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The Arithmetic of the Quasi-Uniserial Semigroups without Zero

Published online by Cambridge University Press:  20 November 2018

Ernst August Behrens
Ernst August Behrens*
Affiliation:
McMaster University, Hamilton, Ontario
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An element a in a partially ordered semigroup T is called integral if

is valid. The integral elements form a subsemigroup S of T if they exist. Two different integral idempotents e and f in T generate different one-sided ideals, because eT = fT, say, implies e = fe ⊆ f and f = ef ⊆ e.

Let M be a completely simple semigroup. M is the disjoint union of its maximal subgroups [4]. Their identity elements generate the minimal one-sided ideals in M. The previous paragraph suggests the introduction of the following hypothesis on M.

Hypothesis 1. Every minimal one-sided ideal in M is generated by an integral idempotent.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 3 , June 1971 , pp. 507 - 516
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Behrens, E. A., Partially ordered completely simple semigroups, to appear in J. Algebra. Preprint: Math. Report McMaster University No. 22, vol. 2 (1970).Google Scholar
2. Behrens, E. A., D*-arithmetic prime rings, Math. Report McMaster University No. 24, vol. 2 (1970).Google Scholar
3. Behrens, E. A., The quasi-uniserial semigroups without zero, their arithmetics and their ϕ*-algebras, Semigroup Forum, to appear.Google Scholar
4. Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups (Vol. 1, Mathematical Surveys, No. 7, American-Mathematical Society, Providence, 1961).Google Scholar
5. Rees, D., On semigroups, Proc. Cambridge Philos. Soc. 36 (1940), 387–400.Google Scholar