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Extension of a Semigroup Embedding Theorem to Semirings
Published online by Cambridge University Press: 20 November 2018
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It is well known [1,3] that a commutative semigroup (S, +) can be embedded in a semigroup which is a union of groups if and only if S is separative (2a = a + b = 2b implies a = b). We extend this result to additively commutative semirings.
A semiring (S, +, ⋅) is a set S with associative addition (+) and multiplication (⋅), the latter distributing over addition from left and right. In what follows (S, +, ⋅) will denote a semiring in which the additive semigroup (S, +) is commutative. An element 0 can be adjoined, where s = s + 0, 0 = 0 ⋅ s = s ⋅ 0 for all s in S, to form S°.
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- Copyright © Canadian Mathematical Society 1975
References
1.
Clifford, A. H. and Preston, G. B., The Algebraic Theory of Semigroups, Volume I, American Mathematical Society (1961).Google Scholar
2.
Grillet, M. O.Poinsignon, Subdivision Rings of a Semiring, Fund. Math.
67 (1970), 67-74.Google Scholar
3.
Tamura, T. and Kimura, N., On Decompositions of a Commutative Semigroup, Kodai Math. Sem. Rep.
1954 (1954), 109-112.Google Scholar
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