  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] $\times$
[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:
$\bullet$ 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).
$\bullet$ For every group G, there is a poset P of length 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 finite iff G is finitely presented.
$\bullet$ For every finite poset P , the monoid $\Upsilon$(P) can be embedded
into a free monoid. $\bullet$ Some of the results above, and many related ones,
can be extended from interval monoids to the universal monoid Umon(S) of any
category S. This enables us, in particular, to characterize the embeddability
of 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.
