The group of self-distributivity is bi-orderable

We prove that the group of left self-distributivity, a cousin of Thompson's group F and of Artin's braid group B_￥ B_infty that describes the geometry of the identity x(yz) = (xy)(xz), admits a bi-invariant linear ordering. To this end, we define a partial action of this group on finite binary trees that preserves a convenient linear ordering.     

The dual braid monoid

We study a new monoid structure for Artin groups associated with finite Coxeter systems. Like the classical positive braid monoid, the new monoid is a Garside monoid. We give several equivalent constructions: algebraically, the new monoid arises when studying Coxeter systems in a “dual” way, replacing the pair (W,S) by (W,T), with T the set of all reflections; geometrically, it arises when looking at the reflection arrangement from a certain basepoint. In the type A case, we recover the monoid constructed by Birman, Ko and Lee.     

The shifted plactic monoid

We introduce a shifted analog of the plactic monoid of Lascoux and Schützenberger, the shifted plactic monoid. It can be defined in two different ways: via the shifted Knuth relations, or using Haiman’s mixed insertion. Applications include: a new combinatorial derivation (and a new version of) the shifted Littlewood–Richardson Rule; similar results for the coefficients in the Schur expansion of a Schur P-function; a shifted counterpart of the Lascoux–Schützenberger theory of noncommutative Schur functions in plactic variables; a characterization of shifted tableau words; and more.     

The inverse braid monoid

We introduce an inverse monoid which plays a similar role with respect to the symmetric inverse semigroup that the braid group plays with respect to the symmetric group.     

The monoid of queue actions

We model the behavior of a fifo-queue as a monoid of transformations that are induced by sequences of writing and reading. We describe this monoid by means of a confluent and terminating semi-Thue system and study some of its basic algebraic properties such as conjugacy. Moreover, we show that while several properties concerning its rational subsets are undecidable, their uniform membership problem is \({{\mathsf {N}}}{{\mathsf {L}}}\)-complete. Furthermore, we present an algebraic characterization of this monoid’s recognizable subsets. Finally, we prove that it is not Thurston-automatic.     

The syntactic monoid of a hypercode

This paper gives a characterization of the syntactic monoid of a hypercode H over a finite alphabet X, a hupercode being a non empty set of non empty words over X, which are pairwise incomparable relatively to the embedding partial order of X. This research has been supported by Grant A7877 of the National Research Council of Canada.     

The radical-annihilator monoid of a ring

Kuratowski’s closure-complement problem gives rise to a monoid generated by the closure and complement operations. Consideration of this monoid yielded an interesting classification of topological spaces, and subsequent decades saw further exploration using other set operations. This article is an exploration of a natural analogue in ring theory: a monoid produced by “radical” and “annihilator” maps on the set of ideals of a ring. We succeed in characterizing semiprime rings and commutative dual rings by their radical-annihilator monoids, and we determine the monoids for commutative local zero-dimensional (in the sense of Krull dimension) rings.     

The dimension monoid of a lattice

We introduce the dimension monoid of a lattice L, denoted by Dim L. The monoid Dim L is commutative and conical, the latter meaning that the sum of any two nonzero elements is nonzero. Furthermore, Dim L is given along with the dimension map, D\Delta from L2L to Dim L, which has the intuitive meaning of a distance function. The maximal semilattice quotient of Dim L is isomorphic to the semilattice Con_c L of compact congruences of L; hence Dim L is a precursor of the congruence lattice of L. Here are some additional features of this construction: ¶¶ (1) Our dimension theory provides a generalization to all lattices of the von Neumann dimension theory of continuous geometries. In particular, if L is an irreducible continuous geometry, then Dim L is either isomorphic to \Bbb Z⁺\Bbb Z^+ or to \Bbb R⁺\Bbb R^+.¶ (2) If L has no infinite bounded chains, then Dim L embeds (as an ordered monoid) into a power of \Bbb Z⁺è{￥}.{\Bbb Z}^{+}\cup \{\infty\}.¶ (3) If L is modular or if L has no infinite bounded chains, then Dim L is a refinement monoid.¶ (4) If L is a simple geometric lattice, then Dim L is isomorphic to \Bbb Z⁺\Bbb Z^+, if L is modular, and to the two-element semilattice, otherwise.¶ (5) If L is an à₀\aleph_0-meet-continuous complemented modular lattice, then both Dim L and the dimension function D\Delta satisfy (countable) completeness properties.¶¶ If R is a von Neumann regular ring and if L is the lattice of principal right ideals of the matrix ring M₂ (R), then Dim L is isomorphic to the monoid V (R) of isomorphism classes of finitely generated projective right R-modules. Hence the dimension theory of lattices provides a wide lattice-theoretical generalization of nonstable K-theory of regular rings.     

The rook monoid is lexicographically shellable

We prove that the Bruhat–Chevalley–Renner order on the rook monoid is EL-shellable. We determine the homeomorphism type of the associated order complex.     

The core congruence on a monoid

The core congruence is defined and its properties studied. It is shown that every fundamental regular monoid can be faithfully represented by its action by translations on its set of left congruences.This research was partially supported by a grant from Tulane University.     

The monoid of strong endomorphisms of a graph

We explicitly find all the idempotents in eachL(R)-class and all the inverses of each element of the strong endomorphism monoid of a graph. The number of these idempotents and inverses is also obtained. The author is deeply indebted to Professor Dr. T. E. Hall for his stimulating questions about this theme and much improvement made to an earlier version of this paper. The author would like to thank Professor Dr. U. Knauer and Dr. E. Wilkeit for helpful comments.     

The monoid of orientation-preserving mappings on a chain

We introduce new kinds of regular monoids of transformations consisting of mappings which preserve or reverse the orientation of a finite cycle.     

Commutative monoid homology

Semigroup Forum - The aim of this paper is to exploit the structure of strongly continuous operator semigroups in order to formulate a categorical framework in which a fresh perspective can be...     

The monoid of ordered partitions of a natural number

The sequences of non-negative integers form a monoid, a natural submonoid of which has elements corresponding to order-preserving transformations of a finite chain. This, in turn, has a submonoid whose elements are ordered partitions of a natural number. A presentation for the last monoid is given, and the inclusion poset of principal right ideals is described. The poset of principal left ideals has a recursive structure that gives rise to an interesting sequence of numbers.