Some varieties of residuated lattices with countably many subvarieties
by
Jeffrey S. Olson
Norwich University

The variety of residuated lattices (in the sense established by C. Tsinakis and his collaborators) contains uncountably many subvarieties. More than this, many of its known subvarieties have uncountably many subvarieties, even in cases where considerable structural conditions have been imposed. For example, the variety of residuated lattices which are idempotent (i.e., which satisfy x ·x ≈ x) and semilinear (i.e., subdirect products of linearly ordered residuated lattices) has uncountably many subvarieties, as was shown by N. Galatos. The proof relies on algebras whose monoid operation is non-commmutative. It is of interest then, to determine if the variety of idempotent semilinear residuated lattices which are commutative (SLIC for short) has uncountably many subvarieties. SLIC has received attention lately, in part because it provides an algebraic semantics for an important extension of a fragment of linear logic.

We present a construction for residuated lattices which have a linearly ordered lattice reduct and a commutative monoid operation (not necessarily idempotent), which offers some transparency with respect to characterizations of subalgebras and homomorphic images. The finitely generated subdirectly irreducible members of the varieties we investigate may be produced by applying this construction to simple members of the variety. SLIC is generated in this fashion. We provide natural conditions on a class of simple algebras which guarantee that the generated variety has a countable number of subvarieties. This allows us to identify many varieties of residuated lattices with a countably infinite number of subvarieties. In particular, SLIC is among them.

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Date received: May 20, 2008