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Supertropical monoids are a structure slightly more general than thesupertropical semirings, which have been introduced and used by the first andthe 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] inconnection with "supervaluations".  In the present paper we establish a category $\STROP_m$ of supertropicalmonoids by choosing as morphisms the "transmissions", defined in the same wayas done in [IKR1] for supertropical semirings. The previously investigatedcategory $STROP$ of supertropical semirings is a full subcategory of $STROP_m.$Moreover, there is associated to every supertropical monoid $V$ a supertropicalsemiring $\hat V$ in a canonical way.  A central problem in [IKR1]-[IKR3] has been to find for a supertropicalsemiring $U$ the quotient $U/E$ by a "TE-relation", which is a certain kind ofequivalence 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 theright quotient in $\STROP.$ Otherwise, analyzing $(U/E)^\wedge,$ we obtain amild 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] andgain further insight into the structure of transmissions and supervaluations.Via supertropical monoids we also obtain new results on totally orderedsupervaluations and monotone transmissions studied in [IKR3].

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