The semigroup of nonempty finite subsets of integers (Q1084518)

This is a continuation of the author's work reviewed above. Let Z be the group of integers with the usual addition and \({\mathcal P}(Z)\) the power semigroup of Z. Let \({\mathcal S}\) be the subsemigroup of \({\mathcal P}(Z)\) consisting of all nonempty finite subsets of Z. In this paper the author studies archimedean components of \({\mathcal S}\). If \(X\in {\mathcal S}\), \(A_ X\) denotes the basis of the semigroup generated by X-min(X), and \(B_ X\) the basis of the semigroup generated by max(X)-X. The main theorem is: X and Y are in the same archimedean component if and only if \(A_ X=A_ Y\) and \(B_ X=B_ Y\). The structure of archimedean components is also studied. One of those results is: If X is a non- singleton, the component \({\mathcal A}(X)\) containing X is isomorphic to the direct product of an idempotent-free power joined semigroup and the group Z. \{Reviewer's note: Lemma 2.1 comes from Theorem 3.2.3 of the reviewer's book ''Semigroup theory'' (in Japanese, Kyoritsusha 1977 second edition).\}