Abstract

We study complementation in bounded posets. It is known and easy to see that every complemented distributive poset is uniquely complemented. The converse statement is not valid, even for lattices. In the present paper we provide conditions that force a uniquely complemented poset to be distributive. For atomistic resp. atomic posets as well as for posets satisfying the descending chain condition we find sufficient conditions in the form of so-called LU-identities. It turns out that for finite posets these conditions are necessary and sufficient.

Highlights

  • It is well-known and a fairly elementary result that every complemented distributive lattice is uniquely complemented and Boolean

  • Dilworth ([6]) disproved the conjecture that every uniquely complemented lattice is distributive by showing that every lattice is, a sublattice of a uniquely complemented lattice

  • A number of conditions forcing a uniquely complemented lattice to be distributive was formulated by several authors

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Summary

Introduction

It is well-known and a fairly elementary result that every complemented distributive lattice is uniquely complemented and Boolean. For finite or even atomic lattices the converse assertion is valid, i.e. every atomic uniquely complemented lattice is distributive. No lattice property which is preserved under the formation of sublattices will be valid in all uniquely complemented lattices. There exists a uniquely complemented lattice containing N5 as a sublattice and it must be non-modular. A number of conditions forcing a uniquely complemented lattice to be distributive was formulated by several authors. A necessary and sufficient condition formulated in the form of an identity in two variables was presented in [4], for several other such identities see e.g. Several other results concerning uniquely complemented posets are obtained in the case of atomicity

Basic Concepts
Two LU-identities
Distributivity of Uniquely Complemented Posets

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