Essential arities in algebras of finite type and arity trees Original Research Article

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 in A. Questions concerning the set S(A) for various types of algebras were raised in the early 1960s. Among still open questions, one of the most important is the problem of characterizing the sets S(A) for algebras of finite type. In this paper we give a characterization of these sets in terms of the so-called arity trees. Using this we prove several concrete results and formulate some particular questions of a combinatorial nature, which seems especially interesting in this connection.