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Finitely generated algebraic structures with various divisibility conditions. (English) Zbl 1254.16041

Let \((P,+,\cdot)\) be a parasemifield, i.e. a commutative semiring such that \((P,\cdot)\) is an Abelian group. It is shown that no such parasemifield can be generated as a semiring by a single element. Now, assume that \(P\) is generated by two elements \(x\neq y\). If \((P,+)\) is not idempotent, denote by \(Q\cong\mathbb Q^+\) the prime subparasemifield of \(P\) and define \(S=\{u\in P\mid Q\cap (P+u)\neq\emptyset\}\) and \(T=Q+S\). Then \(T\subset S\subset P\), \(S\) is a subsemiring of \(P\) and \(T\) an additively cancellative subparasemifield of \(P\). Moreover, the sets \(A=\{(k,l)\in\mathbb N_0^2\mid x^ky^l\in S\}\) and \(B=\{(k,l)\in\mathbb N_0^2\mid x^ky^l\in T\}\) of integral lattice points are subsemigroups of \((\mathbb N_0^2,+)\). By analyzing specific algebraic and topological properties of these subsemigroups, the main result is proved: any parasemifield that is two-generated as a semiring is additively idempotent. By direct constructions it is also shown that every countable groupoid is a subgroupoid of a one-generated congruence-simple division groupoid, and that every countable cancellative groupoid is a subgroupoid of a one-generated quasigroup.

MSC:

16Y60 Semirings
12K10 Semifields
20N02 Sets with a single binary operation (groupoids)
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References:

[1] DOI: 10.1006/jabr.2000.8483 · Zbl 0976.16034
[2] Kala V., Commentationes Math. Univ. Carolinae 49 pp 1– (2008)
[3] DOI: 10.1007/BF01879738 · Zbl 0896.12001
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