Finitely many primitive positive clones

Abstract

Given a finite set A A there are only finitely many sequences of the form ⟨ Con ⁡ ( A n ) ⟩ n ≥ 1 {\left \langle {\operatorname {Con}({{\mathbf {A}}^n})} \right \rangle _{n \geq 1}} or ⟨ Hom ⁡ ( A n , A ) ⟩ n ≥ 1 {\left \langle {\operatorname {Hom}({{\mathbf {A}}^n},{\mathbf {A}})} \right \rangle _{n \geq 1}} , where A {\mathbf {A}} is any algebra on A A . From this we derive the fact that there are only finitely many primitive positive clones on A A , which solves a problem posed by A. F. Danil’čenko in the 1970s. Consequently there are only finitely many model companions for universal Horn classes generated by an algebra of a given finite size.