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- ItemDominance and transmissions in supertropical valuation theory(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Izhakian, Zur; Knebusch, Manfred; Rowen, LouisThis paper is a sequel of [IKR1], where we defined supervaluations on a commutative ring R and studied a dominance relation Φ>=v between supervaluations φ and υ on R, aiming at an enrichment of the algebraic tool box for use in tropical geometry. A supervaluation φ:R→U is a multiplicative map from R to a supertropical semiring U, cf. [IR1], [IR2], [IKR1], with further properties, which mean that φ is a sort of refinement, or covering, of an m-valuation (= monoid valuation) υ:R→M. In the most important case, that R is a ring, m-valuations constitute a mild generalization of valuations in the sense of Bourbaki [B], while φ>=υ means that υ:R→V is a sort of coarsening of the supervaluation φ. If φ(R) generates the semiring U, then φ>=υ if there exists a "transmission" α:U→V with φ=α∘φ. Transmissions are multiplicative maps with further properties, cf. [IKR1, §55]. Every semiring homomorphism α:U→V is a transmission, but there are others which lack additivity, and this causes a major difficulty. In the main body of the paper we study surjective transmissions via equivalence relations on supertropical semirings, often much more complicated than congruences by ideals in usual commutative algebra.
- ItemSupertropical semirings and supervaluations(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2010) Izhakian, Zur; Knebusch, Manfred; Rowen, LouisWe interpret a valuation v on a ring R as a map v : R ! M into a so called bipotent semiring M (the usual max-plus setting), and then de¯ne a supervaluation ' as a suitable map into a supertropical semiring U with ghost ideal M (cf. [IR1], [IR2]) covering v via the ghost map U ! M. The set Cov(v) of all supervaluations covering v has a natural ordering which makes it a complete lattice. In the case that R is a field, hence for v a Krull valuation, we give a complete explicit description of Cov(v). The theory of supertropical semirings and supervaluations aims for an algebra fitting the needs of tropical geometry better than the usual max-plus setting. We illustrate this by giving a supertropical version of Kapranov's lemma.
- ItemMonoid valuations and value ordered supervaluations(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2011) Izhakian, Zur; Knebusch, Manfred; Rowen, LouisWe complement two papers on supertropical valuation theory ([IKR1], [IKR2]) by providing natural examples of m-valuations (= monoid valuations), after that of supervaluations and transmissions between them. The supervaluations discussed have values in totally ordered supertropical semirings, and the transmissions discussed respect the orderings. Basics of a theory of such semirings and transmissions are developed as far as needed.