Some fundamental properties of tiling semigroups

At first, we determine the Greenʹs relations of a tiling semigroup. Then we analyze some congruences, which lead to a variety of properties characterizing tiling semigroups. It is proved that any tiling semigroup is 0-E-reflexive but is not 0-simple. We have found out certain necessary conditions in which tiling semigroups are E-reflexive and E-disjunctive respectively. Also we introduce a new relation on the tiling semigroup which is based on properties inherent to a tiling. This relation is shown to be an idempotent pure congruence. Finally, we investigate the least semilattice congruence on a tiling semigroup.