Face posets of tropical polyhedra and monomial ideals

Abstract

We exhibit several posets arising from commutative algebra, order theory, tropical convexity as potential face posets of tropical polyhedra, and we clarify their inclusion relations. We focus on monomial tropical polyhedra, and deduce how their geometry reflects properties of monomial ideals. Their vertex-facet lattice is homotopy equivalent to a sphere and encodes the Betti numbers of an associated monomial ideal.

50 pages, 14 figures. Revised version with various improvements and clarifications, particularly in Sections 5 and 6

References in corpus (6)

• The Minkowski Theorem for Max-plus Convex Sets
• Minimal resolutions of monomial ideals
• Minimal external representations of tropical polyhedra
• The Betti poset in monomial resolutions
• Slicing and dicing polytopes
• Lcm-lattice, Taylor Bases and Minimal Free Resolutions of a Monomial ideal

Cited by in corpus (2)

• Asymmetric tropical distances and power diagrams
• Tropical convex hulls of polyhedral sets