbl 0043.25601
[42] Grood, C., The rook partition algebra, J. Comb. Theory, Ser. A, 113, 2, 325-351 (2006) · Zbl 1082.05095
[43] Halverson, T.; Ram, A., Partition algebras, Eur. J. Comb., 26, 6, 869-921 (2005) · Zbl 1112.20010
[44] Hirsch, R.; Hodkinson, I., Representability is not decidable for finite relation algebras, Trans. Am. Math. Soc., 353, 4, 1403-1425 (2001) · Zbl 0965.03079
[45] Hirsch, R.; Jackson, M., Undecidability of representability as binary relations, J. Symb. Log., 77, 4, 1211-1244 (2012) · Zbl 1279.03084
[46] 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
[47] Howie, J. M., Idempotent generators in finite full transformation semigroups, Proc. R. Soc. Edinb. A, 81, 3-4, 317-323 (1978) · Zbl 0403.20038
[48] 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
[49] Howie, J. M.; McFadden, R. B., Idempotent rank in finite full transformation semigroups, Proc. R. Soc. Edinb. A, 114, 3-4, 161-167 (1990) · Zbl 0704.20050
[50] Jackson, M.; Stokes, T., An invitation to C-semigroups, Semigroup Forum, 62, 2, 279-310 (2001) · Zbl 0982.20051
[51] Jones, V. F.R., The Potts model and the symmetric group, (Subfactors. Subfactors, Kyuzeso, 1993 (1994), World Sci. Publ.: World Sci. Publ. River Edge, NJ), 259-267 · Zbl 0938.20505
[52] Jónsson, B., Representation of modular lattices and of relation algebras, Trans. Am. Math. Soc., 92, 449-464 (1959) · Zbl 0105.25302
[53] Kauffman, L. H., An invariant of regular isotopy, Trans. Am. Math. Soc., 318, 2, 417-471 (1990) · Zbl 0763.57004
[54] Klimov, V. N., Congruences of globally idempotent semigroups, Ural. Gos. Univ. Mat. Zap., 10, 3, 73-105 (1977), 217 · Zbl 0434.20035
[55] Kudryavtseva, G.; Lawson, M. V., A perspective on non-commutative frame theory, Adv. Math., 311, 378-468 (2017) · Zbl 1423.06041
[56] Kudryavtseva, G.; Maltcev, V., Two generalisations of the symmetric inverse semigroups, Publ. Math. (Debr.), 78, 2, 253-282 (2011) · Zbl 1232.20072
[57] Kudryavtseva, G.; Maltcev, V.; Umar, A., Presentation for the partial dual symmetric inverse monoid, Commun. Algebra, 43, 4, 1621-1639 (2015) · Zbl 1322.20052
[58] Lawson, M. V., Rees matrix semigroups, Proc. Edinb. Math. Soc. (2), 33, 1, 23-37 (1990) · Zbl 0668.20049
[59] Lawson, M. V., Semigroups and ordered categories. I. The reduced case, J. Algebra, 141, 2, 422-462 (1991) · Zbl 0747.18007
[60] Lawson, M. V., Inverse semigroups, (The Theory of Partial Symmetries (1998), World Scientific Publishing Co., Inc.: World Scientific Publishing Co., Inc. River Edge, NJ)
[61] Lipscomb, S., Symmetric Inverse Semigroups, Mathematical Surveys and Monographs, vol. 46 (1996), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0857.20047
[62] Lyndon, R. C., The representation of relational algebras, Ann. Math. (2), 51, 707-729 (1950) · Zbl 0037.29302
[63] Lyndon, R. C., The representation of relation algebras. II, Ann. Math. (2), 63, 294-307 (1956) · Zbl 0070.24601
[64] Mal’cev, A. I., Symmetric groupoids, Mat. Sb. (N.S.). (Twelve Papers in Logic and Algebra. Twelve Papers in Logic and Algebra, Amer. Math. Soc. Translations Ser 2, vol. 113 (1979), AMS), 31, 73, 235-250 (1952), English translation · Zbl 0404.20054
[65] Margolis, S.; Stein, I., Ehressman semigroups whose categories are EI and their representation theory (2020), Preprint
[66] Martin, P., Temperley-Lieb algebras for nonplanar statistical mechanics—the partition algebra construction, J. Knot Theory Ramif., 3, 1, 51-82 (1994) · Zbl 0804.16002
[67] Martin, P., The structure of the partition algebras, J. Algebra, 183, 2, 319-358 (1996) · Zbl 0863.20009
[68] Martin, P. P., The partition algebra and the Potts model transfer matrix spectrum in high dimensions, J. Phys. A, 33, 19, 3669-3695 (2000) · Zbl 0951.82006
[69] McCune, W., Prover9 and Mace4 (2005-2010)
[70] Mishra, A.; Srivastava, S., Jucys-Murphy elements of partition algebras for the rook monoid (2019), Preprint
[71] Mitchell, J. D., Semigroups - GAP package, Version 3.2.4 (Feb. 2020)
[72] Montague, J. S.; Plemmons, R. J., Maximal subgroups of the semigroup of relations, J. Algebra, 13, 575-587 (1969) · Zbl 0184.03703
