This question arose from another one of mine, Homotopy type of some lattices with top and bottom removed.

An element $d$ of a bounded lattice $L$ is called $\mathit{dense}$ if $$ \forall x\in L\ (d\land x=\bot)\Rightarrow(x=\bot) $$ holds.

It is well known that a pseudocomplemented distributive lattice is Boolean if and only if $\top$ is the only dense element: in this case dense elements are precisely those of the form $a\lor\neg a$ where $\neg$ is the pseudocomplement.

Is a characterization of general (non-distributive) lattices with the same property known? That is, which bounded lattices have the property that $\top$ is the unique dense element?

Variations: this property + the only codense element is $\bot$; not necessarily bounded lattices such that all intervals have these properties; just the finite case; etc., etc.

Also, what about those distributive lattices which are neither pseudocomplemented nor copseudocomplemented? And what about (co)pseudocomplemented non-distributive ones?

This question arose from another one of mine, Homotopy type of some lattices with top and bottom removed.

An element $d$ of a bounded lattice $L$ is called $\mathit{dense}$ if $$ \forall x\in L\ (d\land x=\bot)\Rightarrow(x=\bot) $$ holds.

It is well known that a pseudocomplemented distributive lattice is Boolean if and only if $\top$ is the only dense element: in this case dense elements are precisely those of the form $a\lor\neg a$ where $\neg$ is the pseudocomplement.

Is a characterization of general (non-distributive) lattices with the same property known? That is, which bounded lattices have the property that $\top$ is the unique dense element?

Variations: this property + the only codense element is $\bot$; not necessarily bounded lattices such that all intervals have these properties; just the finite case; etc., etc.

Also, what about those distributive lattices which are neither pseudocomplemented nor copseudocomplemented? And what about (co)pseudocomplemented non-distributive ones?

This question arose from another one of mine, Homotopy type of some lattices with top and bottom removed.

An element $d$ of a bounded lattice $L$ is called $\mathit{dense}$ if $$ \forall x\in L\ (d\land x=\bot)\Rightarrow(x=\bot) $$ holds.

It is well known that a finite distributive lattice (more generally, a (not necessarily finite) Heyting algebra (still more generally, a pseudocomplemented distributive lattice))) is Boolean if and only if $\top$ is the only dense element: in this case dense elements are precisely those of the form $a\lor\neg a$ where $\neg$ is the Heyting implication.

Is a characterization of general (non-distributive) lattices with the same property known? That is, which bounded lattices have the property that $\top$ is the unique dense element?

Variations: this property + the only codense element is $\bot$; not necessarily bounded lattices such that all intervals have these properties; just the finite case; etc., etc.

Also, what about those distributive lattices which are neither pseudocomplemented nor copseudocomplemented? And what about (co)pseudocomplemented non-distributive ones?

This question arose from another one of mine, Homotopy type of some lattices with top and bottom removed.

An element $d$ of a bounded lattice $L$ is called $\mathit{dense}$ if $$ \forall x\in L\ (d\land x=\bot)\Rightarrow(x=\bot) $$ holds.

It is well known that a pseudocomplemented distributive lattice is Boolean if and only if $\top$ is the only dense element: in this case dense elements are precisely those of the form $a\lor\neg a$ where $\neg$ is the pseudocomplement.

Is a characterization of general (non-distributive) lattices with the same property known? That is, which bounded lattices have the property that $\top$ is the unique dense element?

Variations: this property + the only codense element is $\bot$; not necessarily bounded lattices such that all intervals have these properties; just the finite case; etc., etc.

Also, what about those distributive lattices which are neither pseudocomplemented nor copseudocomplemented? And what about (co)pseudocomplemented non-distributive ones?