Sandwich set of idempotents in the semigroup of 2 × 2 tropical matrices

Biordered sets are structures representing the set of idempotents of a semigroup. Sandwich sets are special subsets associated with a pair of elements in a biordered set which contains a good amount of information on the structure of the associated semigroup. Sandwich sets are non-empty whenever the associated semigroup is regular. In this paper, we describe the sandwich sets for every pair of idempotents in the semigroup of [Formula: see text] tropical matrices, denoted by [Formula: see text] because of the regularity of [Formula: see text]. For listing the sandwich sets easily, we introduce a [Formula: see text]-valued function [Formula: see text] for a pair of matrices in [Formula: see text]. Further, using this function [Formula: see text], we describe regular biordered subsets of [Formula: see text] as well as some subsemigroups of [Formula: see text].