On the lattice of congruences on inverse semirings. (English) Zbl 1196.16039

Summary: Let \(S\) be a semiring whose additive reduct \((S,+)\) is an inverse semigroup. The relations \(\theta\) and \(\kappa\), induced by tr and ker (resp.), are congruences on the lattice \(\mathcal C(S)\) of all congruences on \(S\). For \(\rho\in\mathcal C(S)\), we introduce four congruences \(\rho_{\min}\), \(\rho_{\max}\), \(\rho^{\min}\) and \(\rho^{\max}\) on \(S\) and show that \(\rho\theta=[\rho_{\min},\rho_{\max}]\) and \(\rho\kappa=[\rho^{\min},\rho^{\max}]\). Different properties of \(\rho\theta\) and \(\rho\kappa\) are considered here. A congruence \(\rho\) on \(S\) is a Clifford congruence if and only if \(\rho_{\max}\) is a distributive lattice congruence and \(\rho^{\max}\) is a skew-ring congruence on \(S\). If \(\eta\) (\(\sigma\)) is the least distributive lattice (resp. skew-ring) congruence on \(S\) then \(\eta\cap\sigma\) is the least Clifford congruence on \(S\).


16Y60 Semirings
08A30 Subalgebras, congruence relations
20M18 Inverse semigroups
Full Text: DOI Link