Related papers: Schreier split extensions of preordered monoids
We consider monoids equipped with a compatible quantale valued relation, to which we call quantale enriched monoids, and study semidirect products of such structures. It is well-known that semidirect products of monoids are closely related…
We give an overview of a number of Schreier-type extensions of monoids and discuss the relation between them. We begin by discussing the characterisations of split extensions of groups, extensions of groups with abelian kernel and finally…
In the context of regular unital categories we introduce an intrinsic version of the notion of a Schreier split epimorphism, originally considered for monoids. We show that such split epimorphisms satisfy the same homological properties as…
In category theory circles it is well-known that the Schreier theory of group extensions can be understood in terms of the Grothendieck construction on indexed categories. However, it is seldom discussed how this relates to extensions of…
Weakly Schreier split extensions are a reasonably large, yet well-understood class of monoid extensions, which generalise some aspects of split extensions of groups. This short note provides a way to define and study similar classes of…
We relate the old and new cohomology monoids of an arbitrary monoid $M$ with coefficients in semimodules over $M$, introduced in the author's previous papers, to monoid and group extensions. More precisely, the old and new second cohomology…
We prove that the category of preordered groups contains two full reflective subcategories that give rise to some interesting Galois theories. The first one is the category of the so-called commutative objects, which are precisely the…
We study the categorical properties of right-preordered groups, giving an explicit description of limits and colimits in this category, and studying some exactness properties. We show that, from an algebraic point of view, the category of…
We give explicit axioms for the algebraic theory of the quasivarieties of right-preordered groups and preordered groups. We then look at lattices of effective equivalence relations, which turn out to be similar to the lattices of…
Motivated by the categorical-algebraic analysis of split epimorphisms of monoids, we study the concept of a special object induced by the intrinsic Schreier split epimorphisms in the context of a regular unital category with binary…
Cosetal extensions of monoids generalise extensions of groups, special Schreier extensions of monoids and Leech's normal extensions of groups by monoids. They share a number of properties with group extensions, including a notion of Baer…
This is the first part of a series of three strongly related papers in which three equivalent structures are studied: - internal categories in categories of monoids; defined in terms of pullbacks relative to a chosen class of spans -…
In this paper we study equivariant crossed modules in its link with strict graded categorical groups. The resulting Schreier theory for equivariant group extensions of the type of an equivariant crossed module generalizes both the theory of…
Split preorders are preordering relations on a domain whose composition is defined in a particular way by splitting the domain into two disjoint subsets. These relations and the associated composition arise in categorial proof theory in…
Given an ordered structure, we study a natural way to extend the order to preorders on type spaces. For definably complete, linearly ordered structures, we give a characterisation of the preorder on the space of 1-types. We apply these…
Prolongations of a group extension can be studied in a more general situation that we call group extensions of the co-type of a crossed module. Cohomology classification of such extensions is obtained by applying the obstruction theory of…
The structure of monoidal categories in which every arrow is invertible is analyzed in this paper, where we develop a 3-dimensional Schreier-Grothendieck theory of non-abelian factor sets for their classification. In particular, we state…
This is the second part of a series of three strongly related papers in which three equivalent structures are studied: - internal categories in categories of monoids; defined in terms of pullbacks relative to a chosen class of spans -…
We investigate the behaviour of split extensions in the category OrdGrp of (pre)\-ordered groups. Namely we show that the lexicographic order plays a key role on the existence of compatible orders for semidirect products, establishing…
It is shown that the category of semi-biproducts in monoids is equivalent to a category of pseudo-actions. A semi-biproduct in monoids is at the same time a generalization of a semi-direct product in groups and a biproduct in commutative…