Blyth, T. S.; Giraldes, Emília Perfect elements in Dubreil-Jacotin regular semigroups. (English) Zbl 0778.06012 Semigroup Forum 45, No. 1, 55-62 (1992). An ordered semigroup \(S\) is strong Dubreil-Jacotin, if there is an ordered group \(G\) and an epimorphism \(f:S\to G\) such that the pre-image of any principal order ideal of \(G\) is a principal order ideal of \(S\). Then the pre-image of the negative cone of \(G\) has the greatest element \(\xi\), called the bimaximum element of \(S\). For any \(x\in S\), denote \(\xi:x\) the greatest element of the order ideal \(\{y\in S;xy\leq\xi\}\); the element \(x\) is called perfect, if \(x=x(\xi:x)x\); \(S\) is perfect, if any element of \(S\) is perfect. The authors give some necessary and sufficient conditions for a regular strong Dubreil-Jacotin semigroup \(S\) to be perfect; one of them is that \(S\) is naturally ordered, i.e. its order is an extension of the natural order of idempotents of \(S\) (if \(ef=fe=e\), then \(e\leq f)\). Generally, the subset \(P(S)\) of perfect elements of \(S\) is a regular strong Dubreil-Jacotin subsemigroup of \(S\); there are given necessary and sufficient conditions for \(P(S)\) to be orthodox. Reviewer: V.Novák (Brno) Cited in 7 Documents MSC: 06F05 Ordered semigroups and monoids 20M19 Orthodox semigroups 20M17 Regular semigroups Keywords:ordered semigroup; regular strong Dubreil-Jacotin semigroup; natural order; perfect elements PDFBibTeX XMLCite \textit{T. S. Blyth} and \textit{E. Giraldes}, Semigroup Forum 45, No. 1, 55--62 (1992; Zbl 0778.06012) Full Text: DOI EuDML References: [1] Blyth, T. S. and M. F. Janowitz,Residuation Theory, Pergamon Press, 1972. · Zbl 0301.06001 [2] Blyth, T. S., Perfect Dubreil-Jacotin semigroups, Proc. Roy. Soc. Edinburgh, 78A, 101-104 (1977) · Zbl 0376.06017 [3] Blyth, T. S.; McAlister, D. B., Split orthodox semigroups, Journal of Algebra, 51, 491-525 (1978) · Zbl 0391.20043 · doi:10.1016/0021-8693(78)90118-7 [4] Blyth, T. S.; McFadden, R., Naturally ordered regular semigroups with a greatest idempotent, Proc. Roy. Soc. Edinburgh, 91A, 107-122 (1981) · Zbl 0503.20025 [5] McAlister, D. B., Regular Rees matrix semigroups and regular Dubreil-Jacotin semigroups, J. Australian Math. Soc., 31, 325-336 (1981) · Zbl 0474.06015 · doi:10.1017/S1446788700019467 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.