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 Gödel 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.

The paper aims at studying, in full generality, logics defined by imposing a variable inclusion condition on a given logic . We prove that the description of the algebraic counterpart of the left variable inclusion companion of a given logic is related to the construction of Płonka sums of the matrix models of . This observation allows to obtain a Hilbert-style axiomatization of the logics of left variable inclusion, to describe the structure of their reduced models, and to locate them in the Leibniz hierarchy.

A notion of interpretation between arbitrary logics is introduced, and the poset of all logics ordered under interpretability is studied. It is shown that in infima of arbitrarily large sets exist, but binary suprema in general do not. On the other hand, the existence of suprema of sets of equivalential logics is established. The relations between and the lattice of interpretability types of varieties are investigated.

It is proved that every finitely subdirectly irreducible De Morgan monoid A (with neutral element e) is either (i) a Sugihara chain in which e covers ¬e or (ii) the union of an interval subalgebra [¬a, a] and two chains of idempotents, (¬a] and [a), where a = (¬e) 2 . In the latter case, the variety generated by [¬a, a] has no nontrivial idempotent member, and A/[¬a) is a Sugihara chain in which ¬e = e. It is also proved that there are just four minimal varieties of De Morgan monoids. This theorem is then used to simplify the proof of a description (due to K.Świrydowicz) of the lower part of the subvariety lattice of relevant algebras. The results throw light on the models and the axiomatic extensions of fundamental relevance logics.

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