Lattices with unique complementation
Michael Adams, Jiří Sichler · Pacific Journal of Mathematics · 1981
Dilworth's theorem that every lattice is a sublattice of a uniquely complemented lattice is shown to hold in 2**o varieties of lattices.1* Introduction• After E. V. Huntington [11], it was conjectured that every uniquely complemented lattice was a distributive lattice; since a uniquely complemented distributive lattice is a Boolean lattice and every Boolean lattice is uniquely complemented, the verification of such a conjecture would have provided a characterization of Boolean lattices.For a uniquely complemented lattice L, G. Birkhoff and J. von Neumann showed that if L is modular or relatively complemented then L is distributive.Subsequently, G. Birkhoff and M. Ward [4] showed that if L is complete, atomic, and dually atomic then L is distributive.Further, R. P. Dilworth [6] verified that if L is finite dimensional then it is distributive.Finally, the conjecture was refuted in, the now famous paper, [7]; R. P. Dilworth proved that every lattice is a sublattice of a uniquely complemented lattice (see also C. C. Chen and G. Gratzer [5]).Since then a number of other sufficient conditions for distributivity of a uniquely complemented lattice have been discovered.For example, if a uniquely complemented lattice L is either atomic (T.Ogasawara and U. Sasaki [151, and J. E. McLaughlin [14]), algebraic (V.N. Saliϊ [17]), or if the function that sends leL to the unique complement of I is order inverting (G.Birkhoff [3]), then L is distributive.The lattices constructed by R. P. Dilworth in [7] contain the free lattice on countably many generators as a sublattice.Hence, in particular, any nontrivial lattice identity fails to hold in any of Dilworth's lattices.(By a nontrivial identity, we mean an identity that does not follow from the lattice axioms.)A growing conjecture has been that any uniquely complemented lattice that satisfies a nontrivial lattice identity is distributive.In this connection (see G. Gratzer [9]), R. Padmanabhan [16] has shown [that a uniquely complemented lattice in the variety M V N 69 or in the variety generated by a finite lattice satisfying one of two implications (namely, (SD A ) or an implication due to E. Fried and G. Gratzer, [9]) is distributive.However, we will show that this is not indicative of the general situation; that is to say, we will show that there are 2Wo varieties of lattices for which Dilworth's theorem holds.Thus, we will prove: