Essential arities in algebras of finite type and arity trees (Q1598815)

Given an algebra \(A\), \(S(A)\) denotes the set of those nonnegative integers \(n\) for which there is a nontrivial essentially \(n\)-ary term operation on \(A\). In 1965, K. Urbanik characterized the sets \(S\) of integers such that \(S= S(A)\) for some idempotent algebra \(A\). The first author characterized such \(S\) for ternary nonidempotent algebras, and S. Fajtlowicz did this for binary algebras. R. Williard characterized such \(S\) for finite algebras. In this paper a characterization of these sets in terms of arity trees is given and several particular concrete results and open questions of a combinatorial nature are established.