3. Questions to ask, given a lattice $L$

1. Is $L$ distributive?

This means that $L$ obeys the distributive law

(D)   $x \wedge (y \vee z) = (x \wedge y) \vee (x \wedge z)$

or the dual law, which is equivalent.

Examples. Chains, Pow$(S)$, Div$(n)$, not $M_3$, not $N_5$.

2. Is $L$ modular?

This means that $L$ obeys the modular law

(M)   $x \leq z \Rightarrow x \vee (y \wedge z) = (x \vee y) \wedge z$, or equivalently,

(M$'$)   $(x \vee y) \wedge (x \vee z) = x \vee (y \wedge (z \vee x))$.

Examples. Normal$(G)$, Subsp$(V)$, any distributive lattice, $M_3$, not $N_5$.

3. Does $L$ have a top element (usually denoted $1$ or $I$) and/or a bottom element (usually denoted 0 or $O$)?

Examples. Any finite lattice has both, R has neither, $\omega$ has a bottom element but not a top element.

4. Is $L$ complemented? (This requires top and bottom elements.)

This means that for each $x$ there is at least one $y$ with $x \wedge y = 0$, $x \vee y = 1$.

Examples. Pow$(S)$, measurable subsets of R.

Package: $L$ is a Boolean lattice if $L$ is distributive, has top and bottom elements, and is complemented.

5. Is $L$ complete?

This means that every subset $S$ of $L$ has a least upper bound and greatest lower bound, not just the two-element subsets. We usually call these sup$\;S$ and inf$\;S$, respectively.

Examples. Pow$(S)$, $[a,b]$ in R, any finite lattice.