One-generated semirings and additive divisibility. (English) Zbl 1358.16038

Korbelář, Miroslav; Landsmann, Günter

J. Algebra Appl. 16, No. 2, Article ID 1750038, 22 p. (2017).

Semirings \((S,+,\cdot)\) in the sense of this paper are commutative with respect to both operations and have an identity \(1\). Such semirings are called additively divisible, if for any \(a\in S\) and every non-negative integer \(n\) there is some \(b\in S\) such that \(a=b+\ldots + b\) (\(n\)-times). It is an open conjecture that every finitely generated additively divisible semiring is additively idempotent. By sophisticated rewriting techniques, similar as in the Buchberger algorithm in the semiring \({\mathbb N}[x]\) of nonzero polynomials over the non-negative integers, the authors prove this conjecture for semirings generated by one single element. As a second result, it is shown that every at most countable commutative semigroup \((A,+)\) can be embedded in the additive semigroup of some one-generated semiring \((S,+,\cdot)\) as above.

Reviewer: Udo Hebisch (Freiberg)

MSC:

16Y60	Semirings
12Y05	Computational aspects of field theory and polynomials (MSC2010)
20M14	Commutative semigroups

Keywords:

commutative semiring; divisible semigroup; numerical semigroup; additively idempotent semiring; embedding

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