2,440 research outputs found
Abstract elementary classes and accessible categories
We compare abstract elementary classes of Shelah with accessible categories
having directed colimits
Counting homomorphisms onto finite solvable groups
We present a method for computing the number of epimorphisms from a
finitely-presented group G to a finite solvable group \Gamma, which generalizes
a formula of G\"aschutz. Key to this approach are the degree 1 and 2 cohomology
groups of G, with certain twisted coefficients. As an application, we count
low-index subgroups of G. We also investigate the finite solvable quotients of
the Baumslag-Solitar groups, the Baumslag parafree groups, and the Artin braid
groups.Comment: 30 pages; accepted for publication in the Journal of Algebr
High-level signatures and initial semantics
We present a device for specifying and reasoning about syntax for datatypes,
programming languages, and logic calculi. More precisely, we study a notion of
signature for specifying syntactic constructions.
In the spirit of Initial Semantics, we define the syntax generated by a
signature to be the initial object---if it exists---in a suitable category of
models. In our framework, the existence of an associated syntax to a signature
is not automatically guaranteed. We identify, via the notion of presentation of
a signature, a large class of signatures that do generate a syntax.
Our (presentable) signatures subsume classical algebraic signatures (i.e.,
signatures for languages with variable binding, such as the pure lambda
calculus) and extend them to include several other significant examples of
syntactic constructions.
One key feature of our notions of signature, syntax, and presentation is that
they are highly compositional, in the sense that complex examples can be
obtained by assembling simpler ones. Moreover, through the Initial Semantics
approach, our framework provides, beyond the desired algebra of terms, a
well-behaved substitution and the induction and recursion principles associated
to the syntax.
This paper builds upon ideas from a previous attempt by Hirschowitz-Maggesi,
which, in turn, was directly inspired by some earlier work of
Ghani-Uustalu-Hamana and Matthes-Uustalu.
The main results presented in the paper are computer-checked within the
UniMath system.Comment: v2: extended version of the article as published in CSL 2018
(http://dx.doi.org/10.4230/LIPIcs.CSL.2018.4); list of changes given in
Section 1.5 of the paper; v3: small corrections throughout the paper, no
major change
The strong profinite genus of a finitely presented group can be infinite
We construct the first example of a finitely-presented, residually-finite
group that contains an infinite sequence of non-isomorphic finitely-presented
subgroups such that each of the inclusion maps induces an isomorphism of
profinite completions.Comment: 10 pages, no figures. Final version to appear in Journal of the
European Math. So
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