Multiplicative structure of 2 × 2 tropical matrices
نویسنده
چکیده
We study the algebraic structure of the semigroup of all 2 × 2 tropical matrices under multiplication. Using ideas from tropical geometry, we give a complete description of Green’s relations and the idempotents and maximal subgroups of this semigroup.
منابع مشابه
Finiteness property of pairs of 2×2 sign-matrices via real extremal polytope norms
This paper deals with the joint spectral radius of a finite set of matrices. We say that a set of matrices has the finiteness property if the maximal rate of growth, in the multiplicative semigroup it generates, is given by the powers of a finite product. Here we address the problem of establishing the finiteness property of pairs of 2× 2 sign-matrices. Such problem is related to the conjecture...
متن کاملUsing Tropical Optimization Techniques to Evaluate Alternatives via Pairwise Comparisons
We describe a new approach based on tropical optimization techniques to solve the problem of rating alternatives from pairwise comparison data. The problem is formulated to approximate, in the logChebyshev sense, pairwise comparison matrices by reciprocal matrices of unit rank, and then represented in general terms of tropical mathematics as a tropical optimization problem. The optimization pro...
متن کاملOn multiplicative (strong) linear preservers of majorizations
In this paper, we study some kinds of majorizations on $textbf{M}_{n}$ and their linear or strong linear preservers. Also, we find the structure of linear or strong linear preservers which are multiplicative, i.e. linear or strong linear preservers like $Phi $ with the property $Phi (AB)=Phi (A)Phi (B)$ for every $A,Bin textbf{M}_{n}$.
متن کاملDeterminants of Multiplicative Toeplitz Matrices1
In this paper we study matrices A = (aij) whose (i, j) -entry is a function of i/j; that is, aij = f(i/j) for some f : Q → C. We obtain a formula for the truncated determinants in the case where f is multiplicative, linking them to determinants of truncated Toeplitz matrices. We apply our formula to obtain several determinants of number-theoretic matrices. Mathematics Subject Classification 200...
متن کاملSemigroup identities in the monoid of two-by-two tropical matrices
We show that the monoid M2(T) of 2 × 2 tropical matrices is a regular semigroup satisfying the semigroup identity A2B4A2A2B2A2B4A2 =A2B4A2B2A2A2B4A2. Studying reduced identities for subsemigroups of M2(T), and introducing a faithful semigroup representation for the bicyclic monoid by 2 × 2 tropical matrices, we reprove Adjan’s identity for the bicyclic monoid in a much simpler way.
متن کامل