Free Monoid Research Articles

Nyldon words

The Chen-Fox-Lyndon theorem states that every finite word over a fixed alphabet can be uniquely factorized as a lexicographically nonincreasing sequence of Lyndon words. This theorem can be used to define the family of Lyndon words in a recursive way. If the lexicographic order is reversed in this definition, we obtain a new family of words, which are called the Nyldon words. In this paper, we show that every finite word can be uniquely factorized into a lexicographically nondecreasing sequence of Nyldon words. Otherwise stated, Nyldon words form a complete factorization of the free monoid with respect to the decreasing lexicographic order. Then we investigate this new family of words. In particular, we show that Nyldon words form a right Lazard set.

Bi-interpretability of some monoids with the arithmetic and applications

We will prove bi-interpretability of the arithmetic $${\mathbb {N}}= \langle N, +,\cdot , 0, 1\rangle $$ and the weak second order theory of $${\mathbb {N}}$$ with the free monoid $$\mathbb {M}_X$$ of finite rank greater than 1 and with a non-trivial partially commutative monoid with trivial center. This bi-interpretability implies that finitely generated submonoids of these monoids are definable. Moreover, any recursively enumerable language in the alphabet X is definable in $$\mathbb {M}_X$$ . Primitive elements, and, therefore, free bases are definable in the free monoid. It has the so-called QFA property, namely there is a sentence $$\phi $$ such that every finitely generated monoid satisfying $$\phi $$ is isomorphic to $$\mathbb {M}_X$$ . The same is true for a partially commutative monoid without center. We also prove that there is no quantifier elimination in the theory of any structure that is bi-interpretable with $$\mathbb {N}$$ to any boolean combination of formulas from $$\Pi _n$$ or $$\Sigma _n$$ .

A characterization of free pairs of upper triangular free monoid morphisms

We study combinatorics on morphisms. More precisely, we study free monoid morphisms and their freeness properties. By definition, a set F of endomorphisms of a free monoid is free, if every product formed of morphims from F can be uniquely factorized. We show that if f and g are endomorphisms of the free monoid A⁎ having upper triangular incidence matrices with diagonal entries at least two, then the pair {f,g} is free if and only if f and g do not commute. This result implies that for such pairs freeness is very easy to decide.

Topos points of quasi-coherent sheaves over monoid schemes

AbstractLet X be a monoid scheme. We will show that the stalk at any point of X defines a point of the topos of quasi-coherent sheaves over X. As it turns out, every topos point of is of this form if X satisfies some finiteness conditions. In particular, it suffices for M/M× to be finitely generated when X is affine, where M× is the group of invertible elements.This allows us to prove that two quasi-projective monoid schemes X and Y are isomorphic if and only if and are equivalent.The finiteness conditions are essential, as one can already conclude by the work of A. Connes and C. Consani [3]. We will study the topos points of free commutative monoids and show that already for ℕ∞, there are ‘hidden’ points. That is to say, there are topos points which are not coming from prime ideals. This observation reveals that there might be a more interesting ‘geometry of monoids’.

A free monoid containing all strongly Bi-singular languages and non-primitive words

ABSTRACTLet , where , be the set of ith powers of primitive words. A language is called strongly bi-singular if the minimal-length words in it are neither prefixes nor suffixes of any other word in the language. Strongly bi-singular languages forms a free monoid with respect to the concatenation of languages. The main result of this paper is that if we start with the basis of this free monoid together with the languages for all , then the resulting family of languages is a code. So we find a free monoid which properly contains the free monoid of all strongly bi-singular languages.

The Generalized Rank of Trace Languages

Given a partially commutative alphabet and a set of words [Formula: see text], the rank of [Formula: see text] expresses the amount of shuffling required to produce a word belonging to [Formula: see text] from two words whose concatenation belongs to the closure of [Formula: see text] with respect to the partial commutation. In this paper, the notion of rank is generalized from concatenations of two words to an arbitrary fixed number of words. In this way, an infinite sequence of non-negative integers and infinity is assigned to every set of words. It is proved that in the case of alphabets defining free commutative monoids, as well as in the more general case of direct products of free monoids, sequences of ranks of regular sets are exactly non-decreasing sequences that are eventually constant. On the other hand, by uncovering a relationship between rank sequences of regular sets and rational series over the min-plus semiring, it is shown that already for alphabets defining free products of free commutative monoids, rank sequences need not be eventually periodic.

