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Bicyclic semirings
Published online by Cambridge University Press: 09 April 2009
Abstract
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Let Bp be the bicyclic semigroup over P=G⋒[1,∞ where G is a subgroup of the multiplicative group of positive real numbers. If + is an addition which makes Bp, with its usual multiplication, into a semiring, then + is idempotent, and P is embedded as a sub-semiring in Bp and for each x in p, 1≦x+1≦x and 1≦1+x≦x. We show that any idempotent addition on P with these inequalities holding is max, min or trivial. The trivial addition on P extends trivially. If addition on P is min, then let 
, 
and 
. We charactertize all additions on Bp in terms of U and U′; and, in particular If U=U′ is a proper subset of R1 we demonstrate a correspondence between all such additions and certain homomorphisms of G to (0,∞)
Subject classification (Amer. math. Soc. (MOS) 1970): primary 16 A 80; secondary 22 A 15.
- Type
- Research Article
- Information
- Journal of the Australian Mathematical Society , Volume 26 , Issue 4 , December 1978 , pp. 419 - 441
- Copyright
- Copyright © Australian Mathematical Society 1978
References
Clifford, A. H. and Preston, G. B. (1961), The Algebraic Theory of Semigroups (Amer. Math. Soc. Math. Surveys No. 7, Vol. I, Providence, R. I.).Google Scholar
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Pearson, K. R. (1966), “Interval semirings on R 1 with ordinary multiplication”, J. Austral. Math. Soc. 6, 273–282.Google Scholar
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