Reflections on lower bounded lattices
by
Kira Adaricheva
Harold Washington College, Chicago, US
Coauthors: J.B. Nation, University of Honolulu, US

The study of bounded lattices and their generalizations began in the early 1970s with the work of Ralph McKenzie and Bjarni Jónsson on projective lattices and lattice varieties. In the ensuing years, finite bounded lattices, and more generally finitely generated lower bounded lattices, came to be well understood. These lattices inherit many structural properties of finitely generated free lattices. Most of the relevant facts concerning finitely generated lower bounded lattices are gathered in the monograph by Freese, Jezek and Nation . On the other hand, much less is known about arbitrary lower bounded lattices, and no characterization of such lattices is found yet.

In our paper we describe the special classes of lower bounded lattices, and our characterization extends the Jónsson's description of finitely generated lower bounded lattices. All these classes are locally finite varieties, while the class of all lower bounded lattices is not even closed under homomorphic images. At the same time we build an example of a lower bounded lattice that is not locally finite, and that does not satisfiy the extended Jónsson's property.

Date received: April 24, 2004