327 research outputs found
On the insertion of n-powers
In algebraic terms, the insertion of -powers in words may be modelled at
the language level by considering the pseudovariety of ordered monoids defined
by the inequality . We compare this pseudovariety with several other
natural pseudovarieties of ordered monoids and of monoids associated with the
Burnside pseudovariety of groups defined by the identity . In
particular, we are interested in determining the pseudovariety of monoids that
it generates, which can be viewed as the problem of determining the Boolean
closure of the class of regular languages closed under -power insertions. We
exhibit a simple upper bound and show that it satisfies all pseudoidentities
which are provable from in which both sides are regular elements
with respect to the upper bound
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
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