On the least distributive lattice congruence on a semiring with a semilattice additive reduct

Abstract

We characterize the least distributive lattice congruence on the semirings S with semilattice additive reduct in different ways arising from the non-symmetry and non-transitivity of the binary relation \({\longrightarrow}\) defined by: for \({a, b{\in}S}\), \({a{\longrightarrow}b}\) if \({b^{n}{\in}\overline{SaS}}\) for some positive integer n. Non-symmetry of the transitive closure \({{\longrightarrow}^{\infty}}\) of \({\longrightarrow}\) gives us \({M(a)=\{x{\in}S {\mid} a{\longrightarrow}^{\infty} x\}}\) and \({N(a)=\{x{\in} S {\mid} x {\longrightarrow}^{\infty} a\}}\) which are the principal completely semiprime k-ideal and the principal filter of S generated by a respectively. Both the relations \({\mathcal{M}}\) and \({\mathcal{N}}\) induced by M(a) and N(a) respectively are the least distributive lattice congruence on S. Again non-transitivity of \({\longrightarrow}\) yields an expanding family \({\{ {\longrightarrow}^{n}\}}\) of binary relations which again associates subsets Mn(a) and Nn(a) for all \({a{\in}S}\), and induces equivalence relations \({\mathcal{M}_{n}}\) and \({\mathcal{N}_{n}}\). Also the n-th power of the symmetric opening \({{\longleftrightarrow}}\) of \({\longrightarrow}\) gives us \({\Sigma_{n}(a)=\{x {\in} S \mid a {\longleftrightarrow}^{n} x\}}\) which induces the equivalence relation \({\sigma_{n}}\). We have characterized the semirings which are distributive lattices of \({\mathcal{M}_{n}(\mathcal{N}_{n}, \sigma_{n})}\) -simple semirings.