Strongly nilpotent finite algebras

Abstract

<p>We investigate the finite algebras with a bound on the essential arities of their terms. This condition was recently shown to be equivalent to strong nilpotence. There are two directions in which our investigation proceeds. We are interested to obtain a satisfactory bound on the essential arities in terms of the size of algebra. We prove there exists a doubly exponential bound in the general case, and a singly exponential bound in the case of finite surjective groupoids. The second is satisfactory, as we exhibit a class of strongly nilpotent finite surjective groupoids for which the bound on essential arities is an exponential function of the size of the groupoids. The second direction is an investigation of the terms and the structure of the strongly nilpotent finite groupoids. We get a full description of the linear terms which depend on the maximal number of variables in a strongly nilpotent surjective finite groupoid. Also, in a special case we are able to prove a structure theorem, namely if the terms of a surjective strongly nilpotent finite groupoid behave in a &quot;nice&quot; way, we have a full description of the groupoid, and we prove that the groupoid is forced to be strongly Abelian in this case.</p>