Nilpotent inverse semigroups with central idempotents
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- by G. Kowol and H. Mitsch PDF
- Trans. Amer. Math. Soc. 271 (1982), 437-449 Request permission
Abstract:
An inverse semigroup $S$ with central idempotents, i.e. a strong semilattice of groups, will be called nilpotent, if it is finite and if for each prime divisor ${p_i}$ of the orders of the structure groups of $S$ the sets ${P_i} = \{ s \in S|o(s) = p_i^{{k_s}}, {k_s} \geqslant 0\}$ are subsemigroups of $S$. If $S$ is a group, then ${P_i}$ are exactly the Sylow ${p_i}$-subgroups of the group. A theory similar to that given by W. Burnside for finite nilpotent groups is developed introducing the concepts of ascending resp. descending central series in an inverse semigroup, and it is shown that almost all of the well-known properties of finite nilpotent groups do hold also for the class of finite inverse semigroups with central idempotents.References
- A. H. Clifford, Semigroups admitting relative inverses, Ann. of Math. (2) 42 (1941), 1037–1049. MR 5744, DOI 10.2307/1968781
- 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
- J. M. Howie, An introduction to semigroup theory, L. M. S. Monographs, No. 7, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0466355
- J. M. Howie, The maximum idempotent-separating congruence on an inverse semigroup, Proc. Edinburgh Math. Soc. (2) 14 (1964/65), 71–79. MR 163976, DOI 10.1017/S0013091500011251
- B. Huppert, Endliche Gruppen. I, Die Grundlehren der mathematischen Wissenschaften, Band 134, Springer-Verlag, Berlin-New York, 1967 (German). MR 0224703
- Gérard Lallement, Congruences et équivalences de Green sur un demi-groupe régulier, C. R. Acad. Sci. Paris Sér. A-B 262 (1966), A613–A616 (French). MR 207872
- H. Mitsch, Congruences on $N$-simple inverse semigroups, Math. Japon. 26 (1981), no. 4, 349–358. MR 634907
- Mario Petrich, Congruences on inverse semigroups, J. Algebra 55 (1978), no. 2, 231–256. MR 523456, DOI 10.1016/0021-8693(78)90219-3
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 271 (1982), 437-449
- MSC: Primary 20M10
- DOI: https://doi.org/10.1090/S0002-9947-1982-0654843-3
- MathSciNet review: 654843