Hello I am doing a bit of reading on lattices right now and I would appreciate some help. A species of algebra with meet and join operations and binary relation which is quasi ordering can be interpreted in different ways. One such interpretation is closure interpretation:
- $\leq$ is relation of inclusion.
- $a$ and $b$ are closed sets.
- $\land$ is set intersection
- Suppose there is a closure operation which assigns to every set $x$ and element $y$ which is its closure, such that $x \leq y$.
- Set $a$ is closed when $y \leq a$, that is $y$ is a least closed set which contains $a$.
- $\lor$ is operation which assigns to a pair of closed sets $a, b$ a least closed set which contains them both.
Now to my questions. If I choose to interpret $a$ and $b$ as vector spaces, then:
- $a \leq b$ means $b$ is a subspace of $a$?
- $a \land b$ means vectors common to both spaces i.e. if $U$ is a set of all vectors of a form $(x, y, 0)$ and $V$ of a form $(0, y, z)$ then $U \land V$ is all vectors $(0, y, 0)$?
- $a \lor b$ means given $U$ of a form $(x, y)$ and $V$ of a form $(x, 0)$, then $U \lor V$ is $R^2$?
- Postulate of a semilattice $a \land b \leq a$ could be interpreted as set of vectors defined in 2. include $a$?
Does this make any sense? Thanks.