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Let $L$ be a lattice and let $\textbf{DM}(\cdot)$ denote the Dedekind-MacNeille completion.

Is there a lattice $L$ that is not a quotient of $\textbf{DM}(L)$? And what if we generalise this question to posets: is every poset $P$ a quotient of $\textbf{DM}(L)$? (I just realise whether the second question makes sens depends on the answer to this postthis post.)

Let $L$ be a lattice and let $\textbf{DM}(\cdot)$ denote the Dedekind-MacNeille completion.

Is there a lattice $L$ that is not a quotient of $\textbf{DM}(L)$? And what if we generalise this question to posets: is every poset $P$ a quotient of $\textbf{DM}(L)$? (I just realise whether the second question makes sens depends on the answer to this post.)

Let $L$ be a lattice and let $\textbf{DM}(\cdot)$ denote the Dedekind-MacNeille completion.

Is there a lattice $L$ that is not a quotient of $\textbf{DM}(L)$? And what if we generalise this question to posets: is every poset $P$ a quotient of $\textbf{DM}(L)$? (I just realise whether the second question makes sens depends on the answer to this post.)

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Dominic van der Zypen
Dominic van der Zypen
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Are lattices quotients of their Dedekind-MacNeille completion?

Let $L$ be a lattice and let $\textbf{DM}(\cdot)$ denote the Dedekind-MacNeille completion.

Is there a lattice $L$ that is not a quotient of $\textbf{DM}(L)$? And what if we generalise this question to posets: is every poset $P$ a quotient of $\textbf{DM}(L)$? (I just realise whether the second question makes sens depends on the answer to this post.)