Mitchell, Sidney S.; Fenoglio, Paul B. Congruence-free commutative semirings. (English) Zbl 0636.16020 Semigroup Forum 37, No. 1, 79-91 (1988). 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 Cited in 15 Documents MSC: 16Y60 Semirings 12K10 Semifields Keywords:congruences on semirings; cancellative semigroup; multiplicatively absorbing element; Boolean semiring; congruence-free semiring; positive cone; archimedean ordered field PDFBibTeX XMLCite \textit{S. S. Mitchell} and \textit{P. B. Fenoglio}, Semigroup Forum 37, No. 1, 79--91 (1988; Zbl 0636.16020) Full Text: DOI EuDML References: [1] Bourne, S., On multiplicative idempotents of a potent semiring, Proc. Nat. Acad. Sci. USA, 42, 632-638 (1956) · Zbl 0071.25602 · doi:10.1073/pnas.42.9.632 [2] Fuchs, L., Partially ordered algebraic systems, Pergamon Press 1963. · Zbl 0137.02001 [3] Hutchins, H., Division Semirings with 1+1=1, Semigroup Forum, 22, 181-188 (1981) · Zbl 0467.16032 · doi:10.1007/BF02572796 [4] Steinfeld, O., Über Semirings mit multiplikativer Kürzungsregel, Acta Sci. Math. Szeged, 24, 190-195 (1963) · Zbl 0123.00804 [5] Weinert, H. J., Über Halbringe und Halbkörper I, Acta Math. Acad. Sci. Hungar., 13, 365-378 (1962) · Zbl 0125.01002 · doi:10.1007/BF02020799 [6] Weinert, H. J., Multiplicative cancellativity of semirings and semigroups, Acta Math. Acad. Sci. Hungar., 35, 335-338 (1980) · Zbl 0454.16024 · doi:10.1007/BF01886303 [7] Weinert, H. J., On O-simple semirings, semigroup semirings and two kinds of division semirings, Semigroup Forum, 28, 313-333 (1984) · Zbl 0526.16031 · doi:10.1007/BF02572492 [8] Wiegandt, R., Über die Struktursätze der Halbringer, Ann. Univ. Sci. Budapest Eötvös Sect. Math., 5, 51-68 (1962) · Zbl 0123.00901 [9] Zeleznekow, J., Regular semirings, Semigroup Forum, 23, 119-136 (1981) · Zbl 0473.16024 · doi:10.1007/BF02676640 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.