Monoid Valuations and Value Ordered Supervaluations

Abstract

We complement two papers on supertropical valuation theory ([11], [12]) by providing natural examples of m-valuations (= monoid valuations), and afterwards of supervaluations and transmissions between them. These supervaluations have values in totally ordered supertropical semirings, and the transmissions discussed respect the orderings. We develop the basics of the theory of such semirings and transmissions.

Key Words:

• Bipotent semirings
• Lattices
• Monoid valuations
• Ordered supertropical semirings
• Supertropical algebra
• Supervaluations
• Transmissive
• and homomorphic equivalence relations
• Valuation theory

2010 Mathematics Subject Classification:

• Primary: 13A18, 13F30, 16W60, 16Y60
• Secondary: 03G10, 06B23, 12K10, 14T05

Notes

Although this does not mean surjectivity in the usual sense, there is no danger of confusion since a supervaluation ϕ: R → U can hardly ever be surjective as a map except in the degenerate case U = M.

The map p given here is analogous to the “ghost map” ν given for semirings with ghosts in [14].

The construction of STR(𝒯, 𝒢, v) can be understood as a special case of [12, Theorem 3.1].

In the examples below, v will be a valuation.

This means that is an upper set in the totally ordered set .

Communicated by M. Cohen.