721 research outputs found
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 . 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 , the set of
all subsets of an element set. Our set-up generalizes previous
groundbreaking work involving left-regular bands (or -trivial
bands) by Brown and Diaconis, extensions to -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
in terms of generators yields again a right Cayley graph. The McCammond
expansion provides normal forms for elements in the expanded . 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 and the
original . 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 -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 ,
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
We consider semigroup algorithmic problems in the Special Affine group
,
which is the group of affine transformations of the lattice 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 . We show that
both problems are decidable and NP-complete. Since , our result
extends that of Bell, Hirvensalo and Potapov (SODA 2017) on the NP-completeness
of both problems in , and contributes a first step
towards the open problems in .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
Algorithmic problems for subsemigroups of infinite groups
This thesis is concerned with several algorithmic problems for subsemigroups of infinite groups. The main objective is to construct algorithms that decide various properties of finitely generated subsemigroups of a given infinite group G (for example, a matrix group). Such problems might not be decidable in general. In fact, they gave rise to some of the earliest undecidability results in algorithmic theory. Nevertheless, when the group G admits additional structures, many algorithmic problems become decidable for its subsemigroups. In this thesis, we study the
decidability and the complexity of these algorithmic problems in the cases where G is nilpotent, metabelian, or represented as a low-dimensional matrix group.
Two of the main problems we consider are the Identity Problem and the Group Problem. Given a finite subset X of a group G, the Identity Problem asks whether the semigroup ⟨X⟩ generated by X contains the neutral element, and the Group Problem asks whether this semigroup is a group. We show that both problems are decidable in finitely generated nilpotent groups of class at most ten, and in PTIME if the input is given as unitriangular matrices. We also show the decidability of these problems in finitely generated metabelian groups, as well as their NP-completeness in the special affine group SA(2,Z).
Apart from the Identity Problem and the Group Problem, we also consider Semigroup Intersection and Orbit Intersection. Given two finite subsets X and H of the group G, Semigroup Intersection asks whether the semigroups ⟨X⟩, ⟨H⟩ generated by X and H have empty intersection, while Orbit Intersection asks whether the sets S · ⟨X⟩ and T · ⟨H⟩ intersect for given elements S, T ∈ G. We show that Semigroup Intersection is decidable in class two nilpotent groups (PTIME if the input is given as unitriangular matrices), and that Orbit Intersection is decidable in the Heisenberg group H3(Q).
We apply a large variety of mathematical tools in the study of these problems, ranging from Lie algebra, algebraic geometry and number theory, to combinatorics, graph theory and convex geometry. By adding these techniques into the toolbox, we are able to significantly advance the current state of art in the algorithmic theory of semigroups
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