  It is proved that epimorphisms are surjective in a range of varieties of
residuated structures, including all varieties of Heyting or Brouwerian
algebras of finite depth, and all varieties consisting of Goedel algebras,
relative Stone algebras, Sugihara monoids or positive Sugihara monoids. This
establishes the infinite deductive Beth definability property for a
corresponding range of substructural logics. On the other hand, it is shown
that epimorphisms need not be surjective in a locally finite variety of Heyting
or Brouwerian algebras of width 2. It follows that the infinite Beth property
is strictly stronger than the so-called finite Beth property, confirming a
conjecture of Blok and Hoogland.

Logic