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The category variant of this question is posted separately here.

For any poset $X$, write $E(X)$ for the poset of endomorphisms of $X$ (with $f\le g$ for $f,g\in E(X)$ if and only if $f(x)\le g(x)$ for all $x\in X$), and consider the following properties a poset $X$ may or may not have:

(P1) $X$ is a singleton,

(P2) $E(X)$ is isomorphic to $X$,

(P3) there is an injective morphism $E(X)\to X$,

(P4) there is a surjective morphism $X\to E(X)$.

Clearly (P1) implies (P2), and (P2) implies (P3) and (P4): $$ \begin{matrix} &&1\\ &&\downarrow\\ &&2\\ &\swarrow&&\searrow\\ 3&&&&4. \end{matrix} $$

Denote by (Qij) the question "Does (Pi) imply (Pj)?".

student9909 asked Question (Q21) here. (student9909 accepted an answer which doesn't answer the question. I find this very confusing. As far as I know, the question is still open.) Let us ask also:

Question (Q31) Does (P3) imply (P1)?

Question (Q41) Does (P4) imply (P1)?

Question (Q32) Does (P3) imply (P2)?

Question (Q42) Does (P4) imply (P2)?

Question (Q34) Does (P3) imply (P4)?

Question (Q43) Does (P4) imply (P3)?

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  • $\begingroup$ @bof - It seems to me that Theorem 3 in ams.org/journals/proc/1971-027-02/S0002-9939-1971-0268091-0/… (one of the links you gave) answers all the questions in this post, as well as the MO question. What am I missing? $\endgroup$ Jul 24 '19 at 13:50
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    $\begingroup$ No reason to think you are missing anything. Although I cited that paper, that doesn't mean I read it closely. Probably I only read bits and pieces that I needed, and by now have completey forgotten the arguments, which explains why I was having a hard time trying to aswer your question. I'm glad that I was able (somehow) to provide a useful reference (without understanding it myself), and I'm glad you can now answer all your questions! $\endgroup$
    – bof
    Jul 24 '19 at 14:10
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    $\begingroup$ Ah, I see. The answer to your questions was not in the Gleason–Dilworth paper but in the subsequent Davies–Hayes–Rousseau paper, which I never read. The only reason I linked to it in my answer was because that's what I found while searching unsuccessfully for a freely available copy of Gleason–Dilworth. Is that what they call serendipity? :-) Thanks for the kind references. $\endgroup$
    – bof
    Jul 24 '19 at 14:59
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In fact the four properties are equivalent. This follows immediately from Theorem 3 in

Roy O. Davies, Allan Hayes and George Rousseau, Complete lattices and the generalized Cantor theorem, Proc. Amer. Math. Soc. 27 (1971), 253–258.

This reference was pointed out to me by user bof in a (now deleted) comment. More precisely bof gave a link to this answer of them.

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