Generation of LLD lattices by Intervals
by
Mark R. Johnson
M.J. Neeley School, Texas Christian University
Coauthors: Richard A. Dean

A finite lattice is an LLD lattice if it has the property that every element in the lattice is uniquely representable as the irredundant join of join irreducibles. Recently these lattices have gained renewed interest, in part, because of the their direct relationship to the Path Independence property frequently assumed in economic models of individual and collective choice behavior. A key property of the relationship between Path Independent choice functions and LLD lattices is the Interval Property identified in Johnson and Dean (1996). The interval property imposes a partition on the Boolean algebra that serves as the domain of the choice function in which each member of the partition can be represented as F(A)/C(A) where F(A) is the top of the interval and C(A) is the bottom of the interval. The mapping A to F(A) is the closure function that characterizes the LLD; the mapping A to C(A) is the path independent choice function that characterizes the LLD. This interval property suggests an alternative view of LLDs. In particular, we present several results. The first of these offers a necessary and sufficient condition on a partition of the Boolean algebra for that partition to be determined by a path independent choice function. Second we address the question of designing LLDs to meet specified requirements such as a restriction on the POS of join irreducibles. In reference to this question, we offer a technique for generating LLDs directly from a set of specified intervals. Finally we offer some open questions about the relationship between properties of the intervals and the properties of the LLD.

Date received: February 13, 2003