atom
Atom
2005-06-14 00:17:55
2006-08-23 16:14:57
Definition
atomic poset
atomic lattice
atomistic lattice
atomistic
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Let $P$ be a poset, partially ordered by $\leq$. An element $a\in P$ is called an \emph{atom} if it covers some minimal element of $P$. As a result, an atom is never minimal. A poset $P$ is called \emph{atomic} if for every element $p\in P$ that is not minimal has an atom $a$ such that $a\leq p$.
\textbf{Examples}.
\begin{enumerate}
\item Let $A$ be a set and $P=2^A$ its power set. $P$ is a poset ordered by $\subseteq$ with a unique minimal element $\varnothing$. Thus, all singleton subsets $\lbrace a \rbrace$ of $A$ are atoms in $P$.
\item $\mathbb{Z}^+$ is partially ordered if we define $a\leq b$ to mean that $a\mid b$. Then $1$ is a minimal element and any prime number $p$ is an atom.
\end{enumerate}
\textbf{Remark.} Given a lattice $L$ with underlying poset $P$, an element $a\in L$ is called an \emph{atom} (of $L$) if it is an atom in $P$. A lattice is a called an \emph{atomic lattice} if its underlying poset is atomic. An \emph{atomistic lattice} is an atomic lattice such that each element that is not minimal is a join of atoms. If $a$ is an atom in a semimodular lattice $L$, and if $a$ is not under $x$, then $a\vee x$ is an atom in any interval lattice $I$ where $x=\bigwedge I$.
\textbf{Examples}.
\begin{enumerate}
\item $P=2^A$, with the usual intersection and union as the lattice operations meet and join, is atomistic: every subset $B$ of $A$ is the union of all the singleton subsets of $B$.
\item $\mathbb{Z}^+$, partially ordered as above, with lattice binary operations defined by $a\wedge b=\operatorname{gcd}(a,b)$, and $a\vee b= \operatorname{lcm}(a,b)$, is a lattice that is atomic, as we have seen earlier. But it is not atomistic: $4$ is not a join of $2$'s; $36$ is not a join of $2$ and $3$ are just two counterexamples.
\end{enumerate}