A geometric study of cores of idempotent stochastic matrices

For each idempotent E in the compact affine semigroup Stn of n×n stochastic matrices the core of E, denoted by Q(E), is defined as the maximal subsemigroup of Stn with zero element E. In this paper we study the algebraic and geometric structures of cores and describe how they are put together inside of Stn. In the process, lattice structures of the idempotents and their cores inside a larger core are revealed and then employed to initiate a possible classification.