In this note we present the
results announced in the Notices of the American Mathematical Society, January,
1969. Algebraic lattices are interesting and important algebraic structures. They
occur in many branches of algebra, e.g. the lattice of all subalgebras of a universal
algebra, the lattice of all filters of a groupoid, and the lattice of all ideals
of a ring are all algebraic lattices. Moreover there is a natural connection
between algebraic lattices and groupoids, since every algebraic lattice is
isomorphic to the lattice of all filters of some groupoid, and in particular of the
groupoid of all compact elements of the lattice. If an algebraic lattice is
distributive, it is relatively pseudo-complemented and is a complete Brouwerian
lattice in the sense of Garrett Birkhoff [1]. Hence it is natural to look for
simple conditions on a groupoid that will insure that the lattice of its filters is
distributive.
We show that the lattice of all filters is distributive if it is a sublattice of the
lattice of all subgroupoids, but this condition is not always necessary for
distributivity. If the groupoid is a semilattice, this condition is both necessary and
sufficient. We then derive some conditions that are both necessary and sufficient for
distributivity for groupoids. One of these is a modification of a condition given by
Grätzer and Schmidt for semilattices.
