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Semigroups over generalized trees

Part of: Connections with other structures, applications Topological and differentiable algebraic systems Semigroups

Published online by Cambridge University Press:  09 April 2009

T. E. Hays
T. E. Hays
Affiliation:
The Ohio State University Newark, OhioU.S.A.
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Abstract

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A semigroup over a generalized tree, denoted by the term ℳL-semigroup, is a compact semigroup S such that Green's relation H is a congruence on S and S/H is an abelian generalized tree with idempotent endpoints and E(S/H) a Lawson semilattice. Each such semigroup is characterized as being constructible from cylindrical subsemigroups of S and the generalized tree S/H in a manner similar to the construction of semigroups over trees and of the hormos. Indeed, semigroups over trees are shown to be particular examples of the construction given herein.

Keywords

compact semigroupgeneralized treesemigroup over a treeLawson semilatticeinverse limit preserving functor.

MSC classification

Secondary: 22A15: Structure of topological semigroups 20M10: General structure theory 54H10: Topological representations of algebraic systems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 25 , Issue 2 , March 1978 , pp. 177 - 188
Copyright
Copyright © Australian Mathematical Society 1978

References

REFERENCES

Bowman, T. T. (1971), “A construction principle and compact Clifford semigroups”, Semigroup Forum 2, 343–353.CrossRefGoogle Scholar
Cohen, H. and Krule, I. S. (1959), “Continuous homomorphic images of real clans with zero”, Proc. Amer. Math. Soc. 10, 106–109.CrossRefGoogle Scholar
Hofmann, K. H. and Mostert, P. S. (1966), Elements of Compact Semigroups (Charles Merrill, Columbus).Google Scholar
Hunter, R. P. (1959), “On the semigroup structure of continua”, Trans. Amer. Math. Soc. 93, 356–368.CrossRefGoogle Scholar
Hunter, R. P. (1961), “On a conjecture of Koch”, Proc. Amer. Math. Soc. 12, 138–139.CrossRefGoogle Scholar
Kelley, J. L. (1955), General Topology (Van Nostrand Company Inc., Princeton, New Jersey).Google Scholar
Koch, R. J. (1957), “Note on weak outpoints in clans”, Duke Math J. 24, 611–616.CrossRefGoogle Scholar
Koch, R. J. and Krule, I. S. (1960), “Weak cutpoint ordering on hereditarily unicoherent continua”, Proc. Amer. Math. Soc. 11, 679–681.CrossRefGoogle Scholar
Mislove, M. W. (1969), “Four problems about compact semigroups”, Dissertation, The University of Tennessee, Knoxville.Google Scholar
Mislove, M. W. (1974), “Semigroups over trees”, Trans. Amer. Math. Soc. 195, 383–400.CrossRefGoogle Scholar
Mitchell, B. (1965), Theory of Categories (Academic Press, New York, 1965).Google Scholar
Phillips, R. C. (1963), “Interval clans with nondegenerate kernel”, Proc. Amer. Math. Soc. 14, 396–400.CrossRefGoogle Scholar
Ward, L. E. (1954), “A note on dendrites and trees”, Proc. Amer. Math. Soc. 5, 992–994.CrossRefGoogle Scholar
Ward, L. E. (1957), “Mobs, trees, and fixed points”, Proc. Amer. Math. Soc. 8, 798–804.CrossRefGoogle Scholar
Ward, L. E. Jr, (1958), “On dendritic sets”, Duke Math. J. 25, 505–514.Google Scholar