Clones, order varieties, near unanimity functions and holes
For an integer \(n\geq 3\) a function m: \(P^ n\to P\) is called a near unanimity function if \(m(x,y,y,...,y)=m(y,x,y,...,y)=...=m(y,y,...,y,x)=y\) for x,y\(\in P\). An ordered set \(P=\{a_ 1,a_ 2,...,a_ n\}\) is called a fence if the ordering in P is given by \(a_ 1<a_ 2>a_ 3<..\). or dually, with no other comparabilities. Theorem 1. A finite ordered set has an order-preserving majority function (i.e. an order-preserving near unanimity function with \(n=3)\) if and only if it is contained in the order variety generated by all fences. The set Cl(P) of all order-preserving functions on an ordered set P forms a clone. The authors put forward the conjecture that a finite bounded ordered set P has an n-ary order-preserving near unanimity function for some \(n\geq 3\) if and only if Cl(P) is finitely generated.
scientific article; zbMATH DE number 4214074
scientific article