260 research outputs found
On some categorical-algebraic conditions in S-protomodular categories
In the context of protomodular categories, several additional conditions have
been considered in order to obtain a closer group-like behavior. Among them are
locally algebraic cartesian closedness and algebraic coherence. The recent
notion of S-protomodular category, whose main examples are the category of
monoids and, more generally, categories of monoids with operations and
Jo\'{o}nsson-Tarski varieties, raises a similar question: how to get a
description of S-protomodular categories with a strong monoid-like behavior. In
this paper we consider relative versions of the conditions mentioned above, in
order to exhibit the parallelism with the "absolute" protomodular context and
to obtain a hierarchy among S-protomodular categories
On the "three subobjects lemma" and its higher-order generalisations
We solve a problem mentioned in an article of Berger and Bourn: we prove that
in the context of an algebraically coherent semi-abelian category, two natural
definitions of the lower central series coincide. In a first, "standard"
approach, nilpotency is defined as in group theory via nested binary
commutators of the form . In a second approach, higher Higgins
commutators of the form are used to define nilpotent objects. The two
are known to be different in general; for instance, in the context of loops,
the definition of Bruck is of the former kind, while the commutator-associator
filtration of Mostovoy and his co-authors is of the latter type. Another
example, in the context of Moufang loops, is given in Berger and Bourn's paper.
In this article, we show that the two streams of development agree in any
algebraically coherent semi-abelian category. Such are, for instance, all
Orzech categories of interest. Our proof of this result is based on a
higher-order version of the Three Subobjects Lemma of Cigoli-Gray-Van der
Linden, which extends the classical Three Subgroups Lemma from group theory to
categorical algebra. It says that any -fold Higgins commutator of normal subobjects of an object may be decomposed into
a join of nested binary commutators.Comment: 20 pages; revised version, with some simplified proof
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Monoid Extensions, Relaxed Actions and Cohomology
In this thesis a particular class of monoid extensions are studied and characterized, the weakly Schreier split extensions. It is demonstrated that both Artin glueings of frames and λ-semidirect products of inverse semigroups are examples of these extensions. The characterization is given in terms of a generalization of an action. This makes the theory amenable to cohomological ideas. Specifically, a new class of extensions called cosetal extensions are introduced and characterized. When parameterized by this new notion of action, a Baer sim may be defined in these extensions giving rise to an analogue of the second cohomology group. Finally, a connection is made to the setting of toposes, exploiting the link between toposes and frames
On the representability of actions for topological algebras
The actions of a group B on a group X correspond bijectively to the group homomorphisms B ⟶ Aut(X), proving that the functor “actions on X” is representable by the group of automorphisms of X. Making the detour through pseudotopological spaces, we generalize this result to the topological case, for quasi-locally compact groups and some other algebraic structures. We investigate next the case of arbitrary topological algebras for a semi-abelian theory and prove that the representability of topological actions reduces to the preservation of coproducts by the functor Act(−,X)
Categorical semi-direct products in varieties of groups with multiple operators
The notion of a categorical semidirect product was introduced by Bourn and Janelidze as a generalization of the classical semidirect product in the category of groups. The main aim of this work is to study the general properties of semidirect products of groups with operators, describe them in various classical varieties of such algebraic structures and apply the results to homological algebra and related areas of modern algebra. The context in which the study is done is a semiabelian category (that is, a pointed, Barr-exact and Bourn-protomodular category). The main result in the thesis is the construction of the semidirect product in a variety -RLoop of right -loops as the product of underlying sets equipped with the -algebra structure. A variety of right -loops is a variety that is pointed, has a binary + (not necessarily associative or commutative) and a binary satisfying the identities 0 + x = x, x + 0 = x, (x + y) y = x and (x - y) + y = x. Thus, -RLoop is a generalization of the variety of -groups introduced by Higgins and the results obtained are valid for varieties of -loops. We also describe precrossed and crossed modules in the variety -RLoop. The theory of crossed modules developed is independent of that developed by Janelidze for crossed modules in an arbitrary semiabelian category and gives simplified explicit formulae for crossed modules in -RLoop. Finally, we mention that our constructions agree with the known ones in the familiar algebraic categories, specifically the categories of groups, rings and Lie algebras
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