Gcd-monoids arising from homotopy groupoids

Abstract

The interval monoid $\Upsilon ({P})$ of a poset P is defined by generators [x, y], where $x\le y$ in P, and relations $[x,x]=1$, $[x,z]=[x,y]\cdot [y,z]$ for $x\le y\le z$. It embeds into its universal group $\Upsilon ^{\pm }({P})$, the interval group of P, which is also the universal group of the homotopy groupoid of the chain complex of P. We prove the following results:

• The monoid $\Upsilon ({P})$ has finite left and right greatest common divisors of pairs (we say that it is a gcd-monoid) iff every principal ideal (resp., filter) of P is a join-semilattice (resp., a meet-semilattice).

• For every group G, there is a connected poset P of height 2 such that $\Upsilon ({P})$ is a gcd-monoid and G is a free factor of $\Upsilon ^{\pm }({P})$ by a free group. Moreover, P can be taken to be finite iff G is finitely presented.

• For every finite poset P, the monoid $\Upsilon ({P})$ can be embedded into a free monoid.

• Some of the results above, and many related ones, can be extended from interval monoids to the universal monoid ${\mathrm{U}_\mathrm{mon}}({S})$ of any category S. This enables us, in particular, to characterize the embeddability of ${\mathrm{U}_\mathrm{mon}}({S})$ into a group, by stating that it holds at the hom-set level. We thus obtain new easily verified sufficient conditions for embeddability of a monoid into a group.

We illustrate our results by various examples and counterexamples.