Related papers: A left topological monoid associated to a topologi…

Let $\A$ be a finitely generated semigroup with 0. An $\A$-module over $\fun$ (also called an $\A$--set), is a pointed set $(M,*)$ together with an action of $\A$. We define and study the Hall algebra $\H_{\A}$ of the category $\C_{\A}$ of…

This unpublished note contains some materials taken from my old study note on groupoids and small categories. It contains a proof for the fact that any groupoid is a group bundle over an equivalence relation. Moreover, the action of a…

We prove a functoriality result for the full C*-algebras of right-LCM monoids with respect to monoid inclusions that are closed under factorization and preserve orthogonality, and use this to show that if a right-LCM monoid is amenable in…

We study algebraic and topological properties of subsemigroups of the hyperspace exp(G) of non-empty compact subsets of a topological group G endowed with the Vietoris topology and the natural semigroup operation. On this base we prove that…

In this paper, we consider topological semigroup actions on compact topological spaces. Under mild assumptions on the semigroup and the action, we construct a semi-direct product groupoid with a Haar system. We also show that it is…

In this paper we develop a new groupoid-based structure theory for the class of regular $*$-semigroups. This class occupies something of a `sweet spot' between the important classes of inverse and regular semigroups, and contains many…

Under the relation of $0$-concordance, the set of knotted 2-spheres in $S^4$ forms a commutative monoid $\mathcal{M}_0$ with the operation of connected sum. Sunukjian has recently shown that $\mathcal{M}_0$ contains a submonoid isomorphic…

The category of strict polynomial functors inherits an internal tensor product from the category of divided powers. To investigate this monoidal structure, we consider the category of representations of the symmetric group which admits a…

We find anti-isomorphic submonoids $\mathscr{C}_{+}(a,b)$ and $\mathscr{C}_{-}(a,b)$ of the bicyclic monoid $\mathscr{C}(a,b)$ with the following properties: every Hausdorff left-continuous (right-continuous) topology on…

In this short note, we construct a right adjoint to the functor which associates to a ring $R$ equipped with a group action its twisted group ring. This right adjoint admits an interpretation as semilinearization, in that it sends an…

The binary products of right, left or double division in semigroups that are semilattices of groups give interesting groupoid structures that are in one to one correspondence with semigroups that are semilattices of groups. This work is…

We define a cohomology for an arbitrary $K$-linear semistrict semigroupal 2-category $(\mathfrak{C},\otimes)$ (called in the paper a Gray semigroup) and show that its first order (unitary) deformations, up to the suitable notion of…

We define Hopf monads on an arbitrary monoidal category, extending the definition given previously for monoidal categories with duals. A Hopf monad is a bimonad (or opmonoidal monad) whose fusion operators are invertible. This definition…

We present some homological properties of a relation $\beta$ on ordered groupoids that generalises the minimum group congruence for inverse semigroups. When $\beta$ is a transitive relation on an ordered groupoid $G$, the quotient $G /…

Let G be a semisimple simply connected group of simply laced type over C. We show that the flag manifold of G has a version defined over the "tropical" semifield Z on which the monoid G(Z) associated to G and the semifield Z acts.

Given a unital action $\theta $ of an inverse monoid $S$ on an algebra $A$ over a filed $K$ we produce (co)homology spectral sequences which converge to the Hochschild (co)homology of the crossed product $A\rtimes_\theta S$ with values in a…

Let $A$ be either a simplicial complex $K$ or a small category $\mathcal C$ with $V(A)$ as its set of vertices or objects. We define a twisted structure on $A$ with coefficients in a simplicial group $G$ as a function $$ \delta\colon…

We study the closures of subgroups, semilattices and different kinds of semigroup extensions in semitopological inverse semigroups with continuous inversion. In particularly we show that a topological group $G$ is $H$-closed in the class of…

We propose a correspondence between topological order in 2+1d and Seifert three-manifolds together with a choice of ADE gauge group $G$. Topological order in 2+1d is known to be characterized in terms of modular tensor categories (MTCs),…

A monoid $S$ is right coherent if every finitely generated subact of every finitely presented right $S$-act is finitely presented. This is a finiteness condition, and we investigate whether or not it is preserved under some standard…