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Congruence-free commutative semirings. (English) Zbl 0636.16020

Let \(S=(S,+,\cdot)\) be a semiring with identity such that both operations are commutative. Assume \(| S| \geq 2\) and call S congruence-free if it has only the two trivial congruences. If S is congruence-free either (S,\(\cdot)\) or (S\(\setminus \{0\},\cdot)\) is a cancellative semigroup, where 0 is a multiplicatively absorbing element of S. In the latter case, S is either a field or the 2-element Boolean semiring or \(\{\) \(S\setminus \{0\},\cdot)\) is any group established with two kinds of addition \((x+y=0\) for all x,y\(\in S\) or \(x+y=0\) if \(x\neq y\) and \(x+x=x)\). If S is a congruence-free semiring without a multiplicatively absorbing element 0, \((S,+)\) is either a band or cancellative. In the latter case, the smallest ring \(F=D(S)\) containing S is an infinite field, and S the strict positive cone of a partial order \(\leq\) of F. The converse is shown in the case that \((F,+,\cdot,\leq)\) is an archimedean ordered field. The proofs are more complicated as these results may suggest and depend on the above assumption on \((S,+,\cdot)\).
Reviewer: H.J.Weinert

MSC:

16Y60 Semirings
12K10 Semifields
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References:

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