On generalized Lyndon words

A generalized lexicographical order on infinite words is defined by choosing for each position a total order on the alphabet. This allows to define generalized Lyndon words. Every word in the free monoid can be factorized in a unique way as a nonincreasing factorization of generalized Lyndon words. We give new characterizations of the first and the last factor in this factorization as well as new characterization of generalized Lyndon words. We also give more specific results on two special cases: the classical one and the one arising from the alternating lexicographical order.

The automorphism group of a shift of slow growth is amenable

Suppose $(X,\unicode[STIX]{x1D70E})$ is a subshift, $P_{X}(n)$ is the word complexity function of $X$, and $\text{Aut}(X)$ is the group of automorphisms of $X$. We show that if $P_{X}(n)=o(n^{2}/\log ^{2}n)$, then $\text{Aut}(X)$ is amenable (as a countable, discrete group). We further show that if $P_{X}(n)=o(n^{2})$, then $\text{Aut}(X)$ can never contain a non-abelian free monoid (and, in particular, can never contain a non-abelian free subgroup). This is in contrast to recent examples, due to Salo and Schraudner, of subshifts with quadratic complexity that do contain such a monoid.

Finiteness Spaces and Generalized Power Series

We consider Ribenboim's construction of rings of generalized power series. Ribenboim's construction makes use of a special class of partially ordered monoids and a special class of their subsets. While the restrictions he imposes might seem conceptually unclear, we demonstrate that they are precisely the appropriate conditions to represent such monoids as internal monoids in an appropriate category of Ehrhard's finiteness spaces. Ehrhard introduced finiteness spaces as the objects of a categorical model of classical linear logic, where a set is equipped with a class of subsets to be thought of as finitary. Morphisms are relations preserving the finitary structure. The notion of finitary subset allows for a sharper analysis of computational structure than is available in the relational model. For example, fixed point operators fail to be finitary. In the present work, we take morphisms to be partial functions preserving the finitary structure rather than relations. The resulting category is symmetric monoidal closed, complete and cocomplete. Any pair of an internal monoid in this category and a ring induces a ring of generalized power series by an extension of the Ribenboim construction based on Ehrhard's notion of linearization of a finiteness space. We thus further generalize Ribenboim's constructions. We give several examples of rings which arise from this construction, including the ring of Puiseux series and the ring of formal power series generated by a free monoid.

Relations between counting functions on free groups and free monoids

We study counting functions on the free groups F_n and free monoids M_n for n \geq 2 , which we introduce for combinatorial approach to famous Brooks quasimorphisms on free groups. Two counting functions are considered equivalent if they differ by a bounded function. We find the complete set of linear relations between equivalence classes of counting functions and apply this result to construct an explicit basis for the vector space of such equivalence classes. Moreover, we provide a simple graphical algorithm to determine whether two given counting functions are equivalent. In particular, this yields an algorithm to decide whether two linear combinations of Brooks quasimorphisms on F_n represent the same class in bounded cohomology.

On finitely generated submonoids of virtually free groups

AbstractWe prove that it is decidable whether or not a finitely generated submonoid of a virtually free group is graded, introduce a new geometric characterization of graded submonoids in virtually free groups as quasi-geodesic submonoids, and show that their word problem is rational (as a relation). We also solve the isomorphism problem for this class of monoids, generalizing earlier results for submonoids of free monoids. We also prove that the classes of graded monoids, regular monoids and Kleene monoids coincide for submonoids of free groups.

Ehresmann monoids: Adequacy and expansions

It is known that an Ehresmann monoid P(T,Y) may be constructed from a monoid T acting via order-preserving maps on both sides of a semilattice Y with identity, such that the actions satisfy an appropriate compatibility criterion. Our main result shows that if T is cancellative and equidivisible (as is the case for the free monoid X⁎), the monoid P(T,Y) not only is Ehresmann but also satisfies the stronger property of being adequate.Fixing T, Y and the actions, we characterise P(T,Y) as being unique in the sense that it is the initial object in a suitable category of Ehresmann monoids. We also prove that the operator P defines an expansion of Ehresmann monoids.

