Semirigid sets of diamond orders (Q1923526)

Let \(A\) be a set. A diamond is an order relation \(\leq_{ab}\) on \(A\) with the least element \(a\) and the greatest element \(b\) such that if \(u,v\in A\) then \(u\leq_{ab}v\) exactly if \(u=a\) or \(v=b\). A set \(R\) of diamonds on \(A\) is called semirigid if the identity and all constant maps are the only maps of \(A\) into \(A\) that are isotone for all diamonds in \(R\). In the paper, the semirigid sets of diamonds are characterized by means of associated graphs. Further, some combinatorial properties of semirigid sets of diamonds on finite sets are investigated. The subject of the paper is in close connection with the study of quasilinear clones.