Projective extensions of bounded semilattices (Q998777)

Let \(S\) and \(K\) be semilattices and \(S\) is an extension of \(K\). Then \(S\) is called a projective extension of \(K\) if there exist a free semilattice \(F\) and homomorphisms \(e: S\rightarrow K\oplus F\), \(q: K\oplus F \rightarrow S\) (where \(K\oplus F\) denotes the free product of \(K\) and \(F\)) such that \(qe = \text{id}_{S}\) and the restrictions of \(e\), \(q\) on \(K\) are equal to id\(_{K}\). In a previous paper [Algebra Univers. 55, No.~1, 45--55 (2006; Zbl 1109.06004)] the authors derived a construction of all dense semilattices \(S\) that are projective extensions of a given dense semilattice \(K\). In this note they describe the projective extensions of bounded semilattices and, as a consequence, a new description of projective extensions of semilattices are obtained.