On semirings whose additive reduct is a semilattice.
An additively commutative and idempotent semiring \((S,+,\cdot)\) is called \(k\)-regular if for every \(a\in S\) there is some \(x\in S\) such that \(a+axa=axa\). In such a semiring define \(E_k(S)=\{e\in S\mid e+e^2=e^2\}\) and \(E^\cdot(S)=\{e\in S\mid e^2=e\}\). It is proved that for a \(k\)-regular semiring the following statements are equivalent: (i) \((S,+,\cdot)\) is a distributive lattice, (ii) \(ef=fe\) for all \(e,f\in E_k(S)\) and \(a+ab=a\) for all \(a,b\in S\), (iii) \(ef=fe\) for all \(e,f\in E^\cdot(S)\) and \(a+ab=a\) for all \(a,b\in S\). Define the relation \(\eta\) for all \(a,b\in S\) in the following way: \(a\eta b\) if and only if \(a+axb=b+axb\) for some \(x\in S\). If \((S,+,\cdot)\) is \(k\)-regular and \(ef=fe\) for all \(e,f\in E_k(S)\) then \(\eta\) is the least distributive lattice congruence on \((S,+,\cdot)\).
scientific article; zbMATH DE number 5872615