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

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].