Multifraction reduction III: The case of interval monoids

We investigate gcd-monoids, which are cancellative monoids in which any two elements admit a left and a right gcd, and the associated reduction of multifractions (arXiv:1606.08991 and 1606.08995), a general approach to the word problem for the enveloping group. Here we consider the particular case of interval monoids associated with finite posets. In this way, we construct gcd-monoids, in which reduction of multifractions has prescribed properties not yet known to be compatible: semi-convergence of reduction without convergence, semi-convergence up to some level but not beyond, non-embeddability into the enveloping group (a strong negation of semi-convergence).Comment: 23 pages ; v2 : cross-references updated ; v3 : one example added, typos corrected; final version due to appear in Journal of Combinatorial Algebr

Gcd-monoids arising from homotopy groupoids

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.Comment: 27 pages (v4). Semigroup Forum, to appea

Garside and locally Garside categories

We define and give axioms for Garside and locally Garside categories. We give an application to Coxeter and Artin groups and Deligne-Lusztig varieties.Comment: We have fixed some errors pointed to us by E. Godelle and P. Dehornoy and added new results in section

The tame-wild principle for discriminant relations for number fields

Consider tuples of separable algebras over a common local or global number field, related to each other by specified resolvent constructions. Under the assumption that all ramification is tame, simple group-theoretic calculations give best possible divisibility relations among the discriminants. We show that for many resolvent constructions, these divisibility relations continue to hold even in the presence of wild ramification.Comment: 31 pages, 11 figures. Version 2 fixes a normalization error: |G| is corrected to n in Section 7.5. Version 3 fixes an off-by-one error in Section 6.