The local loop lemma (Q2302149)

Let \(G=(V,E)\) be a digraph. Its edges are denoted by \((u,v)\), where \(u,v\) are vertices. A walk is a sequence of edges \([v_0, v_1,\dots,v_n]\) such that \((v_i,v_{i+1})\) for all \(i\) . A closed walk of length \(n\) is a walk where \(v_n\) = \(v_0\). A digraph is strongly connected if there is a walk from \(u\) to \(v\) for any pair of vertices \(u,v\). Let \(R\) denote the binary relation on \(G\) determined by the edges of \(G\). A graph \(G\) is called compatible with operation \(t\) if \(t\) is compatible with this relation \(R\). The aim of the paper it to prove the following result. Theorem. Consider a set \(A\) and an idempotent \(n\)-ary operation \(t\) on \(A\), a digraph \(G\) on \(A\) with vertices \(a_{ij}\) where \(i,j\) range over \({\{{1,...,n-1}\}}\) such that \(G\) is compatible with \(t\), \(G\) is either a strongly connected digraph containing directed closed walks of all lenghts with two, or \(G\) is an undirected connected non-bipartite graph, for every \(i\) there is an edge \((a_{i,j},t(a_{i,0},a_{i,1},...,a_{i,n-1}))\). Then \(G\) contains a loop. The proof of this theorem is very sofisticated and is based on the previous results by L.Barto and the author. They use Taylor terms and Taylor algebras. It is a strenghtenning of the previous results and it is the stronger result in this direction.