## 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 $$S$$.

### MSC:

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