Free Monoid Research Articles

A noncommutative extension of Mahler’s interpolation theorem

We prove a noncommutative generalization of Mahler’s theorem on interpolation series, a celebrated result of p -adic analysis. Mahler’s original result states that a function from \mathbb{N} to \mathbb{Z} is uniformly continuous for the p -adic metric d_p if and only if it can be uniformly approximated by polynomial functions. We prove an analogous result for functions from a free monoid A to a free group F(B) where d_p is replaced by the pro- p metric.

Grammic monoids with three generators

Young tableaux are combinatorial objects whose construction can be achieved from words over a finite alphabet by row or column insertion as shown by Schensted sixty years ago. Recently Abram and Reutenauer studied the action of the free monoid on the set of columns defined by the famous insertion algorithm. Since the number of columns is finite, this action yields a finite transformation monoid. Here we consider the action on the set of rows. We investigate the corresponding infinite monoid in the case of a 3 letter alphabet. In particular we show that it is the quotient of the free monoid relative to a congruence generated by the classical Knuth rules plus a unique extra rule.

Affine Completeness of Some Free Binary Algebras

A function on an algebra is congruence preserving if, for any congruence, it maps pairs of congruent elements onto pairs of congruent elements. An algebra is said to be affine complete if every congruence preserving function is a polynomial function. We show that the algebra of (possibly empty) binary trees whose leaves are labeled by letters of an alphabet containing at least one letter, and the free monoid on an alphabet containing at least two letters are affine complete.

Unambiguous injective morphisms in free groups

A morphism g is ambiguous with respect to a word u if there exists a second morphism h≠g such that g(u)=h(u). Otherwise g is unambiguous with respect to u. Thus unambiguous morphisms are those for which the structure of the morphism is preserved in the image. Ambiguity has so far been studied for morphisms of free monoids, where several characterisations exist for the set of words u permitting an (injective) unambiguous morphism. In the present paper, we consider ambiguity of morphisms of free groups, and consider possible analogies to the existing characterisations in the free monoid. While a direct generalisation results in a trivial situation where all morphisms are ambiguous, we discuss some natural and well-motivated reformulations, and provide a characterisation of words in a free group that permit a morphism which is “as unambiguous as possible”.

An algebraic representation of the fixed-point closure of *-continuous Kleene algebras – A categorical Chomsky–Schützenberger theorem

AbstractThe family ${\mathcal{R}} X^*$ of regular subsets of the free monoid $X^*$ generated by a finite set X is the standard example of a ${}^*$ -continuous Kleene algebra. Likewise, the family ${\mathcal{C}} X^*$ of context-free subsets of $X^*$ is the standard example of a $\mu$ -continuous Chomsky algebra, i.e. an idempotent semiring that is closed under a well-behaved least fixed-point operator $\mu$ . For arbitrary monoids M, ${\mathcal{C}} M$ is the closure of ${\mathcal{R}}M$ as a $\mu$ -continuous Chomsky algebra, more briefly, the fixed-point closure of ${\mathcal{R}} M$ . We provide an algebraic representation of ${\mathcal{C}} M$ in a suitable product of ${\mathcal{R}} M$ with $C_2'$ , a quotient of the regular sets over an alphabet $\Delta_2$ of two pairs of bracket symbols. Namely, ${\mathcal{C}}M$ is isomorphic to the centralizer of $C_2'$ in the product of ${\mathcal{R}} M$ with $C_2'$ , i.e. the set of those elements that commute with all elements of $C_2'$ . This generalizes a well-known result of Chomsky and Schützenberger (1963, Computer Programming and Formal Systems, 118–161) and admits us to denote all context-free languages over finite sets $X\subseteq M$ by regular expressions over $X\cup\Delta_2$ interpreted in the product of ${\mathcal{R}} M$ and $C_2'$ . More generally, for any ${}^*$ -continuous Kleene algebra K the fixed-point closure of K can be represented algebraically as the centralizer of $C_2'$ in the product of K with $C_2'$ .

Pronilpotent quotients associated with primitive substitutions

We describe the pronilpotent quotients of a class of projective profinite groups, that we call ω-presented groups, defined using a special type of presentations. The pronilpotent quotients of an ω-presented group are completely determined by a single polynomial, closely related with the characteristic polynomial of a matrix. We deduce that ω-presented groups are either perfect or admit the p-adic integers as quotients for cofinitely many primes. We also find necessary conditions for absolute and relative freeness of ω-presented groups. Our main motivation comes from semigroup theory: the maximal subgroups of free profinite monoids corresponding to primitive substitutions are ω-presented (a theorem due to Almeida and Costa). We are able to show that the composition matrix of a primitive substitution carries partial information on the pronilpotent quotients of the corresponding maximal subgroup. We apply this to deduce that the maximal subgroups corresponding to primitive aperiodic substitutions of constant length are not absolutely free.

