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Assume that we have a (complete) 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.

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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 $v \leq y$.

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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 join-irreducible, $u$ its unique lower cover (w.r.t to $\leq$) and $v \wedge x \leq u$ as well as $v \leq y$(hence, in total, $v \wedge x \leq u \lessdot v \leq y$ where $\lessdot$ denotes the covering relation).

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