Supertropical monoids: Basics and canonical factorization

A supertropical monoid is a monoid U together with a projection onto a totally ordered submonoid eU (where e∈ U is idempotent). Supertropical monoids are slightly more general than the supertropical semirings that were introduced and used by the first and the third authors for refinements of tropical geometry and matrix theory, and then studied systematically by the authors in connection with " supervaluations", and they permit a finer investigation of the supertropical theory.In the present paper we extend our earlier study of the category STROP of supertropical semirings to a category STROP_m of supertropical monoids whose morphisms are "transmissions", defined analogously as for supertropical semirings. Moreover, there is associated to every supertropical monoid V a canonical supertropical semiring V̂.A central problem in Izhakian etal. (2011)[8-10] has been to find the quotient U/E of a supertropical semiring U by a "TE-relation", which is a certain kind of congruence. This quotient always exists in STROP_m, and is the natural quotient in STROP in case U/E happens to be a supertropical semiring. Otherwise, analyzing (U/E)^∧, we obtain a mild modification of E to a TE-relation E^' such that U/E^'=(U/E)^∧ in STROP.In this way we now can solve problems about universality in the category STROP that were left open in our earlier work, and gain further insight into the structure of transmissions and supervaluations which leads to new results on totally ordered supervaluations and monotone transmissions.