On orientation-preserving transformations of a chain. (English) Zbl 1509.20123
Summary: In this paper, we introduce the notion of an orientation-preserving transformation on an arbitrary chain, as a natural extension for infinite chains of the well known concept for finite chains introduced in 1998 by D. B. McAlister [Commun. Algebra 26, No. 2, 515–547 (1998; Zbl 0893.20045)] and, independently, in 1999 by P. M. Catarino and P. M. Higgins [Semigroup Forum 58, No. 2, 190–206 (1999; Zbl 0919.20041)]. We consider the monoid \(\mathcal{POP}(X)(X)\) of all orientation-preserving partial transformations on a finite or infinite chain \(X\) and its submonoids and \(\mathcal{POPJ}\) of all orientation-preserving full transformations and of all orientation-preserving partial permutations on \(X\), respectively. The monoid \(\mathcal{OPPO}\) of all order-preserving partial transformations on \(X\) and its injective counterpart \(\mathcal{POJ}\) are also considered. We study the regularity and give descriptions of the Green’s relations of the monoids \(\mathcal{POP}(X)\), \(\mathcal{PO}(X)\), \(\mathcal{OP}(X)\), \(\mathcal{POPJ}(X)\) and \(\mathcal{POJ}(X)\)
MSC:
| 20M20 | Semigroups of transformations, relations, partitions, etc. |
| 20M10 | General structure theory for semigroups |
Keywords:
order-preserving transformation; orientation-preserving transformation; transformation semigroupsReferences:
| [1] | Adams, M. E.; Gould, M., Posets whose monoids of order-preserving maps are regular, Order., 6, 2, 195-201 (1989) · Zbl 0689.06004 · doi:10.1007/BF02034336 |
| [2] | Aĭzenštat, A. Ya., The defining relations of the endomorphism semigroup of a finite linearly ordered set, Sibirsk. Mat, 3, 161-169 (1962) · Zbl 0114.01702 |
| [3] | Aĭzenštat, A. Ya., Homomorphisms of semigroups of endomorphisms of ordered sets, Uch. Zap., Leningr. Gos. Pedagog. Inst, 238, 38-48 (1962) · Zbl 0208.29602 |
| [4] | Aĭzenštat, A. Ya., Ya. Aĭzenštat, Regular semigroups of endomorphisms of ordered sets, Leningrad. Gos. Ped. Inst. Učen. Zap, 387, 3-11 (1968) |
| [5] | Araújo, J.; Fernandes, V. H.; Jesus, M. M.; Maltcev, V.; Mitchell, J. D., Automorphisms of partial endomorphism semigroups, Publ. Math. Debrecen, 79, 1-2, 23-39 (2011) · Zbl 1235.20055 · doi:10.5486/PMD.2011.4703 |
| [6] | Arthur, R. E.; Ruškuc, N., Presentations for two extensions of the monoid of order-preserving mappings on a finite chain, Sea. Bull. Math, 24, 1, 1-7 (2000) · Zbl 0973.20049 · doi:10.1007/s10012-000-0001-1 |
| [7] | Catarino, P. M., Monoids of orientation-preserving transformations of a finite chain and their presentations, Proc. Conf. Semigroups Appl. St Andrews, Scotland, 1998, 39-46 (1997) · Zbl 0968.20033 |
| [8] | Catarino, P. M.; Higgins, P. M., The monoid of orientation-preserving mappings on a chain, Semigroup Forum, 58, 2, 190-206 (1999) · Zbl 0919.20041 · doi:10.1007/s002339900014 |
| [9] | Dimitrova, I.; Fernandes, V. H.; Koppitz, J., The maximal subsemigroups of semigroups of transformations preserving or reversing the orientation on a finite chain, Publ. Math. Debrecen, 81, 1-2, 11-29 (2012) · Zbl 1257.20061 · doi:10.5486/PMD.2012.4897 |
| [10] | Dimitrova, I.; Fernandes, V. H.; Koppitz, J., A note on generators of the endomorphism semigroup of an infinite countable chain, J. Algebra Appl, 16, 2, 1750031 (2017) · Zbl 1357.20031 · doi:10.1142/S0219498817500311 |
| [11] | Fernandes, V. H., Semigroups of order-preserving mappings on a finite chain: A new class of divisors, Semigroup Forum, 54, 1, 230-236 (1997) · Zbl 0868.20049 · doi:10.1007/BF02676604 |
| [12] | Fernandes, V. H., Semigroups of order-preserving mappings on a finite chain: Another class of divisors, Izvestiya VUZ. Matematika, 3, 478, 51-59 (2002) |
| [13] | Fernandes, V. H.; Gomes, G. M. S.; Jesus, M. M., Congruences on monoids of transformation preserving the orientation on a finite chain, J. Algebra, 321, 3, 743-757 (2009) · Zbl 1167.20035 · doi:10.1016/j.jalgebra.2008.11.005 |
| [14] | Fernandes, V. H.; Gomes, G. M. S.; Jesus, M. M., The cardinal and the idempotent number of various monoids of transformations on a finite chain, Bull. Malays. Math. Sci. Soc, 34, 2, 79-85 (2011) · Zbl 1208.20057 |
