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Numerical isotonies of preordered semigroups through the concept of a scale. (English) Zbl 1090.06011

Summary: Necessary and sufficient conditions are presented for the existence of a continuous and additive real-valued function representing a (not necessarily total) preorder defined on a semigroup. The main purpose of this paper is that of providing a characterization of the existence of a continuous and order-preserving real-valued function defined on a preordered toplogical semigroup such that it also preserves the binary operation. The approach followed to obtain this characterization is based on the existence of particular scales that behave well with respect to the semigroup operation and that we call additive scales. A scale is a device that consists of a nested family of sets indexed by a dense subset of a suitable part of the reals.

MSC:

06F05 Ordered semigroups and monoids
20M15 Mappings of semigroups
22A20 Analysis on topological semigroups
54H10 Topological representations of algebraic systems
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