Kalle Kaarli, Vladimir Kuchmei, Institute of Pure Mathematics, University of Tartu, 50090 Tartu, Estonia, e-mails kaarli@math.ut.ee, kucmei@math.ut.ee
Abstract: This paper grew out from attempts to determine which modular lattices of finite height are locally order affine complete. A surprising discovery was that one can go quite far without assuming the modularity itself. The only thing which matters is that the congruence lattice is finite Boolean. The local order affine completeness problem of such lattices $ L$ easily reduces to the case when $ L$ is a subdirect product of two simple lattices $ L_1$ and $ L_2$. Our main result claims that such a lattice is locally order affine complete if\/f $ L_1$ and $ L_2$ are tolerance trivial and one of the following three cases occurs: \item{1)} $ L= L_1\times L_2$, \item{2)} $ L$ is a maximal sublattice of the direct product, \item{3)} $ L$ is the intersection of two maximal sublattices, one containing $\langle0,1\rangle$ and the other $\langle1,0\rangle$.
Keywords: order affine completeness, congruences of lattices, tolerances of lattices
Classification (MSC 2000): 06B10, 08A40
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