Braids and their monotone clones
The authors investigate properties of monotone clones of ordered sets called braids. For an ordered set \(P\), let \(d(a,b)\) denote the minimal number of edges of the up-fence and the down-fence from \(a\) to \(b\). The reach \(r(P)\) is the supremum of \(d(a,b)\) as \(a\) and \(b\) vary over \(P\). An ordered set \(B\) is called a braid if it has at least three elements, is connected, has finite reach and for any \(a\in B\) there exists a unique \(a'\in B\) with \(d(a,a')= r(B)\). So every \(n\)-crown \(C_ n\) with \(2n\) elements is a braid of reach \(r(C_ n)=n\); if \(L\) is an atomless Boolean lattice, then \(B=L \setminus \{0,1\}\) is a braid with \(r(B)=3\). The main result is the following: If \(B\) is a braid with reach \(r(B)>2\), then the only idempotent order preserving functions \(f: B^ n\to B\) are the \(n\) projections. Hence, using Mal'tsev conditions for congruence modularity, no algebra of monotone functions on a finite braid \(B\) with \(r(B)>2\) generates a congruence modular variety. The main result is also generalized to the so-called near-braids.
scientific article; zbMATH DE number 679600