Zur Izhakian

Tropical Arithmetic and Matrix Algebra

This article introduces a new structure of commutative semiring, generalizing the tropical semiring, and having an arithmetic that modifies the standard tropical operations, i.e., summation and maximum. Although our framework is combinatorial, notions of regularity and invertibility arise naturally for matrices over this semiring; we show that a tropical matrix is invertible if and only if it is regular.

Categories of Layered Semirings

We generalize the constructions of layered domains† to layered semirings, in order to enrich the structure, and in particular to provide finite examples for applications in arithmetic. The layered category theory is extended accordingly, to cover noncancellative monoids, which are examined in their own right.

Monoid Valuations and Value Ordered Supervaluations

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.

Dual spaces and bilinear forms in supertropical linear algebra

Continuing 4, this article investigates finer points of supertropical vector spaces, including dual bases and bilinear forms, with supertropical versions of standard classical results such as the Gram–Schmidt theorem and Cauchy–Schwartz inequality, and change of base. We also present the supertropical version of quadratic forms, and see how they correspond to symmetric supertropical bilinear forms.

Superboolean rank and the size of the largest triangular submatrix of a random matrix

We explore the size of the largest (permuted) triangular submatrix of a random matrix, and more precisely its asymptotical behavior as the size of the ambient matrix tends to infinity. The importance of such permuted triangular submatrices arises when dealing with certain combinatorial algebraic settings in which these submatrices determine the rank of the ambient matrix and thus attract special attention.

A Guide to Supertropical Algebra

This paper describes a new algebraic structure to enrich the algebraic theory underlying “tropical geometry,” an area of mathematics that has developed considerably over the last ten years, with applications to combinatorics, polynomials (Newton’s polytope), linear algebra, and algebraic geometry.