Generalized F -semigroups

E. Giraldes; P. Marques-Smith; Heinz Mitsch

Mathematica Bohemica (2005)

  • Volume: 130, Issue: 2, page 203-220
  • ISSN: 0862-7959

Abstract

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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 ≤ S of S . Using the concept of an anticone, all partially ordered groups which are epimorphic images of a semigroup ( S , · , ≤ 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 , ≤ 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.

How to cite

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Giraldes, E., Marques-Smith, P., and Mitsch, Heinz. "Generalized $F$-semigroups." Mathematica Bohemica 130.2 (2005): 203-220. <http://eudml.org/doc/249592>.

@article{Giraldes2005,
abstract = {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.},
author = {Giraldes, E., Marques-Smith, P., Mitsch, Heinz},
journal = {Mathematica Bohemica},
keywords = {semigroup; natural partial order; group congruence; anticone; pivot elements; partially ordered groups; principal order ideals; semigroups; natural partial orders; group congruences; anticones; pivot elements; partially ordered groups; principal order ideals; idempotents},
language = {eng},
number = {2},
pages = {203-220},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Generalized $F$-semigroups},
url = {http://eudml.org/doc/249592},
volume = {130},
year = {2005},
}

TY - JOUR
AU - Giraldes, E.
AU - Marques-Smith, P.
AU - Mitsch, Heinz
TI - Generalized $F$-semigroups
JO - Mathematica Bohemica
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 130
IS - 2
SP - 203
EP - 220
AB - 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.
LA - eng
KW - semigroup; natural partial order; group congruence; anticone; pivot elements; partially ordered groups; principal order ideals; semigroups; natural partial orders; group congruences; anticones; pivot elements; partially ordered groups; principal order ideals; idempotents
UR - http://eudml.org/doc/249592
ER -

References

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  1. 10.1017/S0305004100075332, Math. Proc. Camb. Phil. Soc. 111 (1992), 241–253. (1992) MR1142743DOI10.1017/S0305004100075332
  2. Sur les images homomorphes d’un demi-groupe ordonné, C. R. Acad. Sci. Paris 260 (1965), 5987–5988. (1965) Zbl0132.01202MR0178080
  3. On the greatest isotone homomorphic group image of an inverse semigroup, J. London Math. Soc. 1 (1969), 260–264. (1969) Zbl0181.32001MR0245705
  4. Residuation Theory, Pergamon Press, Oxford, U.K., 1972. (1972) MR0396359
  5. The Algebraic Theory of Semigroups, Vols. I, II, Amer. Math. Soc. Surveys 7, Providence, USA, 1961/67. (1961/67) MR0132791
  6. Sur les images homomorphes d’un demi-groupe ordonné, Bull. Soc. Math. France 92 (1964), 101–115. (1964) Zbl0129.01502MR0168675
  7. Constructions of trivially ordered semigroups, Pure Math. Appl. 13 (2002), 359–371. (2002) MR1980722
  8. 10.1016/j.jalgebra.2003.09.050, J. Algebra 274 (2004), 491–510. (2004) MR2043359DOI10.1016/j.jalgebra.2003.09.050
  9. An Introduction to Semigroup Theory, Academic Press, London, U.K., 1976. (1976) Zbl0355.20056MR0466355
  10. Inverse Semigroups, World Scientific, Singapore, 1998. (1998) Zbl1079.20505MR1694900
  11. 10.1090/S0002-9947-74-99950-4, Trans. Amer. Math. Soc. 196 (1974), 351–370. (1974) Zbl0297.20072MR0357660DOI10.1090/S0002-9947-74-99950-4
  12. F -inverse semigroups, Proc. Lond. Math. Soc., III. Ser. 22 (1971), 652–666. (1971) MR0292978
  13. 10.1090/S0002-9939-1986-0840614-0, Proc. Amer. Math. Soc. 97 (1986), 384–388. (1986) Zbl0596.06015MR0840614DOI10.1090/S0002-9939-1986-0840614-0
  14. 10.1017/S1446788700035199, J. Austral. Math. Soc. 48 (1990), 66–78. (1990) Zbl0691.20050MR1026837DOI10.1017/S1446788700035199
  15. Semigroups and their natural order, Math. Slovaca 44 (1994), 445–462. (1994) Zbl0816.20057MR1301953
  16. Inverse Semigroups, J. Wiley, New York, 1984. (1984) Zbl0546.20053MR0752899
  17. 10.1007/BF01894689, Acta Math. Acad. Sci. Hung. 19 (1968), 129–146. (1968) MR0227299DOI10.1007/BF01894689
  18. Die natürliche Ordnung auf Halbgruppen, University of Vienna, PhD-Thesis (1994). (1994) 
  19. Generalized grouds and generalized groups with the transitive compatibility relation, Uchenye Zapiski, Mechano-Math. Series, Saratov State University 70 (1961), 25–39. (1961) 

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