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3 backquotes, lattice-theory

A subset $A$ of a complete lattice $L$ is said to be join-dense if $L={\bigvee L=\{\bigvee R|R\subseteq A}$A\}$. An element$a\in L$is said to be join-irreducible if$a\neq 0$and if$a=x\vee y$then$a=x$or$a=y$. Is there a nice necessary and sufficient condition for when the join-irreducible elements of a complete lattice are join-dense, or is there a representation theorem for such lattices? Of course, if$L$is a complete lattice and$A$is the collection of all join-irreducibles, then we may ''shrink'' the lattice$L$to the lattice ${\bigvee \{\bigvee R|R\subseteq A}$A\}$ so that the join-irreducibles are join-dense. I know that the distributive complete lattices where the join-irreducibles are join-dense are precisely the spatial coframes(the spatial coframes are the lattices isomorphic to the closed sets in some topological space). Furthermore, if $L$ is a complete lattice satisfying DCC, then the join-irreducible elements in $L$ are join-dense in $L$. However, none of these ideas characterizes the complete lattices where the join-irreducibles are join-dense.

2 added 95 characters in body

A subset $A$ of a complete lattice $L$ is said to be join-dense if $L={\bigvee R|R\subseteq A}$. An element $a\in L$ is said to be join-irreducible if $a\neq 0$ and if $a=x\vee y$ then $a=x$ or $a=y$. Is there a nice necessary and sufficient condition for when the join-irreducible elements of a complete lattice are join-dense, or is there a representation theorem for such lattices?

Of course, if $L$ is a complete lattice and $A$ is the collection of all join-irreducibles, then we may ''shrink'' the lattice $L$ to the lattice ${\bigvee R|R\subseteq A}$ so that the join-irreducibles are join-dense. I know that the distributive complete lattices where the join-irreducibles are join-dense are precisely the spatial coframes(the spatial coframes are the lattices isomorphic to the closed sets in some topological space). Furthermore, if $L$ is a complete lattice satisfying DCC, then the join-irreducible elements in $L$ are join-dense in $L$. However, none of these ideas characterizes the complete lattices where the join-irreducibles are join-dense.

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# When are the join-irreducibles in a complete lattice join-dense?

A subset $A$ of a complete lattice $L$ is said to be join-dense if $L={\bigvee R|R\subseteq A}$. An element $a\in L$ is said to be join-irreducible if $a\neq 0$ and if $a=x\vee y$ then $a=x$ or $a=y$. Is there a nice necessary and sufficient condition for when the join-irreducible elements of a complete lattice are join-dense, or is there a representation theorem for such lattices?

Of course, if $L$ is a complete lattice and $A$ is the collection of all join-irreducibles, then we may ''shrink'' the lattice $L$ to the lattice ${\bigvee R|R\subseteq A}$ so that the join-irreducibles are join-dense. I know that the distributive complete lattices where the join-irreducibles are join-dense are precisely the spatial coframes. Furthermore, if $L$ is a complete lattice satisfying DCC, then the join-irreducible elements in $L$ are join-dense in $L$. However, none of these ideas characterizes the complete lattices where the join-irreducibles are join-dense.