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
ativity and requiring some of the homomorphisms in the biproduct diagram to be merely identity-preserving maps, we obtain a semi-biproduct. A pseudo-action is a new notion as well. It consists of three ingredients: a pre-action, a factor system and a correction system. In the category of groups all correction systems are trivial. This is perhaps the reason why this notion, to the authors best knowledge, has never been considered before.
We present a new approach to ternary Boolean algebras in which negation is derived from the ternary operation. The key aspect is the replacement of complete commutativity by other axioms that do not require the ternary operation to be symmetric.
The category of mobi algebras has been introduced as a model to the unit interval of real numbers. The notion of mobi space over a mobi algebra has been proposed as a model for spaces with geodesic paths. In this paper we analyse the particular case
of affine mobi spaces and show that there is an isomorphism of categories between R-modules and pointed affine mobi spaces over a mobi algebra R as soon as R is a unitary ring in which 2 is an invertible element.
The structure of topological spaces is analysed here through the lenses of fibrous preorders. Each topological space has an associated fibrous preorder and those fibrous preorders which return a topological space are called spacial. A special class o
f spacial fibrous preorders consisting of an interconnected family of preorders indexed by a unitary magma is called cartesian and studied here. Topological spaces that are obtained from those fibrous preorders, with a unitary magma emph{I}, are called emph{I}-cartesian and characterized. The characterization reveals a hidden structure of such spaces. Several other characterizations are obtained and special attention is drawn to the case of a monoid equipped with a topology. A wide range of examples is provided, as well as general procedures to obtain topologies from other data types such as groups and their actions. Metric spaces and normed spaces are considered as well.
We introduce an algebraic system which can be used as a model for spaces with geodesic paths between any two of their points. This new algebraic structure is based on the notion of mobility algebra which has recently been introduced as a model for th
e unit interval of real numbers. We show that there is a strong connection between modules over a ring and affine mobility spaces over a mobility algebra. However, geodesics in general fail to be affine thus giving rise to the new algebraic structure of mobility space. We show that the so called formula for spherical linear interpolation, which gives geodesics on the n-sphere, is an example of a mobility space over the unit interval mobility algebra.
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
of preordered sets and monoids. Schreier split extensions are described in the full subcategory of preordered monoids whose preorder is determined by the corresponding positive cone.
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 monoids. Eve
ry Schreier extension of monoids can be seen as an instance of a semi-biproduct; namely a semi-biproduct whose associated pseudo-action has a trivial correction system. A correction system is a new ingredient that must be inserted in order to obtain a pseudo-action out of a pre-action and a factor system. In groups, every correction system is trivial. Hence, semi-biproducts there are the same as semi-direct products with a factor system, which are nothing but group extensions. An attempt to establish a general context in which to define semi-biproducts is made. As a result, a new structure of map-transformations is obtained from a category with a 2-cell structure. Examples and first basic properties are briefly explored.
We introduce an algebraic system which can be used as a model for spaces with geodesic paths between any two of their points. This new algebraic structure is based on the notion of mobility algebra which has recently been introduced as a model for th
e unit interval of real numbers. Mobility algebras consist on a set $A$ together with three constants and a ternary operation. In the case of the closed unit interval $A=[0,1]$, the three constants are 0, 1 and 1/2 while the ternary operation is $p(x,y,z)=x-yx+yz$. A mobility space is a set $X$ together with a map $qcolon{Xtimes Atimes Xto X}$ with the meaning that $q(x,t,y)$ indicates the position of a particle moving from point $x$ to point $y$ at the instant $tin A$, along a geodesic path within the space $X$. A mobility space is thus defined with respect to a mobility algebra, in the same way as a module is defined over a ring. We introduce the axioms for mobility spaces, investigate the main properties and give examples. We also establish the connection between the algebraic context and the one of spaces with geodesic paths. The connection with affine spaces is briefly mentioned.
A classification theorem for three different sorts of Maltsev categories is proven. The theorem provides a classification for Maltsev category, naturally Maltsev category, and weakly Maltsev category in terms of classifying classes of spans. The clas
s of all spans characterizes naturally Maltsev categories. The class of relations (i.e. jointly monomorphic spans) characterizes Maltsev categories. The class of strong relations (i.e. jointly strongly monomorphic spans) characterizes weakly Maltsev categories. The result is based on the uniqueness of internal categorical structures such as internal category and internal groupoid (Lawvere condition). The uniqueness of these structures is viewed as a property on their underlying reflexive graphs, restricted to the classifying spans. The class of classifying spans is combined, via a new compatibility condition, with split squares. This is analogous to orthogonality between spans and cospans. The result is a general classifying scheme which covers the main characterizations for Maltsev like categories. The class of positive relations has recently been shown to characterize Goursat categories and hence it is a new example that fits in this general scheme.
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 cod
omain 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.
