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Congruence-simple semirings. (English) Zbl 1155.16034

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.

MSC:

16Y60 Semirings
08A30 Subalgebras, congruence relations
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