Notions of size in a semigroup: an update from a historical perspective

Previous papers have investigated relationships among several notions of largeness in a semigroup, some of which have their origins in topological dynamics, others with pure combinatorial roots, and still others based on the algebraic structure of the Stone–Čech compactification of a discrete semigroup. Here we consider 52 distinct notions of largeness, giving to the extent possible a description of the origins and why the notions are of interest. We establish implications that must hold among these notions. In the event the semigroup is commutative, these reduce to 24 distinct notions. We give examples in ( N , + ) showing that the notions satisfy only the implications which we have established for commutative semigroups.