Finitely Many Primitive Positive Clones
S. Burris, R. Willard · Proceedings of the American Mathematical Society · 1987
Given a finite set $A$ there are only finitely many sequences of the form ${\left \langle {\operatorname {Con}({{\mathbf {A}}^n})} \right \rangle _{n \geq 1}}$ or ${\left \langle {\operatorname {Hom}({{\mathbf {A}}^n},{\mathbf {A}})} \right \rangle _{n \geq 1}}$, where ${\mathbf {A}}$ is any algebra on $A$. From this we derive the fact that there are only finitely many primitive positive clones on $A$, 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.