[73] Munn, W. D., Matrix representations of inverse semigroups, Proc. Lond. Math. Soc., 3, 14, 165-181 (1964) · Zbl 0135.04103
[74] Nambooripad, K. S.S., Structure of regular semigroups. I, Mem. Am. Math. Soc., 22, 224 (1979), vii+119 · Zbl 0457.20051
[75] Nordahl, T. E.; Scheiblich, H. E., Regular ⁎-semigroups, Semigroup Forum, 16, 3, 369-377 (1978) · Zbl 0408.20043
[76] Plemmons, R. J.; West, M. T., On the semigroup of binary relations, Pac. J. Math., 35, 743-753 (1970) · Zbl 0208.01801
[77] Resende, P., Étale groupoids and their quantales, Adv. Math., 208, 1, 147-209 (2007) · Zbl 1116.06014
[78] Rhodes, J.; Steinberg, B., The q-Theory of Finite Semigroups, Springer Monographs in Mathematics (2009), Springer: Springer New York · Zbl 1186.20043
[79] Schein, B. M., Relation algebras and function semigroups, Semigroup Forum, 1, 1, 1-62 (1970) · Zbl 0197.29404
[80] Schweizer, B.; Sklar, A., The algebra of functions, Math. Ann., 139, 366-382 (1960) · Zbl 0095.10101
[81] Schweizer, B.; Sklar, A., The algebra of functions. II, Math. Ann., 143, 440-447 (1961) · Zbl 0099.31901
[82] Schweizer, B.; Sklar, A., The algebra of functions. III, Math. Ann., 161, 171-196 (1965) · Zbl 0134.12602
[83] Schweizer, B.; Sklar, A., Function systems, Math. Ann., 172, 1-16 (1967) · Zbl 0163.01403
[84] Stein, I., The representation theory of the monoid of all partial functions on a set and related monoids as EI-category algebras, J. Algebra, 450, 549-569 (2016) · Zbl 1337.20070
[85] Stein, I., Algebras of Ehresmann semigroups and categories, Semigroup Forum, 95, 3, 509-526 (2017) · Zbl 1422.20030
[86] Stein, I., Erratum to: algebras of Ehresmann semigroups and categories, Semigroup Forum, 96, 3, 603-607 (2018) · Zbl 1422.20031
[87] Stein, I., The global dimension of the algebra of the monoid of all partial functions on an n-set as the algebra of the EI-category of epimorphisms between subsets, J. Pure Appl. Algebra, 223, 8, 3515-3536 (2019) · Zbl 1454.20112
[88] Stein, I., Representation theory of order-related monoids of partial functions as locally trivial category algebras, Algebr. Represent. Theory, 23, 4, 1543-1567 (2020) · Zbl 1457.20050
[89] Steinberg, B., Möbius functions and semigroup representation theory, J. Comb. Theory, Ser. A, 113, 5, 866-881 (2006) · Zbl 1148.20049
[90] Steinberg, B., Möbius functions and semigroup representation theory. II. Character formulas and multiplicities, Adv. Math., 217, 4, 1521-1557 (2008) · Zbl 1155.20057
[91] Steinberg, B., A groupoid approach to discrete inverse semigroup algebras, Adv. Math., 223, 2, 689-727 (2010) · Zbl 1188.22003
[92] Steinberg, B., Representation Theory of Finite Monoids, Universitext (2016), Springer: Springer Cham · Zbl 1428.20003
[93] T. Stokes, Left restriction monoids from left E-completions, in preparation. · Zbl 1516.20124
[94] Tarski, A., On the calculus of relations, J. Symb. Log., 6, 73-89 (1941) · JFM 67.0973.02
[95] Tarski, A., Contributions to the theory of models. III, Ned. Akad. Wet. Proc. Ser. A. Ned. Akad. Wet. Proc. Ser. A, Indag. Math., 17, 56-64 (1955)
[96] Tarski, A., Logic, Semantics, Metamathematics (1983), Hackett Publishing Co.: Hackett Publishing Co. Indianapolis, IN, Papers from 1923 to 1938, Translated by J.H. Woodger, Edited and with an introduction by John Corcoran
[97] Temperley, H. N.V.; Lieb, E. H., Relations between the “percolation” and “colouring” problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the “percolation” problem, Proc. R. Soc. Lond. Ser. A, 322, 1549, 251-280 (1971) · Zbl 0211.56703
[98] Vagner, V. V., Representation of ordered semigroups, Mat. Sb. (N.S.). Mat. Sb. (N.S.), Transl. Am. Math. Soc. (2), 36, 36, 295-336 (1979), English translation in · Zbl 0285.06011
[99] Šaĭn, B. M., On the theory of generalized groups, Dokl. Akad. Nauk SSSR, 153, 296-299 (1963) · Zbl 0138.25501
[100] Šaĭn, B. M., On certain classes of semigroups of binary relations, Sib. Mat. Zh., 6, 616-635 (1965) · Zbl 0237.20059
[101] Zareckiĭ, K. A., The semigroup of binary relations, Mat. Sb. (N.S.), 61, 103, 291-305 (1963) · Zbl 0125.28003
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.