Theory Of Monoids Research Articles

How to twist pointers without breaking them

Using the theory of monoids and monoid actions, we give a unified framework that handles three common pointer manipulation tasks, namely, data serialisation, deserialisation, and memory allocation. Our main theoretical contribution is the formulation of the notion of a twisted functor , a generalisation of the semi-direct product construction for monoids. We show that semi-direct products and twisted functors are particularly well suited as an abstraction for many pointer manipulation tasks. We describe the implementation of these abstractions in the context of a cryptographic library for Haskell. Twisted functors allow us to abstract all pointer arithmetic and size calculations into a few lines of code, significantly reducing the opportunities for buffer overflows.

Divisibility Theory of Arithmetical Rings with One Minimal Prime Ideal

Continuing the study of divisibility theory of arithmetical rings started in [1] and [2], we show that the divisibility theory of arithmetical rings with one minimal prime ideal is axiomatizable as Bezout monoids with one minimal m-prime filter. In particular, every Bezout monoid with one minimal m-prime filter is order-isomorphic to the partially ordered monoid with respect to inverse inclusion, of principal ideals in a Bezout ring with a smallest prime ideal. Although this result can be considered as a satisfactory answer to the divisibility theory of both semihereditary domains and valuation rings, the general representation theory of Bezout monoids is still open.

Towards Compositional Graph Theory

Decomposing graphs into simpler graphs is one of the central concerns of graph theory. Investigations have revealed deep concepts such as modular decomposition, tree width or rank width, which measure—in different ways—the structural complexity of a graph's topology. Courcelle and others have shown that such concepts can be used to obtain efficient algorithms for families of graphs that are amenable to decomposition (e.g. those that have bounded tree-width). These algorithms, in turn, are of course of use in computer science, where graphs are ubiquitous. In this paper we take the first steps towards understanding notions of decomposition in graph theory compositionally, and more generally, in a categorical setting: category theory, after all, is the mathematics of compositionality.We introduce the concept of ∪–matrices (cup-matrices). Like ordinary matrices, ∪-matrices are the arrows of a PROP: we give a presentation, extending the work of Lafont, and Bonchi, Zanasi and the second author. A variant of ∪-matrices is then used in the development of a novel algebra of simple graphs, the lingua franca of graph theory. The algebra is that of a certain symmetric monoidal theory: ∪-matrices—akin to adjacency matrices—encode the graphs' topology.

A finite variant of the Toom Model

We present results for a finite variant of the one-dimensional Toom model with closed boundaries. We show that the steady state distribution is not of product form, but is nonetheless simple. In particular, we give explicit formulas for the densities and some nearest neighbour correlation functions. We also give exact results for eigenvalues and multiplicities of the transition matrix using the theory of R-trivial monoids in joint work with A. Schilling, B. Steinberg and N. M. Thiéry.

Character theory of monoids over an arbitrary field

The basic character theory of finite monoids over the complex numbers was developed in the sixties and seventies based on work of Munn, Ponizovskiĭ, McAlister, Rhodes and Zalcstein. In particular, McAlister determined the space of functions spanned by the irreducible characters of a finite monoid over C and the ring of virtual characters. In this paper, we present the corresponding results over an arbitrary field.As a consequence, we obtain a quick proof of the theorem of Berstel and Reutenauer that the characteristic function of a regular cyclic language is a virtual character of the free monoid. This is a crucial ingredient in their proof of the rationality of the zeta function of a sofic shift in symbolic dynamics.

Directed Nonabelian Sandpile Models on Trees

We define two general classes of nonabelian sandpile models on directed trees (or arborescences) as models of nonequilibrium statistical phenomena. These models have the property that sand grains can enter only through specified reservoirs, unlike the well-known abelian sandpile model. In the Trickle-down sandpile model, sand grains are allowed to move one at a time. For this model, we show that the stationary distribution is of product form. In the Landslide sandpile model, all the grains at a vertex topple at once, and here we prove formulas for all eigenvalues, their multiplicities, and the rate of convergence to stationarity. The proofs use wreath products and the representation theory of monoids.

Functor homology and homology of commutative monoids

The aim of this work is to clarify the relationship between homology theory of commutative monoids constructed 'a la Quillen and technology of Gamma-modules.

Automata with Counters that Recognize Word Problems of Free Products

An M-automaton is a finite automaton with a blind counter that mimics a monoid M. The finitely generated groups whose word problems (when viewed as formal languages) are accepted by M-automata play a central role in understanding the family 𝔏(M) of all languages accepted by M-automata. If G1 and G2 are finitely generated groups whose word problems are languages in 𝔏(M), in general, the word problem of the free product G1 * G2 is not necessarily in 𝔏(M). However, we show that if M is enlarged to the free product M*P2, where P2 is the polycyclic monoid of rank two, then this closure property holds. In fact, we show more generally that the special word problem of M1 * M2 lies in 𝔏(M * P2) whenever M1 and M2 are finitely generated monoids with special word problems in 𝔏(M * P2). We also observe that there is a monoid without zero, denoted by CF2, that can be used in place of P2 for this purpose. The monoid CF2 is the rank two case of what we call a monoid with right invertible basis and its Rees quotient by its maximal ideal is P2. The fundamental theory of monoids with right invertible bases is completely analogous to that of free groups, and thus they are very convenient to use. We also investigate the questions of whether there is a group that can be used instead of the monoid P2 in the above result and under what circumstances P1 (or the bicyclic monoid) is enough to do the job of P2.

