Globally determined lattices and semilattices (Q801948)

Let S be a semigroup and \({\mathcal P}(S)\) the power semigroup of S (i.e. the semigroup of all non-empty subsets of S in which the operation is defined in the ordinary sense). In this paper the authors prove that the class of all meet-semilattices with 1, the class of all chains and the class of all lattices are globally determined, that is, if S, T are in the class then \({\mathcal P}(S)\cong {\mathcal P}(T)\) implies \(S\cong T\). In the proof, ''irreducible'' elements play an important part. An element \(x\in S\) is called irreducible if y,z\(\in S\) and \(x=yz\) implies \(x=y\) or \(x=z\).