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semigroup; natural partial order; group congruence; anticone; pivot elements; partially ordered groups; principal order ideals
A semigroup $S$ is called a generalized $F$-semigroup if there exists a group congruence on $S$ such that the identity class contains a greatest element with respect to the natural partial order $\le _{S}$ of $S$. Using the concept of an anticone, all partially ordered groups which are epimorphic images of a semigroup $(S,\cdot ,\le _{S})$ are determined. It is shown that a semigroup $S$ is a generalized $F$-semigroup if and only if $S$ contains an anticone, which is a principal order ideal of $(S,\le _{S})$. Also a characterization by means of the structure of the set of idempotents or by the existence of a particular element in $S$ is given. The generalized $F$-semigroups in the following classes are described: monoids, semigroups with zero, trivially ordered semigroups, regular semigroups, bands, inverse semigroups, Clifford semigroups, inflations of semigroups, and strong semilattices of monoids.
[1] J. Almeida, J. E. Pin, P. Weil: Semigroups whose idempotents form a subsemigroup. Math. Proc. Camb. Phil. Soc. 111 (1992), 241–253. DOI 10.1017/S0305004100075332 | MR 1142743
[2] A. Bigard: Sur les images homomorphes d’un demi-groupe ordonné. C. R. Acad. Sci. Paris 260 (1965), 5987–5988. MR 0178080 | Zbl 0132.01202
[3] T. Blyth: On the greatest isotone homomorphic group image of an inverse semigroup. J. London Math. Soc. 1 (1969), 260–264. MR 0245705 | Zbl 0181.32001
[4] T. Blyth, M. Janowitz: Residuation Theory. Pergamon Press, Oxford, U.K., 1972. MR 0396359
[5] A. Clifford, G. Preston: The Algebraic Theory of Semigroups, Vols. I, II. Amer. Math. Soc. Surveys 7, Providence, USA, 1961/67. MR 0132791
[6] M. Dubreil-Jacotin: Sur les images homomorphes d’un demi-groupe ordonné. Bull. Soc. Math. France 92 (1964), 101–115. MR 0168675 | Zbl 0129.01502
[7] A. Fidalgo Maia, H. Mitsch: Constructions of trivially ordered semigroups. Pure Math. Appl. 13 (2002), 359–371. MR 1980722
[8] E. Giraldes, P. Marques-Smith, H. Mitsch: $F$-regular semigroups. J. Algebra 274 (2004), 491–510. DOI 10.1016/j.jalgebra.2003.09.050 | MR 2043359
[9] J. Howie: An Introduction to Semigroup Theory. Academic Press, London, U.K., 1976. MR 0466355 | Zbl 0355.20056
[10] M. Lawson: Inverse Semigroups. World Scientific, Singapore, 1998. MR 1694900 | Zbl 1079.20505
[11] D. McAlister: Groups, semilattices and inverse semigroups II. Trans. Amer. Math. Soc. 196 (1974), 351–370. DOI 10.1090/S0002-9947-74-99950-4 | MR 0357660 | Zbl 0297.20072
[12] R. McFadden, L. O’Carroll: $F$-inverse semigroups. Proc. Lond. Math. Soc., III. Ser. 22 (1971), 652–666. MR 0292978
[13] H. Mitsch: A natural partial order for semigroups. Proc. Amer. Math. Soc. 97 (1986), 384–388. DOI 10.1090/S0002-9939-1986-0840614-0 | MR 0840614 | Zbl 0596.06015
[14] H. Mitsch: Subdirect products of $E$-inversive semigroups. J. Austral. Math. Soc. 48 (1990), 66–78. DOI 10.1017/S1446788700035199 | MR 1026837 | Zbl 0691.20050
[15] H. Mitsch: Semigroups and their natural order. Math. Slovaca 44 (1994), 445–462. MR 1301953 | Zbl 0816.20057
[16] M. Petrich: Inverse Semigroups. J. Wiley, New York, 1984. MR 0752899 | Zbl 0546.20053
[17] J. Querré: Plus grand groupe image homomorphe et isotone d’un monoide ordonné. Acta Math. Acad. Sci. Hung. 19 (1968), 129–146. DOI 10.1007/BF01894689 | MR 0227299
[18] S. Reither: Die natürliche Ordnung auf Halbgruppen. University of Vienna, PhD-Thesis (1994).
[19] V. Wagner: Generalized grouds and generalized groups with the transitive compatibility relation. Uchenye Zapiski, Mechano-Math. Series, Saratov State University 70 (1961), 25–39.
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