Algebraic structures of tropical mathematics

Tropical mathematics often is defined over an ordered cancellative monoid \tM, usually taken to be (\RR, +) or (\QQ, +). Although a rich theory has arisen from this viewpoint, cf. [L1], idempotent semirings possess a restricted algebraic structure theory, and also do not reflect certain valuation-theoretic properties, thereby forcing researchers to rely often on combinatoric techniques. In this paper we describe an alternative structure, more compatible with valuation theory, studied by the authors over the past few years, that permits fuller use of algebraic theory especially in understanding the underlying tropical geometry. The idempotent max-plus algebra $A$ of an ordered monoid \tM is replaced by R: = L\times \tM, where $L$ is a given indexing semiring (not necessarily with 0). In this case we say $R$ layered by $L$. When $L$ is trivial, i.e, $L=\{1\}$, $R$ is the usual bipotent max-plus algebra. When $L=\{1,\infty\}$ we recover the "standard" supertropical structure with its "ghost" layer. When L = \NN we can describe multiple roots of polynomials via a "layering function" $s: R \to L$. Likewise, one can define the layering $s: R^{(n)} \to L^{(n)}$ componentwise; vectors $v_1, \dots, v_m$ are called tropically dependent if each component of some nontrivial linear combination \sum \a_i v_i is a ghost, for "tangible" \a_i \in R. Then an $n\times n$ matrix has tropically dependent rows iff its permanent is a ghost. We explain how supertropical algebras, and more generally layered algebras, provide a robust algebraic foundation for tropical linear algebra, in which many classical tools are available. In the process, we provide some new results concerning the rank of d-independent sets (such as the fact that they are semi-additive),put them in the context of supertropical bilinear forms, and lay the matrix theory in the framework of identities of semirings.Comment: 19 page

The set of realizations of a max-plus linear sequence is semi-polyhedral

We show that the set of realizations of a given dimension of a max-plus linear sequence is a finite union of polyhedral sets, which can be computed from any realization of the sequence. This yields an (expensive) algorithm to solve the max-plus minimal realization problem. These results are derived from general facts on rational expressions over idempotent commutative semirings: we show more generally that the set of values of the coefficients of a commutative rational expression in one letter that yield a given max-plus linear sequence is a semi-algebraic set in the max-plus sense. In particular, it is a finite union of polyhedral sets

Methods and Applications of (max,+) Linear Algebra

Projet META2Exotic semirings such as the ``$(\max,+)$ semiring'' $(\mathbb{R}\cup\{-\infty\},\max,+)$, or the ``tropical semiring'' $(\mathbb{N}\cup\{+\infty\},\min,+)$, have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete event system theory; graph theory (path algebra) and Markov decision processes, Hamilton-Jacobi theory; asymptotic analysis (low temperature asymptotics in statistical physics, large deviations, WKB method); language theory (automata with multiplicities). Despite this apparent profusion, there is a small set of common, non-naive, basic results and problems, in general not known outside the $(\max,+)$ community, which seem to be useful in most applications. The aim of this short survey paper is to present what we believe to be the minimal core of $(\max,+)$ results, and to illustrate these results by typical applications, at the frontier of language theory, control, and operations research (performance evaluation of discrete event systems, analysis of Markov decision processes with average cost). Basic techniques include: solving all kinds of systems of linear equations, sometimes with exotic symmetrization and determinant techniques; using the $(\max,+)$ Perron-Frobenius theory to study the dynamics of $(\max,+)$ linear maps. We point out some open problems and current developments

Algebraic Structures using Natural Class of Intervals

This book has eleven chapters. Chapter one describes all types of natural class of intervals and the arithmetic operations on them. Chapter two introduces the semigroup of natural class of intervals using R or Zn and study the properties associated with them. Chapter three studies the notion of rings constructed using the natural class of intervals. Matrix theory using the special class of intervals is analyzed in chapter four of this book. Chapter five deals with polynomials using interval coefficients. New types of rings of natural intervals are introduced and studied in chapter six. The notion of vector space using natural class of intervals is built in chapter seven. In chapter eight fuzzy natural class of intervals are introduced and algebraic structures on them is built and described. Algebraic structures using natural class of neutrosophic intervals are developed in chapter nine.Chapter ten suggests some possible applications. The final chapter proposes over 200 problems of which some are at research level and some difficult and others are simple.Comment: 170 pages; Published by The Educational Publisher Inc in 201

Group actions on central simple algebras

Let G be a group, F a field, and A a finite-dimensional central simple algebra over F on which G acts by F-algebra automorphisms. We study the subalgebras and ideals of A which are preserved by the group action. We prove a structure theorem and two classification theorems for invariant subalgebras under suitable hypotheses on A. We illustrate these results in the case of compact connected Lie groups and give some other applications. We also classify invariant ideals. © 2002 Elsevier Science (USA)