## monoid

Overview page. Subjects: Computing.

A semigroup that possesses an identity element, *e*. If *S* is a semigroup on which there is defined a dyadic operation ∘, then *x* ∘ *e* = *e* ∘ *x* = *x*for all elements *x* in *S*. Monoids play an...

## monoid

Overview page. Subjects: Computing.

A semigroup that possesses an identity element, *e*. If *S* is a semigroup on which there is defined a dyadic operation ∘, then *x* ∘ *e* = *e* ∘ *x* = *x*for all elements *x* in *S*. Monoids play an...

## free monoid

Overview page. Subjects: Computing.

A particular kind of monoid, usually involving strings. Note first that concatenation is an associative operation and also that, if Λ is the empty string, then Λ*w* = *w* = *w*Λ for all strings *w*null...

## A Geometric Characterization of Automatic Monoids

in** The Quarterly Journal of Mathematics**

September 2004; p ublished online September 2004 .

Journal Article. Subjects: Pure Mathematics. 0 words.

It is well known that automatic groups can be characterized using geometric properties of their Cayley graphs. Along the same line of thought, we provide a geometric characterization of...

## syntactic monoid

Overview page. Subjects: Computing.

Of a formal language *L*. The semigroup of the minimal machine for *L*.

## Solutions to Some Open Problems on Totally Ordered Monoids

in** Journal of Logic and Computation**

August 2010; p ublished online November 2008 .

Journal Article. Subjects: Computing; Logic. 0 words.

In this article, solutions to three open problems on ordered commutative monoids posed in Evans *et al.* (2001, *Semigroup forum*, 62, 249-278) [4] are presented. By an ordered monoid, we...

## The Moduli Scheme of Affine Spherical Varieties with a Free Weight Monoid

in** International Mathematics Research Notices**

January 2016; p ublished online October 2015 .

Journal Article. Subjects: Mathematics; Pure Mathematics. 18682 words.

We study Alexeev and Brion's moduli scheme [math] of affine spherical varieties with weight monoid [math] under the assumption that [math] is free. We describe the tangent space to [math]...

## Graded Leinster Monoids and Generalized Deligne Conjecture for 1-Monoidal Abelian Categories

in** International Mathematics Research Notices**

October 2018; p ublished online March 2017 .

Journal Article. Subjects: Mathematics; Pure Mathematics. 30942 words.

**Abstract**

In our recent paper Shoikhet [47], a version of the “generalized Deligne conjecture” for abelian [math]-fold monoidal categories is proven. For [math]...

## The Class of Representable Ordered Monoids has a Recursively Enumerable, Universal Axiomatisation but it is Not Finitely Axiomatisable

in** Logic Journal of the IGPL**

March 2005; p ublished online March 2005 .

Journal Article. Subjects: Logic. 0 words.

An ordered monoid is a structure with an identity element (1′), a binary composition operator (;) and an antisymmetric partial order (≤), satisfying certain axioms. A representation of an...

## free semigroup

Overview page. Subjects: Computing.

A free monoid, but without the identity element. See also semigroup.

## transformation semigroup

Overview page. Subjects: Computing.

A semigroup consisting of a collection *C* of transformations of a set *S* into itself (see function), the dyadic operation ∘ being the composition of functions; it is essential that the set *C* ...

## Generalized Virtual Braid Groups, Quasi-Shuffle Product and Quantum Groups

in** International Mathematics Research Notices**

January 2015; p ublished online January 2014 .

Journal Article. Subjects: Mathematics. 3689 words.

We introduce in this notice the generalized virtual braid group on *n* strands *GVB*
_{
n
}, generalizing simultaneously the braid groups and their virtual...

## Basics

in** From Christoffel Words to Markoff Numbers**

November 2018; p ublished online January 2019 .

Chapter. Subjects: Pure Mathematics. 1225 words.

Definitions and basic results about words: alphabet, length, free monoid, concatenation, prefix, suffix, factor, conjugation, reversal, palindrome, commutative image, periodicity, ultimate...

## Higher-Order Signature Cocycles for Subgroups of Mapping Class Groups and Homology Cylinders

in** International Mathematics Research Notices**

January 2012; p ublished online January 2012 .

Journal Article. Subjects: Mathematics. 13724 words.

We define families of invariants for elements of the mapping class group of *Σ*, a compact-orientable surface. For any characteristic subgroup , let *J*(*H*) denote the subgroup of mapping...

## Polynomial Functors from Algebras Over a Set-Operad and Nonlinear Mackey Functors

in** International Mathematics Research Notices**

January 2015; p ublished online December 2013 .

Journal Article. Subjects: Mathematics. 24964 words.

In this paper, we give a description of polynomial functors from (finitely generated free) groups to abelian groups in terms of nonlinear Mackey functors, generalizing those given in a...

## discrete structure

Overview page. Subjects: Computing.

A set of discrete elements on which certain operations are defined. Discrete implies noncontinuous and therefore discrete sets include finite and countable sets but not uncountable sets...

## An Application of Model Theory to Semimodules

in** Logic Journal of the IGPL**

February 2008; p ublished online February 2008 .

Journal Article. Subjects: Logic. 1102 words.

In this note, we prove that the theory *T* of cancellative semimodules over a semiring *R* has the amalgamation property. If *R* is an entire cancellative zerosumfree semiring, then *T* has no...

## semiring

Overview page. Subjects: Computing.

A set *S* (containing a 0 and a 1) on which there are defined two dyadic operations that are denoted by + and • and that obey certain properties: the set *S*, regarded as a set with a zero on...

## Scalars, Monads, and Categories

in** Quantum Physics and Linguistics**

February 2013; p ublished online May 2013 .

Chapter. Subjects: Applied Mathematics. 13626 words.

This chapter describes interrelations between: algebraic structure on sets of scalars, properties of monads associated with such sets of scalars, and structure in categories (especially...

## CATEGORIES OF DIAGRAMS

in** Category Theory**

May 2006; p ublished online September 2007 .

Chapter. Subjects: Algebra. 6900 words.

This chapter presents the proof for the Yoneda Lemma, which is probably the single most used result in category theory. It is interesting how often it comes up, especially in view of the...

## Coxeter Groups

in** Descent in Buildings (AM-190)**

September 2015; p ublished online October 2017 .

Chapter. Subjects: Combinatorics and Graph Theory. 4887 words.

This chapter develops a theory of descent for buildings by assembling various results about Coxeter groups. It begins with the notation stating that *W* is an arbitrary group with a...