Embedding of the Atomic Theory of Subsets of Free Semigroups to the Atomic Theory of Subsets of Free Monoids

Embedding of the Atomic Theory of Subsets of Free Semigroups to the Atomic Theory of Subsets of Free Monoids

On Minimal Realization of ILF-Languages: A Categorical Approach

This work is toward application of categorical concept to produce a minimal ILF-automaton for a given ILF-language. The ILF-equivalence relation over free monoid formed by a nonempty set together with categorical concepts is used for such construction.

On Morphisms Preserving Palindromic Richness

It is known that each word of length n contains at most n + 1 distinct palindromes. A finite rich word is a word with maximal number of palindromic factors. The definition of palindromic richness can be naturally extended to infinite words. Sturmian words and Rote complementary symmetric sequences form two classes of binary rich words, while episturmian words and words coding symmetric d-interval exchange transformations give us other examples on larger alphabets. In this paper we look for morphisms of the free monoid, which allow us to construct new rich words from already known rich words. We focus on morphisms in Class Pret. This class contains morphisms injective on the alphabet and satisfying a particular palindromicity property: for every morphism ϕ in the class there exists a palindrome w such that ϕ(a)w is a first complete return word to w for each letter a. We characterize Pret morphisms which preserve richness over a binary alphabet. We also study marked Pret morphisms acting on alphabets with more letters. In particular we show that every Arnoux-Rauzy morphism is conjugated to a morphism in Class Pret and that it preserves richness.

From distributive ℓ-monoids to ℓ-groups, and back again

We prove that an inverse-free equation is valid in the variety LG of lattice-ordered groups (ℓ-groups) if and only if it is valid in the variety DLM of distributive lattice-ordered monoids (distributive ℓ-monoids). This contrasts with the fact that, as proved by Repnitskiĭ, there exist inverse-free equations that are valid in all Abelian ℓ-groups but not in all commutative distributive ℓ-monoids, and, as we prove here, there exist inverse-free equations that are valid in all totally ordered groups but not in all totally ordered monoids. We also prove that DLM has the finite model property and a decidable equational theory, establish a correspondence between the validity of equations in DLM and the existence of certain right orders on free monoids, and provide an effective method for reducing the validity of equations in LG to the validity of equations in DLM.

On Undecidability of Finite Subsets Theory for Torsion Abelian Groups

Let M be a commutative cancellative monoid with an element of infinite order. The binary operation can be extended to all finite subsets of M by the pointwise definition. So, we can consider the theory of finite subsets of M. Earlier, we have proved the following result: in the theory of finite subsets of M elementary arithmetic can be interpreted. In particular, this theory is undecidable. For example, the free monoid (the sets of all words with concatenation) has this property, the corresponding algebra of finite subsets is the theory of all finite languages with concatenation. Another example is an arbitrary Abelian group that is not a torsion group. But the method of proof significantly used an element of infinite order, hence, it can’t be immediately generalized to torsion groups. In this paper we prove the given theorem for Abelian torsion groups that have elements of unbounded order: for such group, the theory of finite subsets allows interpreting the elementary arithmetic.

Monads for Measurable Queries in Probabilistic Databases

We consider a bag (multiset) monad on the category of standard Borel spaces, and show that it gives a free measurable commutative monoid. Firstly, we show that a recent measurability result for probabilistic database queries (Grohe and Lindner, ICDT 2020) follows quickly from the fact that queries can be expressed in monad-based terms. We also extend this measurability result to a fuller query language. Secondly, we discuss a distributive law between probability and bag monads, and we illustrate that this is useful for generating probabilistic databases.

Free inverse monoids are notFP2

We give a topological proof that a free inverse monoid on one or more generators is neither of type left-FP 2 nor right-FP 2 . This strengthens a classical result of Schein that such monoids are not finitely presented as monoids.

Free objects in triangular matrix varieties and quiver algebras over semirings

We study the free objects in the variety of semigroups and variety of monoids generated by the monoid of all n×n upper triangular matrices over a commutative semiring. We obtain explicit representations of these, as multiplicative subsemigroups of quiver algebras over polynomial semirings. A case of particular interest is where n=2 and the semiring is the tropical semifield, where the variety coincides with that generated by the bicyclic monoid (or equivalently, by the free monogenic inverse monoid); here we are obtain faithful representations of the free objects in this variety inside a semidirect product of a commutative monoid acting on a semilattice. We apply these representations to answer several questions, including that of when the given varieties are locally finite.

