Free Semigroup Research Articles

Bowen's equations for packing BS dimension of subsets for free semigroup actions

We introduce and study the relations between the packing BS dimension of an arbitrary subset and topological pressure for free semigroup actions. We show that the packing BS dimension is the unique root of the topological pressure function for semigroup actions. Also, we establish a variational principle between the packing BS dimension and the measure-theoretic BS dimension for an arbitrary non-invariant compact set of an action S .

Density of the Free Additive Convolution of Multi-cut Measures

Abstract We consider the free additive convolution semigroup $\lbrace \mu ^{\boxplus t}:\,t\ge 1\rbrace $ and determine the local behavior of the density of $\mu ^{\boxplus t}$ at the endpoints and at any singular point of its support. We then study the free additive convolution of two multi-cut probability measures and show that its density decays either as a square root or as a cubic root at any endpoint of its support. The probability measures considered in this paper satisfy a power law behavior with exponents strictly between $-1$ and $1$ at the endpoints of their supports.

Variational Principles for BS Dimension of Subsets for Free Semigroup Actions

Variational Principles for BS Dimension of Subsets for Free Semigroup Actions

Correlation Entropy of Free Semigroup Actions

Correlation Entropy of Free Semigroup Actions

Topological entropy of i.i.d. case and free semigroup actions

In this paper, firstly we give a new condition to prove that classical topological entropy is equal to Pesin topological entropy in i.i.d. case. After that, we generalize the concepts of topological entropy of free semigroup actions and study the relationship between the entropies of i.i.d. case and the entropies of free semigroup actions. Finally, for any non-empty subset A⊆Σm+, we introduce the concepts of entropies of any subset A actions and give an analogue of Befutov's formula (the Theorem 1 in [6]) under any subset A actions, which is a supplement of [7].

Shadowable Points of Free Semigroup Actions

Shadowable Points of Free Semigroup Actions

Metric Mean Dimension of Free Semigroup Actions for Non-Compact Sets

Metric Mean Dimension of Free Semigroup Actions for Non-Compact Sets

Monogenic free inverse semigroups and partial automorphisms of regular rooted trees

For a one-to-one partial mapping on an infinite set, we present a criterion in terms of its cycle-chain decomposition that the inverse subsemigroup generated by this mapping is monogenic free inverse.
 We also give a sufficient condition for a regular rooted tree partial automorphism to extend to a partial automorphism of another regular rooted tree so that the inverse semigroup gene\-ra\-ted by this extended partial automorphism is monogenic free inverse. The extension procedure we develop is then applied to $n$-ary adding machines.

Thompson’s semigroup and the first Hochschild cohomology

Abstract In this paper, we apply the theory of algebraic cohomology to study the amenability of Thompson’s group $\mathcal {F}$ . We introduce the notion of unique factorization semigroup which contains Thompson’s semigroup $\mathcal {S}$ and the free semigroup $\mathcal {F}_n$ on n ( $\geq 2$ ) generators. Let $\mathfrak {B}(\mathcal {S})$ and $\mathfrak {B}(\mathcal {F}_n)$ be the Banach algebras generated by the left regular representations of $\mathcal {S}$ and $\mathcal {F}_n$ , respectively. We prove that all derivations on $\mathfrak {B}(\mathcal {S})$ and $\mathfrak {B}(\mathcal {F}_n)$ are automatically continuous, and every derivation on $\mathfrak {B}(\mathcal {S})$ is induced by a bounded linear operator in $\mathcal {L}(\mathcal {S})$ , the weak-operator closed Banach algebra consisting of all bounded left convolution operators on $l^2(\mathcal {S})$ . Moreover, we prove that the first continuous Hochschild cohomology group of $\mathfrak {B}(\mathcal {S})$ with coefficients in $\mathcal {L}(\mathcal {S})$ vanishes. These conclusions provide positive indications for the left amenability of Thompson’s semigroup.

Specification and Shadowing Properties of Free Semigroup Actions

Specification and Shadowing Properties of Free Semigroup Actions

Chain Recurrence Rates and Topological Entropy of Free Semigroup Actions

Chain Recurrence Rates and Topological Entropy of Free Semigroup Actions

Novel DNA Numerical Mapping via Pseudo-Metric Free Semigroups for Machine Learning-Enabled Sequence Comparison Tasks

This article proposes a revolutionary idea that seamlessly blends algebraic structures and machine learning techniques. The primary goal is to explore the pseudo-metric free semigroup's structure because it has been potential to be used in more intriguing ways in the future.

