The category of affine algebraic regular monoids

<abstract><p>The main aim of this study is to characterize affine weak $ k $-algebra $ H $ whose affine $ k $-variety $ S = M_{k}(H, k) $ admits a regular monoid structure. As preparation, we determine some results of weak Hopf algebras morphisms, and prove that the anti-function from the category $ \mathcal{C} $ of weak Hopf algebras whose weak antipodes are anti-algebra morphisms is adjoint. Then, we prove the main result of this study: the anti-equivalence between the category of affine algebraic $ k $-regular monoids and the category of finitely generated commutative reduced weak $ k $-Hopf algebras.</p></abstract>

The Hopf Monoid of Hypergraphs and its Sub-Monoids: Basic Invariant and Reciprocity Theorem

In arXiv:1709.07504 Aguiar and Ardila give a Hopf monoid structure on hypergraphs as well as a general construction of polynomial invariants on Hopf monoids. Using these results, we define in this paper a new polynomial invariant on hypergraphs. We give a combinatorial interpretation of this invariant on negative integers which leads to a reciprocity theorem on hypergraphs. Finally, we use this invariant to recover well-known invariants on other combinatorial objects (graphs, simplicial complexes, building sets, etc) as well as the associated reciprocity theorems.

(L)-Semigroup Sums

An (L)-semigroup S is a compact n-manifold with connected boundary B together with a monoid structure on S such that B is a subsemigroup of S. The sum S + T of two (L)-semigroups S and T having boundary B is the quotient space obtained from the union of S × { 0 } and T × { 1 } by identifying the point ( x , 0 ) in S × { 0 } with ( x , 1 ) in T × { 1 } for each x in B. It is shown that no (L)-semigroup sum of dimension less than or equal to five admits an H-space structure, nor does any (L)-semigroup sum obtained from (L)-semigroups having an Abelian boundary. In particular, such sums cannot be a retract of a topological group.

Open projections do not form a right residuated lattice

The lattice of open projection of a C*-algebra has a partial monoid structure on compatible elements. However, this does not extend to a total right residuated operation.

Monoid and Topological Groupoid

<p>Here we introduce some new results which are relative to the concept of topological monoid-groupoid and prove that the category of topological monoid coverings of X is equivalent to the category covering groupoids of the monoid-groupoid <span lang="EN-US">&amp;#960;</span><span lang="EN-US">&lt;sub&gt;</span>1&lt;/sub&gt;(X). Also, it is shown that the monoid structure of monoid-groupoid lifts to a universal covering groupoid.</p>

Graphs of Morphisms of Graphs

This is an account for the combinatorially minded reader of various categories of directed and undirected graphs, and their analogies with the category of sets. As an application, the endomorphisms of a graph are in this context not only composable, giving a monoid structure, but also have a notion of adjacency, so that the set of endomorphisms is both a monoid and a graph. We extend Shrimpton's (unpublished) investigations on the morphism digraphs of reflexive digraphs to the undirected case by using an equivalence between a category of reflexive, undirected graphs and the category of reflexive, directed graphs with reversal. In so doing, we emphasise a picture of the elements of an undirected graph, as involving two types of edges with a single vertex, namely 'bands' and 'loops'. Such edges are distinguished by the behaviour of morphisms with respect to these elements.

THE STRUCTURE OF RESIDUATED LATTICES

A residuated lattice is an ordered algebraic structure [Formula: see text] such that <L,∧,∨> is a lattice, <L,·,e> is a monoid, and \ and / are binary operations for which the equivalences [Formula: see text] hold for all a,b,c ∈ L. It is helpful to think of the last two operations as left and right division and thus the equivalences can be seen as "dividing" on the right by b and "dividing" on the left by a. The class of all residuated lattices is denoted by ℛℒ The study of such objects originated in the context of the theory of ring ideals in the 1930s. The collection of all two-sided ideals of a ring forms a lattice upon which one can impose a natural monoid structure making this object into a residuated lattice. Such ideas were investigated by Morgan Ward and R. P. Dilworth in a series of important papers [15, 16, 45–48] and also by Krull in [33]. Since that time, there has been substantial research regarding some specific classes of residuated structures, see for example [1, 9, 26] and [38], but we believe that this is the first time that a general structural theory has been established for the class ℛℒ as a whole. In particular, we develop the notion of a normal subalgebra and show that ℛℒ is an "ideal variety" in the sense that it is an equational class in which congruences correspond to "normal" subalgebras in the same way that ring congruences correspond to ring ideals. As an application of the general theory, we produce an equational basis for the important subvariety ℛℒC that is generated by all residuated chains. In the process, we find that this subclass has some remarkable structural properties that we believe could lead to some important decomposition theorems for its finite members (along the lines of the decompositions provided in [27]).