Density of discrete semigroup flows

We generalize the notions of upper and lower density on semigroups to semigroup flows. We then show that when the Strong Følner Condition holds, there exist upper densities on the flow satisfying classical properties such as partition regularity of positive sets, finite additivity on translates, and invariance under semigroup action. We also introduce the concept of Følner density of a flow and show that under a certain condition, it is multiplicative on products.