Permutation-pattern algebras. (Q1771851)

Given a \(k\)-ary relation \(\rho \) on a set \(A\), two \(n\)-tuples \((x_1,\dots , x_n), (y_1,\dots ,y_n)\) of elements of \(A\) are said to be of the same pattern with respect to \(\rho \) if for all \(i_1,\dots ,i_k\in \{ 1,\dots , n\}, \, (x_{i_1},\dots ,x_{i_k})\in \rho \) iff \((y_{i_1},\dots , y_{i_k})\in \rho \). An \(n\)-ary operation \(f\) on \(A\) is a \(\rho \)-pattern function if \(f(x_1,\dots ,x_n)\) is always equal to some \(x_i\), where \(i\) depends only on the \(\rho \)-pattern of \((x_1,\dots ,x_n)\). The author examines functional completeness of algebras with \(\rho \)-pattern fundamental operations in the case when \(\rho \) is the graph of some permutation of \(A\). The paper is a continuation of the author's research on functional completeness of finite \(\rho \)-pattern algebras.