On generators and relations for unions of semigroups

Finite generation and presentability of general unions of semigroups, as well as of bands of semigroups, bands of monoids, semilattices of semigroups and strong semilattices of semigroups, are investigated. For instance, it is proved that a band Y of monoids S alpha (alpha is an element of Y) is finitely generated/presented if and only if Y is finite and all S-alpha are finitely generated/presented. By way of contrast, an example is exhibited of a finitely generated semigroup which is not finitely presented, but which is a disjoint union of two finitely presented subsemigroups.