Abstract
Two “basis properties” are considered for semigroups and associated algebras. These were introduced by the author to study inverse semigroups and groups, motivated by well-known properties of vector spaces. First these basis properties are studied in the abstract, from the point of view of exchange properties; then those semigroups with the “strong” basis property are determined for many classes of semigroups, including all regular and all periodic semigroups. The main method of proof is to eliminate undesirable types of semigroups by showing that each contains a certain special sub-semigroup, for instance the bicyclic semigroup. This prompts the study of analogs of a theorem of Andersen on embeddings of the bicyclic semigroup: such analogs are found for the semigroups A = 〈a,b|a2b = a〉 and C = 〈a,b|a2b = a,ab2 = b〉.
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References
O. Andersen, ‘Ein Bericht über die Struktur abstrakter Halbgruppen’, Thesis (Staatsexamensarbeit), Hamburg, 1952.
S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, New York, 1981.
A. H. Clifford and G. B. Preston, ‘The Algebraic Theory of Semigroups’, Amer. Math. Soc., Providence, Vol. I, 1961, Vol. II, 1967.
P. M. Cohn, Universal Algebra, Reidel, Boston, 1981.
N. K. Dickson and P. R. Jones, ‘Finite groups and the basis property’, manuscript, 1976.
J. Doyen, ‘Equipotence et unicité de systemes générateurs minimaux dans certains monoides’, Semigroup Forum 28, (1984), 341–346.
K. Głazek, ‘Some old and new problems in independence theory’, Colloq. Math. 42 (1979), 127–189.
P. R. Jones, ‘Inverse Subsemigroups of Free Inverse Semigroups’, Thesis, Monash University, 1975.
P. R. Jones, ‘A basis theorem for free inverse semigroups’, J. Algebra 49 (1977), 172–190.
P. R. Jones, ‘Basis properties for inverse semigroups’, J. Algebra 50 (1978), 135–152.
P. R. Jones, ‘Embedding semigroups in D-trivial semigroups’.
P. R. Jones, ‘Analogs of the bicyclic semigroup in simple semigroups without idempotents’.
P. R. Jones, ‘Exchange properties and basis properties for closure operators’.
P. R. Jones, ‘Basis properties for semigroups’.
L. Megyesi and G. Pollák, ‘On simple principal ideal semigroups’, Studia Sci. Math. Hung. 16 (1981), 437–448.
M. Petrich, Inverse Semigroups, Wiley, New York, 1984.
S. A. Rankin and C. M. Reis, ‘Semigroups with quasi-zeroes’, Canadian J. Math. 32 (1980), 511–530.
L. Rédei, ‘Halbgruppen und Ringe mit Linkseinheiten ohne Linkseinselemente’, Acta Math. Acad. Sci. Hungar. 11 (1960), 217–222.
W. Scott, Group Theory, Prentice-Hall, New Jersey, 1964.
E. G. Šutov, ‘Embedding semigroups in simple and complete semigroups’, Mat. Sbornik 62 (1963), No. 4; 496–511 (Russian).
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© 1987 D. Reidel Publishing Company
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Jones, P.R. (1987). Basis Properties, Exchange Properties and Embeddings in Idempotent-Free Semigroups. In: Goberstein, S.M., Higgins, P.M. (eds) Semigroups and Their Applications. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3839-7_10
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DOI: https://doi.org/10.1007/978-94-009-3839-7_10
Publisher Name: Springer, Dordrecht
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