Some combinatorially defined subsets of and their relation to the idempotents

Members of idempotents in ( β N , + ) , especially those in the smallest ideal, have strong combinatorial properties. And the closure Γ of the set of idempotents has a simple combinatorial description. We investigate here the relationships among several subsets of β N that have simple combinatorial descriptions, as well as the semigroups they generate and their closures.