The membership problem for subsemigroups of [formula omitted] is NP-complete

We show that the problem of determining if the identity matrix belongs to a finitely generated semigroup of 2×2 matrices from the modular group PSL2(Z), the Special Linear group SL2(Z) and the General Linear Group GL2(Z) is solvable in NP. We extend this to prove that the membership problem is decidable in NP for GL2(Z) and for any arbitrary regular expression over matrices from SL2(Z). We then derive that the problems of whether a given finite set of matrices from SL2(Z) or PSL2(Z) generates a group or a free semigroup are both decidable in NP. The previous algorithm for these problems, shown in 2005 by Choffrut and Karhumäki, was in EXPSPACE. Our algorithm is based on new techniques allowing us to operate on compressed word representations of matrices without explicit expansions. When combined with the known NP-hard lower bound, this proves that the identity (and thus membership) problem over GL2(Z) is NP-complete, and the group problem and the non-freeness problem in SL2(Z) are NP-complete. Thus the paper answer the long standing open question on the complexity of the membership problem in semigroups generated by matrices from GL2(Z). We develop novel techniques that can be used for solving numerical matrix problems in symbolic form, which are applicable for solving compressed word problems for groups and semigroups, bridging the gap between combinatorial group theory, computational problems on matrices and complexity theory.1

Variational principle for neutralized Bowen topological entropy on subsets of free semigroup actions

Variational principle for neutralized Bowen topological entropy on subsets of free semigroup actions

A Tits alternative for endomorphisms of the projective line

We prove an analog of the Tits alternative for endomorphisms of \mathbb{P}^1 . In particular, we show that if S is a finitely generated semigroup of endomorphisms of \mathbb{P}^1 over \mathbb{C} , then either S has polynomially bounded growth or S contains a nonabelian free semigroup. We also show that if f and g are polarizable maps over any field of any characteristic and \operatorname{Prep}(f) \not= \operatorname{Prep}(g) , then for all sufficiently large j , the semigroup \langle f^j, g^j \rangle is a free semigroup on two generators.

Universally free numerical semigroups

A numerical semigroup is said to be universally free if it is free for any possible arrangement of its minimal generating set. In this work, we establish that toric ideals associated with universally free numerical semigroups can be generated by their set of circuits. Additionally, we provide a characterization of universally free numerical semigroups in terms of Gröbner bases. Specifically, a numerical semigroup is universally free if and only if all initial ideals of its corresponding toric ideal are complete intersections. Furthermore, we establish several equalities among the toric bases of a universally free numerical semigroup.We provide a complete characterization of 3-generated universally free numerical semigroups in terms of their minimal generating sets, and by proving the equality of certain toric bases. We compute exactly all the toric bases of a toric ideal defined by a 3-generated universally free numerical semigroup. Notably, we answer some questions posed by Tatakis and Thoma by demonstrating that toric ideals defined by 3-generated universally free numerical semigroups have a set of circuits and a universal Gröbner basis of size 3, while the universal Markov basis and the Graver basis can be arbitrarily large. We present partial results and propose several conjectures regarding universally free numerical semigroups with more than three generators.

Equationally defined classes of semigroups

We apply, in the context of semigroups, the main theorem from the authors’ paper “Algebras defined by equations” (Higgins and Jackson in J Algebra 555:131–156, 2020) that an elementary class {mathscr {C}} of algebras which is closed under the taking of direct products and homomorphic images is defined by systems of equations. We prove a dual to the Birkhoff theorem in that if the class is also closed under the taking of containing semigroups, some basis of equations of {mathscr {C}} is free of the forall quantifier. We also observe the decidability of the class of equation systems satisfied by semigroups, via a link to systems of rationally constrained equations on free semigroups. Examples are given of EHP-classes for which neither (forall cdots )(exists cdots ) equation systems nor (exists cdots )(forall cdots ) systems suffice.

Regular semigroups weakly generated by one element

In this paper we study the regular semigroups weakly generated by a single element x, that is, with no proper regular subsemigroup containing x. We show there exists a regular semigroup F_{1} weakly generated by x such that all other regular semigroups weakly generated by x are homomorphic images of F_{1}. We define F_{1} using a presentation where both sets of generators and relations are infinite. Nevertheless, the word problem for this presentation is decidable. We describe a canonical form for the congruence classes given by this presentation, and explain how to obtain it. We end the paper studying the structure of F_{1}. In particular, we show that the ‘free regular semigroup {textrm{FI}}_2 weakly generated by two idempotents’ is isomorphic to a regular subsemigroup of F_{1} weakly generated by {xx',x'x}.

Publisher Correction to: Variational principle for BS-dimension of subsets of finitely generated free semigroup actions

Publisher Correction to: Variational principle for BS-dimension of subsets of finitely generated free semigroup actions

Variational principle for BS-dimension of subsets of finitely generated free semigroup actions

Variational principle for BS-dimension of subsets of finitely generated free semigroup actions