Finitely Presented Lattices: Canonical Forms and the Covering Relation

A canonical form for elements of a lattice freely generated by a partial lattice is given. This form agrees with Whitman’s canonical form for free lattices when the partial lattice is an antichain. The connection between this canonical form and the arithmetic of the lattice is given. For example, it is shown that every element of a finitely presented lattice has only finitely many minimal join representations and that every join representation can be refined to one of these. An algorithm is given which decides if a given element of a finitely presented lattice has a cover and finds them if it does. An example is given of a nontrivial, finitely presented lattice with no cover at all. Trans. Amer. Math. Soc. 312 (1989), 841–860 c © 1989 Amer. Math. Soc. This paper studies finitely presented lattices. We introduce a canonical form for the elements of such a lattice which agrees with Whitman’s canonical form for free lattices in the case the finitely presented lattice is free. This canonical form, like Whitman’s, has several nice theoretical properties. In addition it allows one to efficiently calculate in such a lattice. We have programs, written in both Common Lisp and muLisp, for dealing with finitely presented lattices. We also investigate the covering relation in finitely presented lattices. We show that there is an effective proceedure for determining if an element of such a lattice has any lower covers and for finding them if there are any. Alan Day [2] has shown that every finitely generated free lattice is weakly atomic, i.e., every interval contains a covering. It was conceivable that such a theorem could extend to all finitely presented lattices. We show that this is not the case. In fact we show that there is a (nontrivial) finitely presented lattice without any coverings. There is a close connection between finitely presented lattices and lattices freely generated by a finite partial lattice (see below). Because of this, most of our theorems will be phrased in terms of FL(P), the free lattice generated by the partial lattice P. 1991 Mathematics Subject Classification. 06B25, 06B05.