2
$\begingroup$

Assume that we have a complete lattice $(L,\leq)$.

I would like to know whether the following property has a specific name and whether lattices with this property have been studied somewhere:

For each $x,y \in L$ with $x < y$, there exist $u, v \in L$ such that $v$ is completely join-irreducible, $u$ its unique lower cover (w.r.t to $\leq$) and $v \wedge x \leq u$ as well as $v \leq y$.

It might be helpful to point out that this is equivalent to the condition that each element of $L$ is the join of completely join-irreducible elements. My feeling is that there should be a name for such a property.

Improve this question
$\endgroup$

1 Answer 1

Reset to default
3
$\begingroup$

Faigle and Herrmann call them point-lattices. They are useful in the modular, algebraic case (Faigle embedding theorem) and also more generally in the (strongly) semi-modular algebraic case (generalized matroid lattices).

Improve this answer
$\endgroup$
3
  • $\begingroup$ Thank you very much. If that is not too much trouble, could you give me a reference? $\endgroup$
    – Niemi
    Commented Jun 27, 2012 at 8:51
  • 1
    $\begingroup$ springerlink.com/content/h38122511085g364 $\endgroup$
    – user24527
    Commented Jun 27, 2012 at 14:13
  • $\begingroup$ Great, thank you very much for this answer and the reference. $\endgroup$
    – Niemi
    Commented Jun 27, 2012 at 14:24

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .