Algebra Seminar talk
Maximal clones of bounded essential arity
In the study of universal algebras, one is often interested in the set of term operations rather than just the basic operations themselves, which leads to the study of clones: composition-closed sets of operations on a fixed domain. An interesting question in clone theory is whether a given clone can be generated by finitely many operations or not. Even for clones over finite domains, this problem is still not well understood. In a sense, a clone is not finitely generated when we are somehow restricted in increasing essential arity via composition. This motivates asking what clones with operations of bounded essential arity look like.
In the first part of the talk, I will give a short overview of known examples of finitely and non-finitely generated clones. In the second part, I will talk about clones with bounded essential arity, presenting a construction of maximal essentially binary clones. Although it is not clear whether we obtain all maximal essentially binary clones, we will see that every essentially binary operation $f$ which cannot generate operations of higher essential arities is contained in one of them. The results can be also partially generalized to higher arities, which yields examples of non-finitely generated clones.