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Residuated mappings. (English) Zbl 0553.06001

This interesting well written survey article deserves the attention of anyone having an interest in partially ordered sets, or more generally in partially ordered algebraic structures. The central theme is that of a residuated mapping. If P, Q are posets, this is a mapping \(\Phi\) : \(P\to Q\) having the property that the preimage of every principal ideal of Q is a principal ideal of P. The author defines residuated mappings, shows how the Adjoint Functor Theorem bears on them, and then presents a potpourri of examples. Following this, there is a discussion of their role in lattice theory. A central theme in the study of residuated mappings has been the investigation of the semigroup RES(P) of such mappings on a poset P with a view toward relating order properties of P to semigroup properties of RES(P). The author’s exposition of this topic serves to give the reader a nice introduction to this topic together with some feeling for the underlying issues. Following this, there is a survey of residuated algebraic structures with mention given to Dubreil-Jacotin semigroups and their role in the study of fractionary ideals of a commutative integral domain. Some mention is then given to the work of Crown on the category of posets with residuated mappings as the morphisms, and the paper concludes with a list of 10 open questions.
Reviewer: M.F.Janowitz

MSC:

06A06 Partial orders, general
06B05 Structure theory of lattices
20M20 Semigroups of transformations, relations, partitions, etc.
06A15 Galois correspondences, closure operators (in relation to ordered sets)
06F05 Ordered semigroups and monoids
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References:

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