How Do You Measure Primality?

In commutative monoids, the ω-value measures how far an element is from being prime. This invariant, which is important in understanding the factorization theory of monoids, has been the focus of much recent study. This paper provides detailed examples and an overview of known results on ω-primality, including several recent and surprising contributions in the setting of numerical monoids. As many questions related to ω-primality remain, we provide a list of open problems accessible to advanced undergraduate students and beginning graduate students.

Ehrhart theory of polytopes and Seiberg–Witten invariants of plumbed 3–manifolds

Let M be a rational homology sphere plumbed 3-manifold associated with a connected negative definite plumbing graph. We show that its Seiberg-Witten invariants equal certain coefficients of an equivariant multivariable Ehrhart polynomial. For this, we construct the corresponding polytopes from the plumbing graphs together with an action of the first homology of M, and we develop Ehrhart theory for them. At an intermediate level we define the `periodic constant' of multivariable series and establish their properties. In this way, one identifies the Seiberg-Witten invariant of a plumbed 3-manifold, the periodic constant of its `combinatorial zeta-function', and a coefficient of the associated Ehrhart polynomial. We make detailed presentations for graphs with at most two nodes. The two node case has surprising connections with the theory of affine monoids of rank two.

Factorization properties of Leamer monoids

The Huneke-Wiegand conjecture has prompted much recent research in Commutative Algebra. In studying this conjecture for certain classes of rings, Garc\'ia-S\'anchez and Leamer construct a monoid S_\Gamma^s whose elements correspond to arithmetic sequences in a numerical monoid \Gamma of step size s. These monoids, which we call Leamer monoids, possess a very interesting factorization theory that is significantly different from the numerical monoids from which they are derived. In this paper, we offer much of the foundational theory of Leamer monoids, including an analysis of their atomic structure, and investigate certain factorization invariants. Furthermore, when S_\Gamma^s is an arithmetical Leamer monoid, we give an exact description of its atoms and use this to provide explicit formulae for its Delta set and catenary degree.

Open-graphs and monoidal theories

String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks and many other compositional structures. The distinguishing feature of these diagrams is that edges need not be connected to vertices at both ends, and these unconnected ends can be interpreted as the inputs and outputs of a diagram. In this paper, we give a concrete construction for string diagrams using a special kind of typed graph called an open-graph. While the category of open-graphs is not itself adhesive, we introduce the notion of a selective adhesive functor, and show that such a functor embeds the category of open-graphs into the ambient adhesive category of typed graphs. Using this functor, the category of open-graphs inherits ‘enough adhesivity’ from the category of typed graphs to perform double-pushout (DPO) graph rewriting. A salient feature of our theory is that it ensures rewrite systems are ‘type safe’ in the sense that rewriting respects the inputs and outputs. This formalism lets us safely encode the interesting structure of a computational model, such as evaluation dynamics, with succinct, explicit rewrite rules, while the graphical representation absorbs many of the tedious details. Although topological formalisms exist for string diagrams, our construction is discrete and finitary, and enjoys decidable algorithms for composition and rewriting. We also show how open-graphs can be parameterised by graphical signatures, which are similar to the monoidal signatures of Joyal and Street, and define types for vertices in the diagrammatic language and constraints on how they can be connected. Using typed open-graphs, we can construct free symmetric monoidal categories, PROPs and more general monoidal theories. Thus, open-graphs give us a tool for mechanised reasoning in monoidal categories.

Partial mirror symmetry, lattice presentations and algebraic monoids

This is the second in a series of papers that develops the theory of reflection monoids, motivated by the theory of reflection groups. Reflection monoids were first introduced in arXiv:0812.2789. In this paper we study their presentations as abstract monoids. Along the way we also find general presentations for certain join-semilattices (as monoids under join) which we interpret for two special classes of examples: the face lattices of convex polytopes and the geometric lattices, particularly the intersection lattices of hyperplane arrangements. Another spin-off is a general presentation for the Renner monoid of an algebraic monoid, which we illustrate in the special case of the "classical" algebraic monoids.

