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      Supertropical Monoids: Basics, Canonical Factorization, and Lifting Ghosts to Tangibles

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          Abstract

          Supertropical monoids are a structure slightly more general than the supertropical semirings, which have been introduced and used by the first and the third authors for refinements of tropical geometry and matrix theory in [IR1]-[IR3], and then studied by us in a systematic way in [IKR1]-[IKR3] in connection with "supervaluations". In the present paper we establish a category \(\STROP_m\) of supertropical monoids by choosing as morphisms the "transmissions", defined in the same way as done in [IKR1] for supertropical semirings. The previously investigated category \(STROP\) of supertropical semirings is a full subcategory of \(STROP_m.\) Moreover, there is associated to every supertropical monoid \(V\) a supertropical semiring \(\hat V\) in a canonical way. A central problem in [IKR1]-[IKR3] has been to find for a supertropical semiring \(U\) the quotient \(U/E\) by a "TE-relation", which is a certain kind of equivalence relation on the set \(U\) compatible with multiplication (cf. [IK1, Definition 4.5]). It turns out that this quotient always exists in \(\STROP_m\). In the good case, that \(U/E\) is a supertropical semiring, this is also the right quotient in \(\STROP.\) Otherwise, analyzing \((U/E)^\wedge,\) we obtain a mild modification of \(E\) to a TE-relation \(E'\) such that \(U/E' = (U/E)^\wedge\) in \(\STROP.\) In this way we now can solve various problems left open in [IKR1], [IKR2] and gain further insight into the structure of transmissions and supervaluations. Via supertropical monoids we also obtain new results on totally ordered supervaluations and monotone transmissions studied in [IKR3].

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          Most cited references 3

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          Supertropical matrix algebra

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            Supertropical matrix algebra III: Powers of matrices and their supertropical eigenvalues

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              THEM-VALUATION SPECTRUM OF A COMMUTATIVE RING

               Digen Zhang (2002)
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                Author and article information

                Journal
                09 August 2011
                Article
                1108.1880

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                Custom metadata
                13A18, 13F30, 16W60, 16Y60
                48 pages
                math.AC math.RA

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