On quasi completely regular semirings.
A semiring \((S,+,\cdot)\) is called quasi completely regular if for every \(a\in S\) there is some positive integer \(n\) and some \(x\in S\) such that (i) \(na=na+x+na\), (ii) \(na+x=x+na\), (iii) \(na(na+x)=na+x\). Denote by \(n_a\) the smallest positive integer satisfying (i), and by \(\mathcal L^+,\mathcal R^+,\mathcal J^+,\mathcal H^+\) Green's relations on \((S,+)\). If \(\varrho^+\) is any of these relations, define the corresponding relation \(\varrho^{*+}\) by \(a\varrho^{*+}b\Leftrightarrow n_aa\varrho^+n_bb\). Then a quasi completely regular semiring \((S,+,\cdot)\) is called completely Archimedean if \(\mathcal J^{*+}=S\times S\). Moreover, \((S,+,\cdot)\) is called a quasi skew-ring if \(S\) has a subskew-ring \(R\) such that for every \(a\in S\) there is a positive integer \(n\) with \(na\in R\). Then it is proved that the following conditions on a semiring \((S,+,\cdot)\) are equivalent: (i) \(S\) is quasi completely regular, (ii) every \(\mathcal H^{*+}\)-class of \(S\) is a quasi skew-ring, (iii) \(S\) is a (disjoint) union of quasi skew-rings, (iv) \(S\) is a \(b\)-lattice of completely Archimedean semirings, (v) \(S\) is an idempotent semiring of quasi skew-rings.
scientific article; zbMATH DE number 6385110
scientific article