Tropical Arithmetic and Tropical Matrix Algebra

This paper 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. Introduction Traditionally, researchers have been able to frame mathematical theories using formal structures provided by algebra; geometry is often a source for interesting phenomena in the core of these theories. The semiring structure introduced in this paper emerges from the combinatorics within max-plus algebra and its corresponding polyhedral geometry, called tropical geometry. Although our ground structure is a semiring, much of the theory of standard commutative algebra can be formulated on this semiring, leading to application in combinatorics, semigroup theory, polynomials algebra, and algebraic geometry. Tropical mathematics takes place over the tropical semiring (R ∪ {−∞},max,+), the real numbers equipped with the operations of maximum and summation, respectively, addition and multiplication [4, 6, 12], and it interacts with a number of fields of study including algebraic geometry, polyhedral geometry, commutative algebra, and combinatorics. Polyhedral complexes, resembling algebraic varieties over a field with real non-archimedean valuation, are the main objects of the tropical geometry, where their geometric combinatorial structure is a maximal degeneration of a complex structure on a manifold. Over the past few years, much effort has been invested in the attempt to characterize a tropical analogous to classical linear algebra, [3, 7, 12], and to determine connections between the classical and the tropical worlds [11, 13, 14]. Despite the progress that has been achieved in these tropical studies, some fundamental issues have not been settled yet; the idempotency of addition in (R ∪ {−∞},max,+) is maybe one of the main reasons for that. Addressing this reason, and other algebro-geometric needs, our goals are: (a) Introducing a new structure of a partial idempotent semiring having its own arithmetic that generalizes the max-plus arithmetic and also carries a tropical geometric meaning; (b) Presenting a novel approach for a theory of matrix algebra over partial idempotent semirings that includes notions of regularity and semigroup invertibility, analogous as possible to that of matrices over fields. The latter goal is central issue in the study of Green’s relations over semigroups and is essential toward developing a linear representations of semigroups. Our new approach answers these goals and paves a way to treat other needs like having a notions of linear dependency and rank. Our new structure, which we call extended tropical semiring , is built on the disjoint union of two copies of R, denoted R and Rν , together with the formal element −∞ that serves as the gluing point of R and Rν . Thus, T := R ∪ {−∞} ∪ Rν Date: Febuary 2008. 1991 Mathematics Subject Classification. Primary 15A09, 15A15, 16Y60; Secondary 15A33, 20M18, 51M20.