On locally order affine complete lattices
by
Kalle Kaarli
University of Tartu

A lattice L is called (locally) order affine complete (briefly oa-complete), if all order and congruence preserving functions on L are (local) polynomial functions of that lattice.

Several results earlier proved by R. Wille for finite lattices are sharpened and transfered to lattices of finite height. We show that a lattice of finite height is locally oa-complete iff all of its tolerances are obtained in a certain way from congruences. We also prove that the class of locally oa-complete lattices of finite height is closed with respect to homomorphic images. These results reduce the characterization of locally oa-complete lattices of finite height to subdirect products of two SI lattices.

The main object is to characterize modular locally oa-affine complete lattices of finite height. We reduce this problem to subdirect products of two finite dimensional projective geometries. Several series of new examples of such lattices have been found and there is a good hope to obtain a complete description of them.

Date received: May 24, 2000