Endomorphisms of complete heyting algebras

Abstract

Each monoid can be represented as the endomorphism monoid of a complete Heyting algebra. More generally, the category of soberT1-spaces and their open continuous maps, and consequently the category of complete Heyting algebras are universal. On the other hand, it is often impossible to represent monoids using more special (Hausdorff) complete Heyting algebras: for instance, each finite commutative endomorphism monoid of such a Heyting algebra is weakly idempotent.