Epimorphisms in Varieties of Heyting Algebras

Abstrakt

Clearly any surjective homomorphism is an epimorphism. We say that K has the epimorphism surjectivity (ES) property if the converse holds. When a variety K algebraizes a logic `, then K has the ES property if and only if ` has the infinite (deductive) Beth (definability) property, i.e., whenever an arbitrarily large set Z of variables is defined implicitly in terms of other variables by means of some formulas over `, then it can also be defined explicitly [4]. When this is demanded only for finite Z, then ` is said to have the finite Beth property, which corresponds similarly to the so-called weak ES property. This invites us to question which varieties of Heyting algebras have surjective epimorphisms, or equivalently which intermediate logics have the infinite Beth property.