A Syntactic Approach to the MacNeille Completion of Λ∗, the Free Monoid Over an Ordered Alphabet Λ

Let Λ∗ be the free monoid of (finite) words over a not necessarily finite alphabet Λ, which is equipped with some (partial) order. This ordering lifts to Λ∗, where it extends the divisibility ordering of words. The MacNeille completion of Λ∗ constitutes a complete lattice ordered monoid and is realized by the system of “closed” lower sets in Λ∗ (ordered by inclusion) or its isomorphic copy formed of the “closed” upper sets (ordered by reverse inclusion). Under some additional hypothesis on Λ, one can easily identify the closed lower sets as the finitely generated ones, whereas it is more complicated to determine the closed upper sets. For a fairly large class of ordered sets Λ (including complete lattices as well as antichains) one can generate the closure of any upper set of words by means of binary operations (“syntactic rules”) thus obtaining an efficient procedure to test closedness. Closed upper sets of words are involved in an embedding theorem for valuated oriented graphs. In fact, generalized paths (so-called “zigzags”) are encoded by words over an alphabet Λ. Then the valuated oriented graphs which are “isometrically” embeddable in a product of zigzags have the characteristic property that the words corresponding to the zigzags between any pair of vertices form a closed upper set in Λ∗.

Gcd-monoids arising from homotopy groupoids

The interval monoid Υ(P) of a poset P is defined by generators [x, y], where x ≤ y in P , and relations [x, x] = 1, [x, z] = [x, y] · [y, z] for x ≤ y ≤ z. It embeds into its universal group Υ ± (P), the interval group of P , which is also the universal group of the homotopy groupoid of the chain complex of P. We prove the following results: • The monoid Υ(P) has finite left and right greatest common divisors of pairs (we say that it is a gcd-monoid) iff every principal ideal (resp., filter) of P is a join-semilattice (resp., a meet-semilattice). • For every group G, there is a poset P of length 2 such that Υ(P) is a gcd-monoid and G is a free factor of Υ ± (P) by a free group. Moreover, P can be taken finite iff G is finitely presented. • For every finite poset P , the monoid Υ(P) can be embedded into a free monoid. • Some of the results above, and many related ones, can be extended from interval monoids to the universal monoid Umon(S) of any category S. This enables us, in particular, to characterize the embeddability of Umon(S) into a group, by stating that it holds at the hom-set level. We thus obtain new easily verified sufficient conditions for embeddability of a monoid into a group. We illustrate our results by various examples and counterexamples.

The Hopf algebra of skew shapes, torsion sheaves on [formula omitted], and ideals in Hall algebras of monoid representations

We study ideals in Hall algebras of monoid representations on pointed sets corresponding to certain conditions on the representations. These conditions include the property that the monoid act via partial permutations, that the representation possess a compatible grading, and conditions on the support of the module. Quotients by these ideals lead to combinatorial Hopf algebras which can be interpreted as Hall algebras of certain sub-categories of modules. In the case of the free commutative monoid on n generators, we obtain a co-commutative Hopf algebra structure on n-dimensional skew shapes, whose underlying associative product amounts to a “stacking” operation on the skew shapes. The primitive elements of this Hopf algebra correspond to connected skew shapes, and form a graded Lie algebra by anti-symmetrizing the associative product. We interpret this Hopf algebra as the Hall algebra of a certain category of coherent torsion sheaves on A/F1n supported at the origin, where F1 denotes the field of one element. This Hopf algebra may be viewed as an n-dimensional generalization of the Hopf algebra of symmetric functions, which corresponds to the case n=1.

Substitutions, coding prescriptions and integer representation

Coding prescriptions are combinatorial objects linked to a substitution, that is a morphism of the free monoid. Originally they have been introduced in order to code the induced symbolic dynamical system. In the present article we are interested in coding prescriptions of compositions and powers of substitutions. This will provide a very general framework for representing integers. We will study the properties and find several relations with well-known systems of integer numeration.

Equivalence of Deterministic Top-Down Tree-to-String Transducers Is Decidable

We prove that equivalence of deterministic top-down tree-to-string transducers is decidable, thus solving a long-standing open problem in formal language theory. We also present efficient algorithms for subclasses: for linear transducers or total transducers with unary output alphabet (over a given top-down regular domain language), as well as for transducers with the single-use restriction. These results are obtained using techniques from multi-linear algebra. For our main result, we introduce polynomial transducers and prove that for these, validity of a polynomial invariant can be certified by means of an inductive invariant of polynomial ideals. This allows us to construct two semi-algorithms, one searching for a certificate of the invariant and one searching for a witness of its violation. Via a translation into polynomial transducers, we thus obtain that equivalence of general y dt transducers is decidable. In fact, our translation also shows that equivalence is decidable when the output is not in a free monoid but in a free group.

