Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

The minimal degree of a finite inverse semigroup


Author: Boris M. Schein
Journal: Trans. Amer. Math. Soc. 333 (1992), 877-888
MSC: Primary 20M30; Secondary 20M18, 20M20
MathSciNet review: 1072101
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The minimal degree of an inverse semigroup $ S$ is the minimal cardinality of a set $ A$ such that $ S$ is isomorphic to an inverse semigroup of one-to-one partial transformations of $ A$. The main result is a formula that expresses the minimal degree of a finite inverse semigroup $ S$ in terms of certain subgroups and the ordered structure of $ S$. In fact, a representation of $ S$ by one-to-one partial transformations of the smallest possible set $ A$ is explicitly constructed in the proof of the formula. All known and some new results on the minimal degree follow as easy corollaries.


References [Enhancements On Off] (What's this?)

  • [1] A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. II, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1967. MR 0218472
  • [2] David Easdown, Efficient representations of semigroups, Proc. Internat. Sympos. Theory of Regular Semigroups and Applications, University of Kerala, Trivandrum, 1987, pp. 22-46.
  • [3] D. Easdown, The minimal faithful degree of a fundamental inverse semigroup, Bull. Austral. Math. Soc. 35 (1987), no. 3, 373–378. MR 888896, 10.1017/S0004972700013356
  • [4] J. M. Howie, An introduction to semigroup theory, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. L.M.S. Monographs, No. 7. MR 0466355
  • [5] Mario Petrich, Inverse semigroups, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 752899
  • [6] B. M. Shaĭn (B. M. Schein), Predstavleniya obobshchënnykh grupp (Representations of generalized groups), Izvestiya Vysshikh Uchebnykh Zavedeniĭ, Matematika (Izv. Vyssh. Uchebn. Zaved. Matematika) 1962, no. 3, 164-176. (Russian)
  • [7] -, Inversnye polugruppy, ne dopuskayushchie izomorfnykh predstavleniĭ chastichnymi preobrazovaniyami svoikh sobstvennykh podmnozhestv (Inverse semigroups that do not admit representations by partial transformations of their proper subsets), Teoriya Polugrupp i eë Prilozheniya, Izdatelstvo Saratovskogo Universiteta, Saratov (Theory of Semigroups and its Applications, vol. 3, Saratov Univ. Press, Saratov) 1974, pp. 139-148 (Russian; English transl., Amer. Math. Soc. Transl. (2) 139 (1988), 177-182).
  • [8] Boris M. Schein, Noble inverse semigroups with bisimple core, Semigroup Forum 36 (1987), no. 2, 175–178. MR 911054, 10.1007/BF02575014
  • [9] Boris M. Schein, The minimal degree of noble inverse semigroups, Contributions to general algebra, 6, Hölder-Pichler-Tempsky, Vienna, 1988, pp. 247–252. MR 1078045

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 20M30, 20M18, 20M20

Retrieve articles in all journals with MSC: 20M30, 20M18, 20M20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1072101-9
Article copyright: © Copyright 1992 American Mathematical Society