Unified theory for finite Markov chains

We provide a unified framework to compute the stationary distribution of any finite irreducible Markov chain or equivalently of any irreducible random walk on a finite semigroup $S$. Our methods use geometric finite semigroup theory via the Karnofsky-Rhodes and the McCammond expansions of finite semigroups with specified generators; this does not involve any linear algebra. The original Tsetlin library is obtained by applying the expansions to $P(n)$, the set of all subsets of an $n$ element set. Our set-up generalizes previous groundbreaking work involving left-regular bands (or $\mathscr{R}$-trivial bands) by Brown and Diaconis, extensions to $\mathscr{R}$-trivial semigroups by Ayyer, Steinberg, Thi\'ery and the second author, and important recent work by Chung and Graham. The Karnofsky-Rhodes expansion of the right Cayley graph of $S$ in terms of generators yields again a right Cayley graph. The McCammond expansion provides normal forms for elements in the expanded $S$. Using our previous results with Silva based on work by Berstel, Perrin, Reutenauer, we construct (infinite) semaphore codes on which we can define Markov chains. These semaphore codes can be lumped using geometric semigroup theory. Using normal forms and associated Kleene expressions, they yield formulas for the stationary distribution of the finite Markov chain of the expanded $S$ and the original $S$. Analyzing the normal forms also provides an estimate on the mixing time.Comment: 29 pages, 12 figures; v2: Section 3.2 added, references added, revision of introduction, title change; v3: typos fixed and clarifications adde

Almost overlap-free words and the word problem for the free Burnside semigroup satisfying x^2=x^3

In this paper we investigate the word problem of the free Burnside semigroup satisfying x^2=x^3 and having two generators. Elements of this semigroup are classes of equivalent words. A natural way to solve the word problem is to select a unique "canonical" representative for each equivalence class. We prove that overlap-free words and so-called almost overlap-free words (this notion is some generalization of the notion of overlap-free words) can serve as canonical representatives for corresponding equivalence classes. We show that such a word in a given class, if any, can be efficiently found. As a result, we construct a linear-time algorithm that partially solves the word problem for the semigroup under consideration.Comment: 33 pages, submitted to Internat. J. of Algebra and Compu

A proof of Zhil'tsov's theorem on decidability of equational theory of epigroups

Epigroups are semigroups equipped with an additional unary operation called pseudoinversion. Each finite semigroup can be considered as epigroup. We prove the following theorem announced by Zhil'tsov in 2000: the equational theory of the class of all epigroups coincides with the equational theory of the class of all finite epigroups and is decidable. We show that the theory is not finitely based but provide a transparent infinite basis for it.Comment: 32 page

Equilibrium states on right LCM semigroup C*-algebras

We determine the structure of equilibrium states for a natural dynamics on the boundary quotient diagram of $C^*$-algebras for a large class of right LCM semigroups. The approach is based on abstract properties of the semigroup and covers the previous case studies on $\mathbb{N} \rtimes \mathbb{N}^\times$, dilation matrices, self-similar actions, and Baumslag-Solitar monoids. At the same time, it provides new results for large classes of right LCM semigroups, including those associated to algebraic dynamical systems.Comment: 43 pages, to appear in Int. Math. Res. No

The linear nature of pseudowords

Given a pseudoword over suitable pseudovarieties, we associate to it a labeled linear order determined by the factorizations of the pseudoword. We show that, in the case of the pseudovariety of aperiodic finite semigroups, the pseudoword can be recovered from the labeled linear order.The work of the first, third, and fourth authors was partly supported by the Pessoa French-Portuguese project “Separation in automata theory: algebraic, logical, and combinatorial aspects”. The work of the first three authors was also partially supported respectively by CMUP (UID/MAT/ 00144/2019), CMUC (UID/MAT/00324/2019), and CMAT (UID/MAT/ 00013/2013), which are funded by FCT (Portugal) with national (MCTES) and European structural funds (FEDER), under the partnership agreement PT2020. The work of the fourth author was partly supported by ANR 2010 BLAN 0202 01 FREC and by the DeLTA project ANR-16-CE40-000

Groups of Fibonacci type revisited

This article concerns a class of groups of Fibonacci type introduced by Johnson and Mawdesley that includes Conway?s Fibonacci groups, the Sieradski groups, and the Gilbert-Howie groups. This class of groups provides an interesting focus for developing the theory of cyclically presented groups and, following questions by Bardakov and Vesnin and by Cavicchioli, Hegenbarth, and Repov?s, they have enjoyed renewed interest in recent years. We survey results concerning their algebraic properties, such as isomorphisms within the class, the classification of the finite groups, small cancellation properties, abelianizations, asphericity, connections with Labelled Oriented Graph groups, and the semigroups of Fibonacci type. Further, we present a new method of proving the classification of the finite groups that deals with all but three groups

The identity problem in the special affine group of Z2

We consider semigroup algorithmic problems in the Special Affine group SA(2,Z)=Z2⋊SL(2,Z), which is the group of affine transformations of the lattice Z2 that preserve orientation. Our paper focuses on two decision problems introduced by Choffrut and Karhumäki (2005): the Identity Problem (does a semigroup contain a neutral element?) and the Group Problem (is a semigroup a group?) for finitely generated sub-semigroups of SA(2,Z). We show that both problems are decidable and NP-complete. Since SL(2,Z)≤SA(2,Z)≤SL(3,Z), our result extends that of Bell, Hirvensalo and Potapov (SODA 2017) on the NP-completeness of both problems in SL(2,Z), and contributes a first step towards the open problems in SL(3,Z)

The Identity Problem in the special affine group of Z2\mathbb{Z}^2

We consider semigroup algorithmic problems in the Special Affine group $\mathsf{SA}(2, \mathbb{Z}) = \mathbb{Z}^2 \rtimes \mathsf{SL}(2, \mathbb{Z})$, which is the group of affine transformations of the lattice $\mathbb{Z}^2$ that preserve orientation. Our paper focuses on two decision problems introduced by Choffrut and Karhum\"{a}ki (2005): the Identity Problem (does a semigroup contain a neutral element?) and the Group Problem (is a semigroup a group?) for finitely generated sub-semigroups of $\mathsf{SA}(2, \mathbb{Z})$. We show that both problems are decidable and NP-complete. Since $\mathsf{SL}(2, \mathbb{Z}) \leq \mathsf{SA}(2, \mathbb{Z}) \leq \mathsf{SL}(3, \mathbb{Z})$, our result extends that of Bell, Hirvensalo and Potapov (SODA 2017) on the NP-completeness of both problems in $\mathsf{SL}(2, \mathbb{Z})$, and contributes a first step towards the open problems in $\mathsf{SL}(3, \mathbb{Z})$.Comment: 17 pages, 10 figure

Factoriality and the pin-reutenauer procedure

We consider implicit signatures over finite semigroups determined by sets of pseudonatural numbers. We prove that, under relatively simple hypotheses on a pseudovariety V of semigroups, the finitely generated free algebra for the largest such signature is closed under taking factors within the free pro-V semigroup on the same set of generators. Furthermore, we show that the natural analogue of the Pin-Reutenauer descriptive procedure for the closure of a rational language in the free group with respect to the profinite topology holds for the pseudovariety of all finite semigroups. As an application, we establish that a pseudovariety enjoys this property if and only if it is full.European structural funds (FEDER)European Regional Development Fund, through the program COMPET