For any cssc-crossed module a category is constructed, equipped with a structure and proved that this is a coherent categorical group. Together with a result of the previous paper, where to any categorical group the cssc-crossed module is associated, this construction will enable us to prove an equivalence between the categories of categorical groups and of cssc-crossed modules in the sequel to this paper.

We introduce a notion of relative commutator - an important special case being commutators twisted by an action - as a straightforward modification of the definition of the Higgins commutator, establish its relation with a new notion of commutativity - also obtained as a modification of the usual notion - and show how we can use it to characterise internal crossed modules in the context of a semi-abelian category.

Effective codescent morphisms of $n$-quasigroups and of $n$-loops are characterized. To this end, it is proved that, for any $n\geq 1$, every codescent morphism of $n$-quasigroups (resp. $n$-loops) is effective. This statement generalizes our earlier result on quasigroups (resp. loops). Moreover, it is shown that the variety of $n$-quasigroups (resp. $n$-loops) satisfies the strong amalgamation property, and the elements of the amalgamated free products of $n$-quasigroups (resp. $n$-loops) have unique normal forms. The latter two statements generalize the corresponding old results on quasigroups (resp. loops) by T. Evans.

In this paper, we revisit the 1979 work of Isbell on subfactors of groups and their projections, which he uses to establish a stronger formulation of the butterfly lemma and its consequence, the refinement theorem for subnormal series of subgroups. We point out an error in the second part of Isbell's refinement theorem, but show that the rest of his results can be extended to the general self-dual context of a noetherian form, which includes in its scope all semi-abelian categories as well as all Grandis exact categories. Furthermore, we show that Isbell's formulations of the butterfly lemma and the refinement theorem amount to canonicity of isomorphisms established in these results.

We study the extensibility problem for a pair of derivations associated with an abelian extension of Leibniz $n$-algebras, and derive an exact sequence of the Wells type.

We extend the notion of semidirect product from semi-abelian to ideally exact context, and illustrate it by considering the case of unital commutative rings.

The purpose of this paper is to give, in the context of enriched category theory, some equivalent conditions characterising simultaneously reflective and coreflective full subcategories of the category of functors on a given small category. In particular, it is proved that such subcategories are (equivalent to) functor categories.

Compactness and paracompactness have been explored within the framework of partial metric spaces. It has been demonstrated that while a partial metric space may not be paracompact, induced partial metric spaces are indeed paracompact. Additionally, it has been shown that the compactness of a partial metric space can be analyzed through the pairwise compactness of the bitopological space induced by that partial metric.

Let $R$ be a Dedekind ring, ${C_0}(R[\pi ,\rho ])$ the projective class group of the crossed product $R[\pi,\rho]$, $M$ some class of subgroups of a finite group $\pi$, $C_0^M(R[\pi ,\rho ])$ the sum of the images of the maps ${C_0}(R[\pi',\rho])\to{C_0}(R[\pi,\rho])$ induced by $i:\pi'\subset \pi$, $\pi ' \in M$. The main result of the paper is the study of some properties of the exponents of $C_0^M(R[\pi ,\rho ])$ in $C(R[\pi ,\rho ])$ for cyclic, elementary, and hyperelementary subgroups of $\pi$, under natural restrictions on $R$ and $\pi$.