Name of a lattice-property - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T03:14:46Z http://mathoverflow.net/feeds/question/88516 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88516/name-of-a-lattice-property Name of a lattice-property Sebastian 2012-02-15T13:11:52Z 2012-06-26T21:29:23Z <p>Assume that we have a complete lattice $(L,\leq)$.</p> <p>I would like to know whether the following property has a specific name and whether lattices with this property have been studied somewhere:</p> <p>For each $x,y \in L$ with $x &lt; y$, 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 $v \leq y$.</p> <p>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.</p> http://mathoverflow.net/questions/88516/name-of-a-lattice-property/100723#100723 Answer by NN for Name of a lattice-property NN 2012-06-26T21:29:23Z 2012-06-26T21:29:23Z <p>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).</p>