Gcd-monoids arising from homotopy groupoids

TL;DR

This paper investigates the algebraic structure of interval monoids derived from posets, characterizing when they are gcd-monoids and exploring their embeddings into free monoids and groups, with implications for category theory.

Contribution

It provides new characterizations of gcd-monoids from posets, constructs examples with arbitrary groups as free factors, and extends results to universal monoids of categories.

Findings

Upsilon(P) is a gcd-monoid iff principal ideals/filters are join/meet-semilattices.

Every group G can be realized as a free factor of a gcd-monoid from a length-2 poset.

Finite posets yield monoids embeddable into free monoids.

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] $\times$ [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.