On algebras with many symmetric operations
Catarina Carvalho, Andrei Krokhin · International Journal of Algebra and Computation · 2016
An [Formula: see text]-ary operation [Formula: see text] is called symmetric if, for all permutations [Formula: see text] of [Formula: see text], it satisfies the identity [Formula: see text]. We show that, for each finite algebra [Formula: see text], either it has symmetric term operations of all arities or else some finite algebra in the variety generated by [Formula: see text] has two automorphisms without a common fixed point. We also show this two-automorphism condition cannot be replaced by a single fixed-point-free automorphism.