Araújo, João; Konieczny, Janusz Semigroups of transformations preserving an equivalence relation and a cross-section. (English) Zbl 1068.20061 Commun. Algebra 32, No. 5, 1917-1935 (2004). Summary: For a set \(X\), an equivalence relation \(\rho\) on \(X\), and a cross-section \(R\) of the partition \(X/\rho\) induced by \(\rho\), consider the semigroup \(T(X,\rho,R)\) consisting of all mappings \(a\) from \(X\) to \(X\) such that \(a\) preserves both \(\rho\) (if \((x,y)\in\rho\) then \((xa,ya)\in\rho\)) and \(R\) (if \(r\in R\) then \(ra\in R\)). The semigroup \(T(X,\rho,R)\) is the centralizer of the idempotent transformation with kernel \(\rho\) and image \(R\). We determine the structure of \(T(X,\rho,R)\) in terms of Green’s relations, describe the regular elements of \(T(X,\rho,R)\), and determine the following classes of the semigroups \(T(X,\rho,R)\): regular, abundant, inverse, and completely regular. Cited in 1 ReviewCited in 38 Documents MSC: 20M20 Semigroups of transformations, relations, partitions, etc. Keywords:equivalences; cross-sections; semigroups of mappings; centralizers; idempotent transformations; Green relations; regular elements PDFBibTeX XMLCite \textit{J. Araújo} and \textit{J. Konieczny}, Commun. Algebra 32, No. 5, 1917--1935 (2004; Zbl 1068.20061) Full Text: DOI References: [1] Araújo J., J. Algebra 269 pp 227– (2003) · Zbl 1037.20062 · doi:10.1016/S0021-8693(03)00499-X [2] Fountain J., Proc. Edinburgh Math. Soc. (2) 22 pp 113– (1979) · Zbl 0414.20048 · doi:10.1017/S0013091500016230 [3] Fountain J., Proc. London Math. Soc. (3) 44 pp 103– (1982) · Zbl 0481.20036 · doi:10.1112/plms/s3-44.1.103 [4] Higgins P. M., Glasgow Math. J. 30 pp 41– (1988) · Zbl 0634.20034 · doi:10.1017/S0017089500007011 [5] Howie J. M., Fundamentals of Semigroup Theory (1995) [6] Konieczny J., Glasgow Math. J. 41 pp 45– (1999) · Zbl 0924.20049 · doi:10.1017/S0017089599970301 [7] Konieczny J., Algebra Colloq. 9 pp 121– (2002) [8] Konieczny J., Math. Japon. 48 pp 367– (1998) [9] Lallement G., Semigroups and Combinatorial Applications (1979) [10] Liskovec V. A., Dokl. Akad. Nauk BSSR 7 pp 366– (1963) [11] Liskovec V. A., Dokl. Akad. Nauk BSSR 12 pp 596– (1968) [12] Umar A., Proc. Roy. Soc. Edinburgh Sect. A 120 pp 129– (1992) [13] Umar A., Comm. Algebra 25 pp 2987– (1997) · Zbl 0883.20026 · doi:10.1080/00927879708826035 [14] Weaver M. W., Pacific J. Math. 10 pp 705– (1960) 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.