LAUSR

The interval monoid $$\Upsilon ({P})$$Υ(P) of a poset P is defined by generators [x, y], where $$x\le y$$x≤y in P, and relations $$[x,x]=1$$[x,x]=1, $$[x,z]=[x,y]\cdot [y,z]$$[x,z]=[x,y]·[y,z] for $$x\le y\le z$$x≤y≤z. It embeds into its universal group $$\Upsilon ^{\pm }({P})$$Υ±(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 $$\Upsilon ({P})$$Υ(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 $$\Upsilon ({P})$$Υ(P) is a gcd-monoid and G is a free factor of $$\Upsilon ^{\pm }({P})$$Υ±(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 $$\Upsilon ({P})$$Υ(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 $${\mathrm{U}_\mathrm{mon}}({S})$$Umon(S) of any category S. This enables us, in particular, to characterize the embeddability of $${\mathrm{U}_\mathrm{mon}}({S})$$Umon(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.