  It was shown recently that epimorphisms need not be surjective in a variety K
of Heyting algebras, but only one counter-example was exhibited in the
literature until now. Here, a continuum of such examples is identified, viz.
the variety generated by the Rieger-Nishimura lattice, and all of its (locally
finite) subvarieties that contain the original counter-example K. It is known
that, whenever a variety of Heyting algebras has finite depth, then it has
surjective epimorphisms. In contrast, we show that for every integer n greater
or equal than 2, the variety of all Heyting algebras of width at most n has a
non-surjective epimorphism. Within the so-called Kuznetsov-Gerciu variety
(i.e., the variety generated by finite linear sums of one-generated Heyting
algebras), we describe exactly the subvarieties that have surjective
epimorphisms. This yields new positive examples, and an alternative proof of
epimorphism surjectivity for all varieties of Goedel algebras. The results
settle natural questions about Beth-style definability for a range of
intermediate logics.
