Generators of lattice varieties
R. P. Dilworth, Ralph S. Freese · Birkhäuser Boston eBooks · 1990
Although it is well known that the variety of all lattices is generated by the subclass of finite lattices, there are lattice varieties which are not generated by their finite members. In fact, there are modular varieties which are not even generated by their finite dimensional members [3]. At the present time it is not known if the variety of all modular lattices is generated by its finite or even its finite dimensional members. This raises the question: Do there exist generators for lattice varieties which satisfy some kind of finiteness conditions? In this note we will be concerned with generators satisfying atomicity conditions. Since any lattice variety is generated by its subdirectly irreducible members, we shall be particularly interested in generators which are also subdirectly irreducible. The main results are the following. The notation and terminology for this paper is taken from [1].