[Paper Review] Supertropical Monoids: Basics, Canonical Factorization, and Lifting Ghosts to Tangibles

Zur Izhakian, Manfred Knebusch+1 more|arXiv (Cornell University)|Aug 9, 2011

Polynomial and algebraic computation14 references3 citations

TL;DR

This paper introduces supertropical monoids as a generalization of supertropical semirings, establishing a category STROPₘ of transmissions and proving that quotients under TE-relations always exist in this category. The key contribution is a canonical construction of a supertropical semiring from any supertropical monoid, enabling the solution of open problems in supervaluation theory, including lifting ghosts to tangibles via the almost tangible lift construction.

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].

Motivation & Objective

• To generalize supertropical semirings to supertropical monoids, extending the category STROP to STROPₘ.
• To resolve open problems in supervaluation theory, particularly the existence and structure of quotients under TE-relations.
• To provide a canonical construction of a supertropical semiring from any supertropical monoid, facilitating the lifting of ghosts to tangibles.
• To characterize the almost tangible lift of a supervaluation as the minimal supervaluation dominating a given one.
• To establish a framework for ordered supertropical monoids and m-supervaluations, enriching tropical geometry and valuation theory.

Proposed method

• Define supertropical monoids as structures with a ghost map ν: U → U that is idempotent and satisfies a + a = ν(a), generalizing supertropical semirings.
• Introduce transmissions as morphisms in STROPₘ, preserving ghost maps and multiplicative structure, extending the category STROP.
• Construct the associated supertropical semiring Ŷ from any supertropical monoid Y via a canonical quotient process.
• Use the concept of TE-relations to define quotients in STROPₘ, showing they always exist and yield a supertropical semiring when the quotient is good.
• Define the almost tangible lift φ̌ of a supervaluation φ as (φ̃)̂, the canonical quotient of the tangible lift φ̃, ensuring φ̌ is a supervaluation.
• Prove that the almost tangible lift φ̌ is the minimal supervaluation dominating φ, resolving the problem of lifting ghosts to tangibles.

Experimental results

Research questions

• RQ1 Does every TE-relation on a supertropical semiring admit a quotient in a category where such quotients are guaranteed to exist?
• RQ2 Can the quotient U/E of a supertropical semiring U under a TE-relation E be lifted to a supertropical semiring when U/E is not itself a supertropical semiring?
• RQ3 How can ghost elements in a supervaluation be systematically lifted to tangible elements in a refined supervaluation?
• RQ4 What is the minimal supervaluation dominating a given supervaluation, and how is it characterized?
• RQ5 Under what conditions does the almost tangible lift of a supervaluation coincide with the original supervaluation?

Key findings

• Every quotient U/E under a TE-relation exists in the category STROPₘ of supertropical monoids.
• If U/E is not a supertropical semiring, then (U/E)̂ is a supertropical semiring, and U/E′ = (U/E)̂ for a modified TE-relation E′, solving a long-standing problem from [IKR1]–[IKR3].
• The almost tangible lift φ̌ of a supervaluation φ is a supervaluation and is the minimal such supervaluation dominating φ.
• The almost tangible lift φ̌ satisfies G(φ̌) = v(𝔮′), where 𝔮′ is the preimage of the set of non-cancellative elements under the valuation v.
• When v is a valuation (i.e., M is cancellative), the almost tangible lift φ̌ coincides with the tangible lift φ̃.
• For any surjective tangible m-supervaluation φ: R → U, the image φ(R∖𝔮′) equals the set of tangible elements T(U), and φ(Y(v)) = S(U), the set of tangible NC-products in U.