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We show that the category of cancellative conjugation semigroups is weakly Maltsev and give a characterization of all admissible diagrams there. In the category of cancellative conjugation monoids we describe, for Schreier split epimorphisms with codomain B and kernel X, all morphisms h from X to B which induce a reflexive graph, an internal category or an internal groupoid. We describe Schreier split epimorphisms in terms of external actions and consider the notions of precrossed semimodule, crossed semimodule and crossed module in the context of cancellative conjugation monoids. In this category we prove that a relative version of the so-called Smith is Huq condition for Schreier split epimorphisms holds as well as other relative conditions.
It is shown that the category of emph{semi-biproducts} of monoids is equivalent to the category of emph{pseudo-actions}. A semi-biproduct of monoids is a new notion, obtained through generalizing a biproduct of commutative monoids. By dropping commut
The main objective of the paper is to define the construction of the object of monoids, over a monoidal category object in any 2-category with finite products, as a weighted limit. To simplify the definition of the weight, we use matrices of symmetri
We compute coherent presentations of Artin monoids, that is presentations by generators, relations, and relations between the relations. For that, we use methods of higher-dimensional rewriting that extend Squiers and Knuth-Bendixs completions into a
We introduce a notion of $n$-commutativity ($0le nle infty$) for cosimplicial monoids in a symmetric monoidal category ${bf V}$, where $n=0$ corresponds to just cosimplicial monoids in ${bf V,}$ while $n=infty$ corresponds to commutative cosimplicial
Properties of preordered monoids are investigated and important subclasses of such structures are studied. The corresponding full subcategories of the category of preordered monoids are functorially related between them as well as with the categories