Two Problems of Porton on Reloids: Metamonovalued Reloids Are Monovalued, and S(S(f)) = S(f) Fails

A reloid from a set \(A\) to a set \(B\) is a filter on \(A\times B\). Reloids were introduced by V. Porton as a common generalisation of binary relations and uniformities. We settle two problems on reloids from Porton's book "General Topology as Ordered Semigroup Actions" which are also listed in the Open Problem Garden. First, every metamonovalued reloid is monovalued: if \((g\sqcap h)\circ f=(g\circ f)\sqcap(h\circ f)\) holds for the identity reloid \(g\) of \(B\) and the principal reloid \(h\) of the complement of the diagonal of \(B\), then some member of the filter \(f\) is the graph of a partial function. Together with a theorem of Porton this shows that, for nonempty families, monovalued, metamonovalued and weakly metamonovalued reloids are the same. Second, for a reloid \(f\) from a set to itself let \(S(f)\) be the join of the powers \(f^n\), \(n\ge 0\). We give such an \(f\) on a countable set with \(S(f)\circ S(f)\ne S(f)\) and \(S(S(f))\ne S(f)\). This answers Porton's question negatively and refutes two conjectures of the book. The identities do hold for principal reloids, in particular on finite sets, and for Porton's operator \(S^*\). A funcoid on the same set shows that the corresponding two conjectures for funcoids fail as well. This is an unrefereed note.