Algebraic and combinatorial aspects of sandpile monoids on directed graphs

The sandpile group of a graph is a well-studied object that combines ideas from algebraic graph theory, group theory, dynamical systems, and statistical physics. A graphʼs sandpile group is part of a larger algebraic structure on the graph, known as its sandpile monoid. Most of the work on sandpiles so far has focused on the sandpile group rather than the sandpile monoid of a graph, and has also assumed the underlying graph to be undirected. A notable exception is the recent work of Babai and Toumpakari, which builds up the theory of sandpile monoids on directed graphs from scratch and provides many connections between the combinatorics of a graph and the algebraic aspects of its sandpile monoid.In this paper we primarily consider sandpile monoids on directed graphs, and we extend the existing theory in four main ways. First, we give a combinatorial classification of the maximal subgroups of a sandpile monoid on a directed graph in terms of the sandpile groups of certain easily-identifiable subgraphs. Second, we point out certain sandpile results for undirected graphs that are really results for sandpile monoids on directed graphs that contain exactly two idempotents. Third, we give a new algebraic constraint that sandpile monoids must satisfy and exhibit two infinite families of monoids that cannot be realized as sandpile monoids on any graph. Finally, we give an explicit combinatorial description of the sandpile group identity for every graph in a family of directed graphs which generalizes the family of (undirected) distance-regular graphs. This family includes many other graphs of interest, including iterated wheels, regular trees, and regular tournaments.

Betti Numbers of Toric Varieties and Eulerian Polynomials

It is well known that the Eulerian polynomials, which count permutations in S n by their number of descents, give the h-polynomial/h-vector of the simple polytopes known as permutohedra, the convex hull of the S n -orbit for a generic weight in the weight lattice of S n . Therefore, the Eulerian polynomials give the Betti numbers for certain smooth toric varieties associated with the permutohedra. In this article we derive recurrences for the h-vectors of a family of polytopes generalizing this. The simple polytopes we consider arise as the orbit of a nongeneric weight, namely, a weight fixed by only the simple reflections J = {s n , s n−1, s n−2,…, s n−k+2, s n−k+1} for some k with respect to the A n root lattice. Furthermore, they give rise to certain rationally smooth toric varieties X(J) that come naturally from the theory of algebraic monoids. Using effectively the theory of reductive algebraic monoids and the combinatorics of simple polytopes, we obtain a recurrence formula for the Poincaré polynomial of X(J) in terms of the Eulerian polynomials.

Eléments d’algébre linéaire tropicale

We define a formal framework for the study of algebras of type Max-plus, Min-Plus, tropical algebras, and more generally algebras over a commutative idempotent semi-field. This work is motivated by the increasingly diversified use of these algebras which occur also in control theory, automata theory as well as in algebraic geometry, and in more specific ways in other parts of mathematics such as the theory of monoids. In this first article, we expecially re-examine linear algebra over idempotent semi-fields: the most delicate, but undoubtedly the most interesting point is the notion of a singular point seen as a generalization of the notion of zero. We thus rediscover many notions of regularity already introduced for matrices, and this allows us to define further notions, new in this context, such as that of the kernel of a linear form, and to apply duality to obtain a good notion of tropical dimension of a submodule.

Partial symmetry, reflection monoids and Coxeter groups

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 monoids, introduce new examples, and determine their orders.

REPRESENTATIONS OF THE RENNER MONOID

We describe irreducible representations and character formulas of the Renner monoids for reductive monoids, which generalize the Munn–Solomon representation theory of rook monoids to any Renner monoids. The type map and polytope associated with reductive monoids play a crucial role in our work. It turns out that the irreducible representations of certain parabolic subgroups of the Weyl groups determine the complete set of irreducible representations of the Renner monoids. An analogue of the Munn–Solomon formula for calculating the character of the Renner monoids, in terms of the characters of the parabolic subgroups, is shown.

THE BRAID ROOK MONOID

In linear algebraic monoid theory, the Renner monoids play the role of the Weyl groups in linear algebraic group theory. It is well known that Weyl groups are Coxeter groups, and that we can associate a Hecke algebra and an Artin–Tits group to each Coxeter group. The question of the existence of a Hecke algebra associated with each Renner monoid has been positively answered. In this paper we discuss the question of the existence of an equivalent of the Artin–Tits groups in the framework of Renner monoids. We consider the seminal case of the rook monoid and introduce a new length function.

Descent systems for Bruhat posets

Let (W,S) be a finite Weyl group and let w∈W. It is widely appreciated that the descent set $$D(w)=\{s\in S\mid l(ws)<l(w)\}$$ determines a very large and important chapter in the study of Coxeter groups. In this paper we generalize some of those results to the situation of the Bruhat poset W J where J⊆S. Our main results here include the identification of a certain subset S J ⊆W J that convincingly plays the role of S⊆W, at least from the point of view of descent sets and related geometry. The point here is to use this resulting descent system (W J ,S J ) to explicitly encode some of the geometry and combinatorics that is intrinsic to the poset W J . In particular, we arrive at the notion of an augmented poset, and we identify the combinatorially smooth subsets J⊆S that have special geometric significance in terms of a certain corresponding torus embedding X(J). The theory of J-irreducible monoids provides an essential tool in arriving at our main results.