| [15] | Fernandes, V. H.; Honyam, P.; Quinteiro, T. M.; Singha, B., On semigroups of endomorphisms of a chain with restricted range, Semigroup Forum, 89, 1, 77-104 (2014) · Zbl 1309.20051 · doi:10.1007/s00233-013-9548-x |
| [16] | Fernandes, V. H.; Honyam, P.; Quinteiro, T. M.; Singha, B., On semigroups of orientation-preserving transformations with restricted range, Commun. Algebra, 44, 1, 253-264 (2016) · Zbl 1339.20060 · doi:10.1080/00927872.2014.975345 |
| [17] | Fernandes, V. H.; Jesus, M. M.; Maltcev, V.; Mitchell, J. D., Endomorphisms of the semigroup of order-preserving mappings, Semigroup Forum, 81, 2, 277-285 (2010) · Zbl 1207.20059 · doi:10.1007/s00233-010-9220-7 |
| [18] | Fernandes, V. H.; Quinteiro, T. M., Bilateral semidirect product decompositions of transformation monoids, Semigroup Forum, 82, 2, 271-287 (2011) · Zbl 1229.20067 · doi:10.1007/s00233-011-9290-1 |
| [19] | Fernandes, V. H.; Quinteiro, T. M., The cardinal of various monoids of transformations that preserve a uniform partition, Bull. Malays. Math. Sci. Soc, 35, 4, 885-896 (2012) · Zbl 1252.20061 |
| [20] | Fernandes, V. H.; Quinteiro, T. M., On the ranks of certain monoids of transformations that preserve a uniform partition, Commun. Algebra, 42, 2, 615-636 (2014) · Zbl 1290.20053 · doi:10.1080/00927872.2012.720322 |
| [21] | Fernandes, V. H.; Volkov, M. V., On divisors of semigroups of order-preserving mappings of a finite chain, Semigroup Forum, 81, 3, 551-554 (2010) · Zbl 1211.20054 · doi:10.1007/s00233-010-9257-7 |
| [22] | Gomes, G. M. S.; Howie, J. M., On the ranks of certain semigroups of order-preserving transformations, Semigroup Forum, 45, 1, 272-282 (1992) · Zbl 0769.20029 · doi:10.1007/BF03025769 |
| [23] | Higgins, P. M., Divisors of semigroups of order-preserving mappings on a finite chain, Int. J. Algebra Comput, 05, 6, 725-742 (1995) · Zbl 0842.20054 · doi:10.1142/S0218196795000306 |
| [24] | Higgins, P. M.; Mitchell, J. D.; Ruškuc, N., Generating the full transformation semigroup using order preserving mappings, Glasgow Math. J, 45, 3, 557-566 (2003) · Zbl 1043.20041 · doi:10.1017/S0017089503001460 |
| [25] | Howie, J. M., Products of idempotents in certain semigroups of transformations, Proc. Edinb. Math. Soc, 17, 3, 223-236 (1971) · Zbl 0226.20072 · doi:10.1017/S0013091500026936 |
| [26] | Howie, J. M., Fundamentals of Semigroup Theory (1995), Oxford: Oxford University Press, Oxford · Zbl 0835.20077 |
| [27] | 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 · doi:10.1080/00927879808826160 |
| [28] | Kemprasit, Y.; Changphas, T., Regular order-preserving transformation semigroups, Bull. Austral. Math. Soc, 62, 3, 511-524 (2000) · Zbl 0972.20037 · doi:10.1017/S000497270001902X |
| [29] | Kim, V. I.; Kozhukhov, I. B., Regularity conditions for semigroups of isotone transformations of countable chains (Russian). Fundam. Prikl. Mat. 12 (2006), no. 8,, J. Math. Sci. (N. Y.), 152, 2, 203-208 (2008) · Zbl 1149.20049 |
| [30] | McAlister, D. B., Semigroups generated by a group and an idempotent, Commun. Algebra, 26, 515-547 (1998) · Zbl 0893.20045 |
| [31] | Mora, W.; Kemprasit, Y., Regular elements of some order-preserving transformation semigroups, Int. J. Algebra, 4, 631-641 (2010) · Zbl 1218.20044 |
| [32] | Repnitskiĭ, V. B.; Vernitskii, A., Semigroups of order preserving mappings, Commun. Algebra, 28, 3635-3641 (2000) · Zbl 0959.20059 |
| [33] | Repnitskiĭ, V. B.; Volkov, M. V., The finite basis problem for the pseudovariety \(####\) Proc, Roy. Soc. Edinburgh Sect. A, 128, 661-669 (1998) · Zbl 0909.20044 |
| [34] | Ruškuc, N., On the rank of completely 0-simple semigroups, Math. Proc. Camb. Phil. Soc, 116, 2, 325-338 (1994) · Zbl 0817.20062 · doi:10.1017/S0305004100072613 |
| [35] | Vernitskii, A.; Volkov, M. V., A proof and generalisation of Higgins’ division theorem for semigroups of order-preserving mappings, Izv.vuzov. Matematika, 1, 38-44 (1995) |
| [36] | Zhao, P.; Fernandes, V. H., The ranks of ideals in various transformation monoids, Commun. Algebra, 43, 2, 674-692 (2015) · Zbl 1320.20061 · doi:10.1080/00927872.2013.847946 |
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.
