El Bashir, Robert; Kepka, Tomas Congruence-simple semirings. (English) Zbl 1155.16034 Semigroup Forum 75, No. 3, 589-609 (2007). The semirings \((S,+,\cdot)\) under consideration are additively commutative but need not have a zero or an identity. Such a semiring is called congruence-simple if it has just two congruence relations. It is shown that every congruence-simple semiring belongs to one of the following three classes: (i) \((S,+)\) is idempotent, (ii) \((S,+)\) is cancellative, (iii) there is some \(o\in S\) such that \(a+a=o\) and \(o+a=a+o=o\) for all \(a\in S\). – Various new examples in each of these classes are given. Moreover, for special kinds of congruence-simple semirings, e.g. when \((S,+)\) is semisubtractive or \((S,\cdot)\) contains a neutral element, refinements of the above classification (i)-(iii) are proved. Reviewer: Udo Hebisch (Freiberg) Cited in 13 Documents MSC: 16Y60 Semirings 08A30 Subalgebras, congruence relations Keywords:congruences; congruence-simple semirings PDFBibTeX XMLCite \textit{R. El Bashir} and \textit{T. Kepka}, Semigroup Forum 75, No. 3, 589--609 (2007; Zbl 1155.16034) Full Text: DOI