On \(q\)-Free Subsets of Operation Sets

A set \(T\) with a binary operation \(+\) is called an operation set and denoted as \((T, +)\). An operation set \((S, +)\) is called \(q\)-free if \(qx \notin S\) for all \(x \in S\). Let \(\psi_q(T)\) be the maximum possible cardinality of a \(q\)-free operation subset \((S, +)\) of \((T, +)\).

We obtain an algorithm for finding \(\psi_q({N}_n)\), \(\psi_q({Z}_n)\) and \(\psi_q(D_n)\), \(q \in {N}\), where \({N}_n = \{1, 2, \ldots, n\}\), \(( {Z}_n, +_n)\) is the group of integers under addition modulo \(n\) and \((D_n, +_n)\) is the dihedral group of order \(2n\).