On characteristically simple conservative algebras (Q2714204)

A finite algebra \(\mathcal A=(A, F)\) is functionally complete if Pol (the clone of all polynomial operations of \(\mathcal A\)) is the set of all operations on \(A\) and it is characteristically simple if (\(A\), \(F\cup \operatorname{Aut}\)) is simple. An \(n\)-ary operation \(f\) on \(A\) is called a \(\varrho\)-pattern operation if for any \((x_1,\dots,x_n)\in A^n\), \(f(x_1,\dots,x_n)=x_i\) for some \(i\), where \(i\) depends on the \(\varrho\)-pattern of \((x_1,\dots,x_n)\) only. By a \(\varrho\)-pattern algebra we mean an algebra on \(A\) whose fundamental operations are all \(\varrho\)-pattern operations. NEWLINENEWLINENEWLINEIn this paper a classification of finite, characteristically simple, conservative algebras is given and several corollaries are derived. Among others it is shown that every nontrivial, at least three-element, finite, conservative algebra with primitive automorphism group is functionally complete, and a nontrivial \(\varrho\)-pattern algebra where \(\varrho\) is a regular relation is functionally complete iff the intersection of the equivalence relations determining \(\varrho\) is the equality relation.