On additively regular seminearrings.
Let \((S,+,\cdot)\) be a (right distributive) seminearring such that \((S,+)\) is an inverse semigroup, and denote by \(E^+(S)\) the set of all idempotent elements in \((S,+)\). For any \(a\in S\) write \(a^*\) for the unique inverse of \(a\) in \((S,+)\) and \(a^0=a+a^*\). Then \((S,+,\cdot)\) satisfies property (D) if \(ab^0=a^0b\) for all \(a,b\in S\). A seminearring \((S,+,\cdot)\) is called distributively generated if it contains a multiplicative semigroup \((D,\cdot)\) of distributive elements which generates \((S,+)\). An ideal \(I\) of \((S,+,\cdot)\) is a \(k\)-ideal if for all \(a,b\in S\) such that \(a+b\in I\) one has \(a\in I\Leftrightarrow b\in I\), and it is full if \(E^+(S)\subseteq I\). A congruence \(\varrho\) on \((S,+,\cdot)\) is called normal, if \(e\varrho f\) for all \(e,f\in E^+(S)\). Then \(S/\varrho\) is a near-ring if and only if \(\varrho\) is normal. Let \(\mathcal I\) denote the set of all normal congruences and \(\mathcal J\) the set of all full \(k\)-ideals of \(S\). The main result is: if \((S,+,\cdot)\) is a distributively generated additively inverse seminearring satisfying (D), then \(\varphi\colon\mathcal I\to\mathcal J\) with \(\varphi(\varrho)=\{a\in S\mid a\varrho e\) for some \(e\in E^+(S)\}\) defines an inclusion preserving bijection. Moreover, the least near-ring congruence on such seminearrings is characterized. Some more results are proved for the special classes of Clifford and Bandelt seminearrings.
scientific article; zbMATH DE number 6343954