Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-17T20:10:31.975Z Has data issue: false hasContentIssue false
  • English
  • Français

Completely simple semigroups: free products, free semigroups and varieties

Published online by Cambridge University Press:  14 November 2011

P. R. Jones
P. R. Jones
Affiliation:
Mathematics Department, Monash University, Clayton, Victoria, Australia3168
Get access Rights & Permissions [Opens in a new window]

Synopsis

The class CS of completely simple semigroups forms a variety under the operations of multiplication and inversion (x−1 being the inverse of x in its ℋ-class). We determine a Rees matrix representation of the CS-free product of an arbitrary family of completely simple semigroups and deduce a description of the free completely simple semigroups, whose existence was proved by McAlister in 1968 and whose structure was first given by Clifford in 1979. From this a description of the lattice of varieties of completely simple semigroups is given in terms of certain subgroups of a free group of countable rank. Whilst not providing a “list” of identities on completely simple semigroups it does enable us to deduce, for instance, the description of all varieties of completely simple semigroups with abelian subgroups given by Rasin in 1979. It also enables us to describe the maximal subgroups of the “free” idempotent-generated completely simple semigroups T(α, β) denned by Eberhart et al. in 1973 and to show in general the maximal subgroups of the “V-free” semigroups of this type (which we define) need not be free in any variety of groups.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 88 , Issue 3-4 , 1981 , pp. 293 - 313
Copyright
Copyright © Royal Society of Edinburgh 1981

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Clifford, A. H.. The free completely regular semigroup on a set. J. Algebra 59 (1979), 434–451.CrossRefGoogle Scholar
2Clifford, A. H. and Preston, G. B.. The algebraic theory of semigroups, Vol. I. (Amer. Math. Soc. Surveys No. 7, Providence R. I., 1961).Google Scholar
3Eberhart, C., Williams, W. and Kinch, L.. Idempotent-generated regular semigroups. J. Austral. Math. Soc. 15 (1973), 27–34.CrossRefGoogle Scholar
4Evans, T.. The lattice of semigroup varieties. Semigroup Forum 2 (1971), 1–43.CrossRefGoogle Scholar
5Fitzgerald, D. G.. On inverses of products of idempotents in regular semigroups. J. Austral. Math. Soc. 13 (1972), 335–337.CrossRefGoogle Scholar
6Grätzer, G.. Universal algebra (Princeton, N. J.: Von Nostrand, 1968).Google Scholar
7Hall, T. E. and Jones, P. R.. On the lattice of varieties of bands of groups. Pacific J. Math. 92 (1981), to appear.Google Scholar
8Howie, J. M.. An introduction to semigroup theory (London: Academic Press, 1976).Google Scholar
9Howie, J. M.. Idempotents in completely 0-simple semigroups. Glasgow Math. J. 19 (1978), 109–113.CrossRefGoogle Scholar
10Magnus, W., Karrass, A. and Solitar, D.. Combinatorial group theory (New York: Interscience 1966).Google Scholar
11McAlister, D. B.. A homomorphism theorem for semigroups. J. London Math. Soc. 43 (1968), 355–366.CrossRefGoogle Scholar
12Neumann, B. H.. Universal algebra (Lecture notes, Courant Inst. of Math. Sci., New York University).Google Scholar
13Neumann, H., Varieties of groups (New York: Springer Verlag, 1967).CrossRefGoogle Scholar
14Pastijn, F.. Idempotent-generated completely 0-simple semigroups. Semigroup Forum 15 (1977), 41–50.CrossRefGoogle Scholar
15Pastijn, F.. The biorder on the partial groupoid of idempotents of a semigroup, to appear.Google Scholar
16Petrich, M.. Varieties of orthodox bands of groups. Pacific J. Math 58 (1975), 209–217.CrossRefGoogle Scholar
17Petrich, M.. Certain varieties and quasi-varieties of completely regular semigroups. Canad. J. Math 29 (1977), 1171–1197.CrossRefGoogle Scholar
18Rasin, V. V.. On the lattice of varieties of completely simple semigroups. Semigroup Forum 17 (1979), 113–122.CrossRefGoogle Scholar