Janowitz, M. F. Regularity of residuated mappings. (English) Zbl 0725.06002 Semigroup Forum 42, No. 3, 313-332 (1991). A principal ideal of a poset P is a subset of the form \(J_ x=\{t\in P\); \(t\leq x\}\) for some fixed element \(x\in P\). If P and Q are posets, a mapping f: \(P\to Q\) is said to be residuated if it has the property that for each \(q\in Q\), \(f^{-1}(q)\) is necessarily a principal ideal of P. Let Res(P) denote the monoid of all residual mappings f: \(P\to P\) with composition as the operation. The author studies complete lattices L for which Res(L) is regular. He shows that Res(L) is regular provided L is a complete chain or the horizontal sum of a certain class of complete chains. A complete characterization of lattices L of finite length with regular Res(L) is given. Reviewer: E.Fuchs (Brno) Cited in 1 Document MSC: 06B05 Structure theory of lattices 06B23 Complete lattices, completions 20M99 Semigroups Keywords:lattices of finite length; regular monoid; principal ideal; residual mappings; complete lattices; complete chains PDFBibTeX XMLCite \textit{M. F. Janowitz}, Semigroup Forum 42, No. 3, 313--332 (1991; Zbl 0725.06002) Full Text: DOI EuDML References: [1] Adams, M. E. and Gould, M.,Posets whose monoids of order-preserving maps are regular, Order6 (1989) 195–201. · Zbl 0689.06004 · doi:10.1007/BF02034336 [2] Adams, M. E. and Gould. M.,Finite semilattices whose monoids of endomorphisms are regular. to appear in Trans. Amer. Math. Soc. · Zbl 0766.06008 [3] Andréka, H., Greechie, R. J., and Strecker, G. E.,On residuated approximations, to appear in the ”Proceedings of the International workshop on categorical methods in computer science with aspects from topology”. [4] Blyth, T. S. and Janowitz, M. F., ”Residuation Theory”, Pergamon Press, 1972. · Zbl 0301.06001 [5] Habib, M. and Möhring, R. H.,On some complexity properties of N-free posets and posets with bounded decomposition diameter, Discrete Math.63 (1987), 157–182. · Zbl 0608.06004 · doi:10.1016/0012-365X(87)90006-9 [6] Janowitz, M. F.,Baer semigroups, Duke Math. J.32 (1963), 85–96. · Zbl 0158.02303 · doi:10.1215/S0012-7094-65-03206-0 [7] Janowitz, M. F.,A semigroup approach to lattices, Canad. J. Math.18 (1966), 1212–1223. · Zbl 0154.01003 · doi:10.4153/CJM-1966-119-5 [8] Schweizer, B. and Sklar. A., ”Probabilistic Metric Spaces”, North-Holland, 1983. · Zbl 0546.60010 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.