Splittability for partially ordered sets (Q5935837)

This paper deals with an extension of the notion of splittability (cleavability) of topological spaces, introduced by Arkhangel'skij (1985), to partially ordered sets [\textit{D. J. Marron} and \textit{T. B. M. McMaster}, Math. Proc. R. Ir. Acad. 99A, 189-194 (1999; Zbl 0966.06002)] as follows: If \(A\) is a subset of a poset \(X\), we say that \(X\) splits along \(A\) over a poset \(Y\) if there exists an increasing map \(f: X\to Y\) such that \(f^{-1}f(A)= A\). If \(X\) splits along each subset \(A\) of \(X\) over \(Y\), we say that \(X\) is increasing splittable over \(Y\). \textit{D. J. Marron} in his Ph.D. Thesis (1997) characterized posets that are increasing splittable over the \(n\)-point chain. The main results of this paper are characterizations of the subsets \(A\) of a poset \(X\) of height \(n\) or \(m> n\) with the property that \(X\) splits along \(A\) over the \(n\)-point chain. Further, the authors show that the union of two splittability classes need not be a splittability class and give a necessary condition for a class of topological spaces to be a splittability class. The definition of a splittability class is as follows: Let \({\mathcal Q}\) be a class of topological spaces. Denote by \(s({\mathcal Q})\) the class of topological spaces \(X\) such that for each subset \(A\) of \(X\) there is \(Y\in{\mathcal Q}\) and a continuous mapping \(f: X\to Y\) with \(f^{-1}f(A)= A\). A class \({\mathcal P}\) of topological spaces is called a splittability class if there exists a class of topological spaces \({\mathcal Q}\) such that \(s({\mathcal Q})={\mathcal P}\).