The Characterization of Partition Lattices
Usa Sasaki, Shigeru Fujiwara · Hiroshima Mathematical Journal · 1951
The problem of giving an .axiomatic characterization of a partition lattice-a lattice of all partiti~ns of a set-can be formulate'.! in different ways.0. Ore [1]1) has first characterized a partition lattice as ~ goemetric system with points and lines with rather •peculiar ptoperties 2 l which are not familiar to us.In this paper a partition lattice is characterized in § 1 as a geometric system with points, lines, and moreover, with planes and parallel lines which satisfy the following properties : G I. Any line contains at most three points.G II.Any two points on a line defines the same line.G III. Any p!ane contains only three, four or six points.G IV.If a line has• a parallel line, then it has iust two ones.G V. If a line contains ,only two points, then it .hasalways a parallel line.By applying the results obtained in § 1, we shall show in § 2 that any partition -lattice is a matroid, planer lattice 3 l with a few properties of its points.In § 3 we shall prove that such a lattice is isomorphic to the lattice of all partitions of a set if and only if it is irreducible (Theorem 3-2).§ 1. Geometrical Characterization of Partition Lattices.DEFINITION 1 • 1.A • partion P of a set S is a decomposition of S into subsets C ,r,( a E I) such that every point in S belongs to one and only one set Ca,(a EI).We shall call the sets Ca,(a EI) the blocks of partition P. Let P 1 <P 2 mean that P 1 is a subpartition of P 2 , that is, the blocks in P 1 are obtained by subdivisions of the blocks of P 2 • Then the system of all partitions of a set S forms a lattice and it is called a partition lattice.DEFINITION 1 • 2. Let G be a set of points.With every pair of different1) The numbers in square brackets refer to the list of references at the paper.2) Cf.Definition 1-2 below.3) Cf.Definition 2.3 and 2-4 below.