SEMIGROUP OF 2 × 2 TROPICAL CIRCULANT MATRICES AND THE BIORDER STRUCTURE OF THEIR IDEMPOTENTS

Abstract

Tropical circulant matrix is a circulant matrix which operates within the tropical algebra framework where addition is replaced by the minimum and multiplication by the ordinary addition. These matrices have found applications in areas like cryptography and graph theory. In this paper we are dealing with C2(R), the multiplicative semigroup of 2 × 2 circulant matrices over tropical semiring R = (R [ f1g; min; +). The semigroup M2(R) of all 2×2 tropical matrices are known to be a regular semigroup. We show that the subsemigroup C2(R) of circulant matrices is an inverse semigroup, that is, a regular semigroup in which every element has a unique inverse. It is known that the set of idempotents of an inverse semigroup is a semi-lattice and this semi-lattice of idempotents provides a way to understand and analyze the structure and behavior of inverse semigroups. So the semi-lattice E(C2) of idempotents of C2(R) plays a crucial role in the structure of C2(R). Also we determine the biorder structure of the set of idempotents and describe the sandwich sets associated with the idempotents. A description of the associated inductive groupoid is also given.