Related papers: The Inverse Semigroup Theory of Elementary Arithme…

We introduce a preorder on an inverse semigroup $S$ associated to any normal inverse subsemigroup $N$, that lies between the natural partial order and Green's ${\mathscr J}$-relation. The corresponding equivalence relation $\simeq_N$ is not…

We summarize the main known results involving subword reversing, a method of semigroup theory for constructing van Kampen diagrams by referring to a preferred direction. In good cases, the method provides a powerful tool for investigating…

This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection…

The monoid of all partial injections on a finite set (the symmetric inverse semigroup) is of particular interest because of the well-known Wagner-Preston Theorem. In this article, we step forward the study of a submonoid of the symmetric…

We study generalized inverses on semigroups by means of Green's relations. We first define the notion of inverse along an element and study its properties. Then we show that the classical generalized inverses (group inverse, Drazin inverse…

We introduce a class of inverse monoids that can be regarded as non-commutative generalizations of Boolean algebras. These inverse monoids are related to a class of \'etale topological groupoids, under a non-commutative generalization of…

The partial automorphism monoid of an inverse semigroup is an inverse monoid consisting of all isomorphisms between its inverse subsemigroups. We prove that a tightly connected fundamental inverse semigroup $S$ with no isolated nontrivial…

The dual symmetric inverse monoid $\mathscr{I}_n^*$ is the inverse monoid of all isomorphisms between quotients of an $n$-set. We give a monoid presentation of $\mathscr{I}_n^*$ and, along the way, establish criteria for a monoid to be…

Using the recent notion of inverse along an element in a semigroup, and the natural partial order on idempotents, we study bicommuting generalized inverses and define a new inverse called natural inverse, that generalizes the Drazin inverse…

In this short note, we prove a formula for the group inverse of a block matrix and consider the pseudo principal pivot transform expressed in terms of group inverses. Extensions of the usual principal pivot transform, where the usual…

We describe a formalism, using groupoids, for the study of rewriting for presentations of inverse monoids, that is based on the Squier complex construction for monoid presentations. We introduce the class of pseudoregular groupoids, an…

A simple observation, showing that every groupoid becomes an inverse semigroup after adding one element. In such inverse semigroups all idempotents are mutually orthogonal. This fact implies that every C*-algebra of a discrete groupoid is a…

For a given inverse semigroup, one can associate an \'etale groupoid which is called the universal groupoid. Our motivation is studying the relation between inverse semigroups and associated \'etale groupoids. In this paper, we focus on…

Every $F$-inverse monoid can be equipped with the unary operation which maps each element to the maximum element of its $\sigma$-class. In this enriched signature, the class of all $F$-inverse monoids forms a variety of algebraic…

Inverse braid monoid describes a structure on braids where the number of strings is not fixed. So, some strings of initial $n$ may be deleted. In the paper we show that many properties and objects based on braid groups may be extended to…

In this short note, we show that the number of monogenic submonoids of the full transformation monoid of degree $n$ for $n > 0$, equals the sum of the number of cyclic subgroups of the symmetric groups on $1$ to $n$ points. We also prove an…

Let $<X>$ be the free monoid on a generating set $X$, and suppose one adjoins to $<X>$ universal 2-sided inverses to a finite set $S$ of its elements. We note an elementary algorithm which yields a normal form for elements of the resulting…

In this paper, we introduce a new concept called left (right) g-MP inverse in a $*$-monoid. The relations of this type of generalized inverse with left inverse along an element are investigated. Also, the reverse order law for the inverse…

We give a thorough structural analysis of the principal one-sided ideals of arbitrary semigroups, and then apply this to full transformation semigroups and symmetric inverse monoids. One-sided ideals of these semigroups naturally occur as…

Let $S$ be a $*$-semigroup and let $a,w,v\in S$. The initial goal of this work is to introduce two new classes of generalized inverses, called the $w$-core inverse and the dual $v$-core inverse in $S$. An element $a\in S$ is $w$-core…