Free monoids and generalized metric spaces

Let A be an ordered alphabet, A∗ be the free monoid over A ordered by the Higman ordering, and let F(A∗) be the set of final segments of A∗. With the operation of concatenation, this set is a monoid. We show that the submonoid F∘(A∗)≔F(A∗)∖{0̸} is free. The MacNeille completion N(A∗) of A∗ is a submonoid of F(A∗). As a corollary, we obtain that the monoid N∘(A∗)≔N(A∗)∖{0̸} is free. We give an interpretation of the freeness of F∘(A∗) in the category of metric spaces over the Heyting algebra V≔F(A∗), with the non-expansive mappings as morphisms. Each final segment F of A∗ yields the injective envelope SF of a two-element metric space over V. The uniqueness of the decomposition of F is due to the uniqueness of the block decomposition of the graph GF associated to this injective envelope.

Idempotent elements and uniquely clean property of skew monoid rings

Let R be a ring with an endomorphism σ and F ∪ {0} the free monoid generated by U = {u1, ..., ut} with 0 added, and M = F ∪ {0}/(I) where I is the set of certain monomial in U such that Mn = 0, for some n. Then we can form the non-semiprime skew monoid ring R[M; σ]. An element a ∈ R is uniquely strongly clean if a has a unique expression as a = e + u, where e is an idempotent and u is a unit with ea = ae. We show that a σ-compatible ring R is uniquely clean if and only if R[M; σ] is a uniquely clean ring. If R is strongly π-regular and uniquely strongly clean, then R[M; σ] is uniquely strongly clean. It is also shown that idempotents of R[M; σ] (and hence the ring R[x; σ]=(xn)) are conjugate to idempotents of R and we apply this to show that R[M; σ] over a projective-free ring R is projective-free. It is also proved that if R is semi-abelian and σ(e) = e for each idempotent e ∈ R, then R[M; σ] is a semi-abelian ring.

Recognizability for sequences of morphisms

We investigate different notions of recognizability for a free monoid morphism $\unicode[STIX]{x1D70E}:{\mathcal{A}}^{\ast }\rightarrow {\mathcal{B}}^{\ast }$ . Full recognizability occurs when each (aperiodic) point in ${\mathcal{B}}^{\mathbb{Z}}$ admits at most one tiling with words $\unicode[STIX]{x1D70E}(a)$ , $a\in {\mathcal{A}}$ . This is stronger than the classical notion of recognizability of a substitution $\unicode[STIX]{x1D70E}:{\mathcal{A}}^{\ast }\rightarrow {\mathcal{A}}^{\ast }$ , where the tiling must be compatible with the language of the substitution. We show that if $|{\mathcal{A}}|=2$ , or if $\unicode[STIX]{x1D70E}$ ’s incidence matrix has rank $|{\mathcal{A}}|$ , or if $\unicode[STIX]{x1D70E}$ is permutative, then $\unicode[STIX]{x1D70E}$ is fully recognizable. Next we investigate the classical notion of recognizability and improve earlier results of Mossé [Puissances de mots et reconnaissabilité des points fixes d’une substitution. Theoret. Comput. Sci. 99(2) (1992), 327–334] and Bezuglyi et al [Aperiodic substitution systems and their Bratteli diagrams. Ergod. Th. & Dynam. Sys. 29(1) (2009), 37–72], by showing that any substitution is recognizable for aperiodic points in its substitutive shift. Finally we define recognizability and also eventual recognizability for sequences of morphisms which define an $S$ -adic shift. We prove that a sequence of morphisms on alphabets of bounded size, such that compositions of consecutive morphisms are growing on all letters, is eventually recognizable for aperiodic points. We provide examples of eventually recognizable, but not recognizable, sequences of morphisms, and sequences of morphisms which are not eventually recognizable. As an application, for a recognizable sequence of morphisms, we obtain an almost everywhere bijective correspondence between the $S$ -adic shift it generates, and the measurable Bratteli–Vershik dynamical system that it defines.