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I've suddenly found myself working with some free monoids $F(S)$ in which the set $S$ is a poset, and the order extends to an order $F(S)$, satisfying if (but not only if) $s_1, s_2, \ldots, s_r, t_1, t_2, \ldots, t_r\in S$ and $s_i \leq t_i$ for all $i$, then $s_1s_2\cdots s_r \leq t_1t_2\ldots t_r$ in $F(S)$.

Roughly speaking, in my context, the partial order reflects the dependency relationships between the cones in a CW complex. It is entirely possible for the same partial order on the generators to be extended in different ways, depending on the homotopy properties of the space.

It seems like there could easily be a well-developed theory of such objects, and I'd appreciate any pointers to the literature. (A google search turned up nothing.)

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    $\begingroup$ What do you mean by "multiplication respects the partial order"? the order is a priori on $S$, not on $F(S)$. $\endgroup$
    – YCor
    Commented Apr 21, 2021 at 14:11
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    $\begingroup$ Do you mean a quotient of the free monoid? $\endgroup$
    – Sam Hopkins
    Commented Apr 21, 2021 at 14:12
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    $\begingroup$ Are you looking for a left adjoint to the forgetful functor from pomonoids to posets? $\endgroup$
    – Benjamin Steinberg
    Commented Apr 21, 2021 at 14:31
  • $\begingroup$ @YCor Clarified! $\endgroup$
    – Jeff Strom
    Commented Apr 21, 2021 at 14:51
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    $\begingroup$ @SamHopkins I said if and did not mean only if -- clarified now. $\endgroup$
    – Jeff Strom
    Commented Apr 21, 2021 at 15:19

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