Abstract
A canonical join representation is a certain minimal "factorization" of an element in a finite lattice $L$ analogous to the prime factorization of an integer from number theory. The expression $\bigvee A =w$ is the canonical join representation of $w$ if $A$ is the unique lowest subset of $L$ satisfying $\bigvee A=w$ (where "lowest" is made precise by comparing order ideals under containment). Canonical join representations appear in many familiar guises, with connections to comparability graphs and noncrossing partitions. When each element in $L$ has a canonical join representation, we define the canonical join complex to be the abstract simplicial complex of subsets $A$ such that $\bigvee A$ is a canonical join representation. We characterize the class of finite lattices whose canonical join complex is flag, and show how the canonical join complex is related to the topology of $L$.
Publisher
The Electronic Journal of Combinatorics
Subject
Computational Theory and Mathematics,Geometry and Topology,Theoretical Computer Science,Applied Mathematics,Discrete Mathematics and Combinatorics
Cited by
11 articles.
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