Partial orders induced by quasilinear clones (Q2837214)

Let \(A\) and \(U\) be sets, let \({\mathcal C}\) be a clone of operations on \(A\), let \(f:\;A^m\to U\), \(g:\;A^n\to U\) be functions. We write \(f\leq_{\mathcal C}g\) if \(f=g(h_1,\dots,h_n)\) for some \(m\)-ary operations \(h_1,\dots,h_n\in{\mathcal C}\). This defines a quasiorder relation on the set \({\mathcal F}(A,U)\) of all finitary functions \(A^n\to U\). After factorizing by the associated equivalence we obtain the partially ordered set \({\mathcal F}(A,U)/\equiv_{\mathcal C}\). As their main result the authors find several types of clones \({\mathcal C}\) for which the following holds: (1) \({\mathcal F}(A,U)/\equiv_{\mathcal C}\) is countable and all its principal ideals are finite; (2) every countable partially ordered set with finite principal ideals embeds into \({\mathcal F}(A,U)/\equiv_{\mathcal C}\).NEWLINENEWLINEFor the entire collection see [Zbl 1234.08003].