The complexity of solution sets to equations in hyperbolic groups

We show that the full set of solutions to systems of equations and inequations in a hyperbolic group, as shortlex geodesic words (or any regular set of quasigeodesic normal forms), is an EDT0L language whose specification can be computed in NSPACE(n2 log n) for the torsion-free case and NSPACE(n4 log n) in the torsion case. Furthermore, in the presence of effective quasi-isometrically embeddable rational constraints, we show that the full set of solutions to systems of equations in a hyperbolic group remains EDT0L. Our work combines the geometric results of Rips, Sela, Dahmani and Guirardel on the decidability of the existential theory of hyperbolic groups with the work of computer scientists including Plandowski, Jeż, Diekert and others on PSPACE algorithms to solve equations in free monoids and groups using compression, and involves an intricate language-theoretic analysis.

Γ-LANGUAGES AND Γ-AUTOMATA

The aim of this paper is to extend the notion of an automaton as a triple made of a set of states, a free monoid on some set, and an action of this monoid on the set of states, to the case where the free monoid is replaced by a free Γ-monoid, and the action is replaced by the action of this Γ-monoid on the set of states. We call the respective triple a Γ-automaton. This concept leads to another new concept, that of a Γ-language, which is a subset of a free Γ-monoid. Also, we define recognizable Γ-languages and prove that they are exactly those Γ-languages that are recognized by a finite Γ-automaton. In the end, in analogy with the standard theory, we relate the recognizability of a Γ-language with the concept of division of semigroups.

Limit varieties of J-trivial monoids

We show that limit varieties of monoids recently discovered by Gusev, Zhang and Luo and their subvarieties are generated by monoids of the form \(M_\tau (W)\) for certain congruences \(\tau \) on the free monoid. The construction \(M_\tau (W)\) is a generalization of widely used Dilworth–Perkins construction. Using this construction, we find explicit generators for Gusev limit varieties and give a short reproof to the fact that Zhang–Luo limit variety is non-finitely based.

On equations and first-order theory of one-relator monoids

We investigate systems of equations and the first-order theory of one-relator monoids. We describe a family F of one-relator monoids of the form 〈A|w=1〉 where for each monoid M in F, the longstanding open problem of decidability of word equations with length constraints reduces to the Diophantine problem (i.e. decidability of systems of equations) in M. We achieve this result by finding an interpretation in M of a free monoid, using only systems of equations together with length relations. It follows that each monoid in F has undecidable positive AE-theory, hence in particular it has undecidable first-order theory. The family F includes many one-relator monoids with torsion 〈A|wn=1〉 (n>1). In contrast, all one-relator groups with torsion are hyperbolic, and all hyperbolic groups are known to have decidable Diophantine problem. We further describe a different class of one-relator monoids with decidable Diophantine problem.

Primitive sets of words

Given a (finite or infinite) subset X of the free monoid A⁎ over a finite alphabet A, the rank of X is the minimal cardinality of a set F such that X⊆F⁎. We say that a submonoid M generated by k elements of A⁎ is k-maximal if there does not exist another submonoid generated by at most k words containing M. We call a set X⊆A⁎primitive if it is the basis of a |X|-maximal submonoid. This definition encompasses the notion of primitive word — in fact, {w} is a primitive set if and only if w is a primitive word. By definition, for any set X, there exists a primitive set Y such that X⊆Y⁎. We therefore call Y a primitive root of X. As a main result, we prove that if a set has rank 2, then it has a unique primitive root. To obtain this result, we prove that the intersection of two 2-maximal submonoids is either the empty word or a submonoid generated by one single primitive word.For a single word w, we say that the set {x,y} is a bi-root of w if w can be written as a concatenation of copies of x and y and {x,y} is a primitive set. We prove that every primitive word w has at most one bi-root {x,y} such that |x|+|y|<|w|. That is, the bi-root of a word is unique provided the word is sufficiently long with respect to the size (sum of lengths) of the root.Our results are also compared to previous approaches that investigate pseudo-repetitions, where a morphic involutive function θ is defined on A⁎. In this setting, the notions of θ-power, θ-primitive and θ-root are defined, and it is shown that any word has a unique θ-primitive root. This result can be obtained with our approach by showing that a word w is θ-primitive if and only if {w,θ(w)} is a primitive set.

Products in a Category with Only One Object

We consider certain decision problems for the free model of the theory of Cartesian monoids. We introduce a model of computation based on the notion of a single stack one-way PDA due to Ginsburg, Greibach and Harrison. This model allows us to solve problems such as (1) Given a finite set B of elements and an element F, is F a product of members of B? (2) Is the submonoid generated by the finite set B infinite? for certain fragments of the free Cartesian monoid. These fragments include the submonoid of right invertible elements and so our results apply to the Thompson-Higman groups.