Tropical Arithmetic & Algebra of Tropical Matrices

The purpose of this paper is to study the tropical algebra – the algebra over the tropical semi-ring. We start by introducing a third approach to arithmetic over the max-plus semi-ring which generalizes the two other concepts in use. Regarding this new arithmetic, matters of tropical matrices are discussed and the properties of these matrices are studied. These are the preceding phases toward the characterization of the tropical inverse matrix which is eventually attained. Further development yields the notion of tropical normalization and the principle of basis change in the tropical sense. Instruction Over the past few years, much effort has been invested in the attempt to characterize the tropical analogue to " classical " linear algebra and to determine the linkage between the " classical " and the tropical " worlds ". However, despite the intensive development and the progress that has been achieved in the study of this field, many fundamental issues have not been settled yet. Motivated by these absences, the purposes of this paper are, • to introduce a new arithmetic defined over the extended tropical semi-ring and to phrase its basics, • to develop the theory of matrices' algebra over the tropical semi-ring and solve the matrix invertibility problem. Referring to semi-rings, the last aim mentioned above is indeed one of the hardest central issues and it appears in many applications, researches etc. Recently, intensive development in the studies of the tropical algebra and tropical geometry has been made, and the special " nature " of tropical objects enables this progress 1 to proceed in varied directions and via different approaches. The main significance of these tropical entities is their being geometric " images " of algebraic objects and concurrently comprise combinatorial attributes. As in the " classical " perception, the tropical algebra is aimed at providing a " pure " description of these geometric objects, which in this case includes their combinatorial properties. In fact, this algebra is a variant of the max-plus algebra which, for this reason, carries the name " the linear algebra of combinatorics " that was outlined by Butkovic [23]. Algebraic objects are formally, elements of the geometry over the fields (K, +, ·), where the tropical ones are those over the geometry of the semi-ring (R, max, +) – the semi-ring which contains the Max-Plus Algebra [5, 25]. The fundamental objects in this geometry are Polyhedral Complexes, where their …