On generators and relations for unions of semigroups (Q5939965)

A semigroup presentation is a pair \((A\mid R)\), where \(A\) is an alphabet and \(R\subseteq A^+\times A^+\), the symbol \(A^+\) denoting the free semigroup over \(A\). (The set \(R\) is the set of defining relations.) If \(A\) can be chosen to be finite, then \(S\) is called finitely generated; if so can both \(A\) and \(R\), it is called finitely presented. Four problems are treated for a semigroup \(S\) which is a union of semigroups \((S_i)_{i\in I}\); whether the fact that all \(S_i\) are finitely generated implies the same for \(S\) and conversely and then the analogous question for being finitely presented. The particular items of the paper study these problems for general unions of semigroups, for bands of semigroups, for bands of monoids and for semilattices and strong semilattices of semigroups.