distributive latticeDistributiveLattice2002-02-24 14:56:122006-03-22 03:58:07Definition\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}\PMlinkescapeword{distributive}
\PMlinkescapeword{examples}
\PMlinkescapeword{lattice}
\PMlinkescapeword{lattices}
A \PMlinkname{lattice}{Lattice} is said to be \emph{distributive} if it satisifes either (and therefore both) of the \PMlinkname{distributive laws}{Distributive}:
\begin{itemize}
\item $x \land (y \lor z) = (x \land y) \lor (x \land z)$
\item $x \lor (y \land z) = (x \lor y) \land (x \lor z)$
\end{itemize}
Every distributive lattice is \PMlinkname{modular}{ModularLattice}.
Examples of distributive lattices include \PMlinkname{Boolean lattices}{BooleanLattice}, totally ordered sets, and the \PMlinkname{subgroup lattices}{LatticeOfSubgroups} of locally cyclic groups.