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Additive Divisibility in Compact Topological Semirings

Published online by Cambridge University Press:  20 November 2018

P. H. Karvellas
P. H. Karvellas*
Affiliation:
Department of Electrical Engineering, University of Alberta, Edmonton, Alberta
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A topological semiring (S, + , ·) is a nonempty Hausdorff space S on which are defined continuous and associative operations, termed addition (+) and multiplication (·), such that the multiplication distributes over addition from left and right. The additive semigroup (S, +) need not be commutative.

We prove that the set A of additively divisible elements of a compact semiring S is a two-sided multiplicative ideal, containing the set E[+] of additive idempotents, with the property that (A.S) ∪ (S.A) ⊂ E[+].

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 3 , June 1974 , pp. 593 - 599
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Anzai, H., On compact topological rings, Proc. Japan Acad. 37 (1943), 613–615.Google Scholar
2. Beidleman, J. C. and Cox, R. H., Topological near-rings, Arch. Math. 18 (1967), 485–492.Google Scholar
3. Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Volume I (American Mathematical Society, Providence, 1961).Google Scholar
4. Mostert, P. S. and Shields, A. L., On the structure of semigroups on a compact manifold with boundary, Ann. of Math. 65 (1957), 117–143.Google Scholar
5. Mycielski, J., Some properties of connected compact groups, Colloq. Math. 5 (1958), 162–166.Google Scholar
6. Paalman de Miranda, A. B., Topological semigroups (Mathematisch Centrum, Amsterdam, 1964).Google Scholar
7. Pearson, K. R., Interval semirings on Ri with ordinary multiplication, J. Austral. Math. Soc. 6 (1966), 273–288.Google Scholar
8. Pearson, K. R., Certain topological semirings on Ri, J. Austral. Math. Soc. 8 (1968), 171–182.Google Scholar
9. Shelden, J., A note on compact semirings, Proc. Amer. Math. Soc. 15 (1964), 882–886.Google Scholar
10. Wallace, A. D., The structure of topological semigroups, Bull. Amer. Math. Soc. 61 (1955), 95–112.Google Scholar
11. Wallace, A. D., Dispersal in compact semimodules, J. Austral. Math. Soc. 18 (1972), 338–342.Google Scholar