47 research outputs found
Monoid presentations of groups by finite special string-rewriting systems
We show that the class of groups which
have monoid presentations by means of finite special
[λ]-confluent string-rewriting systems strictly contains the class of plain groups
(the groups which are free products of a finitely generated free
group and finitely many finite groups),
and that any group
which has an infinite cyclic central subgroup
can be presented by such a string-rewriting system if and only if it is the
direct product of an infinite cyclic group and a finite cyclic group
Some undecidability results concerning the property of preserving regularity
AbstractA finite string-rewriting system R preserves regularity if and only if it preserves Σ-regularity, where Σ is the alphabet containing exactly those letters that have occurrences in the rules of R. This proves a conjecture of Gyenizse and Vágvölgyi (1997). In addition, some undecidability results are presented that generalize results of Gilleron and Tison (1995) from term-rewriting systems to string-rewriting systems. It follows that the property of being regularity preserving is undecidable for term-rewriting systems, thus answering another question of Gyenizse and Vágvölgyi (1997). Finally, it is shown that it is undecidable in general whether a finite, lengthreducing, and confluent string-rewriting system yields a regular set of normal forms for each regular language
On Certain Subshifts and their Associated Monoids
Within a subclass of monoids (with zero) a structural characterization is
given of those that are associated to topologically transitive subshifts with
Property (A).Comment: 11 page
Partial monoids: associativity and confluence
A partial monoid is a set with a partial multiplication (and
total identity ) which satisfies some associativity axiom. The partial
monoid may be embedded in a free monoid and the product is
simulated by a string rewriting system on that consists in evaluating the
concatenation of two letters as a product in , when it is defined, and a
letter as the empty word . In this paper we study the profound
relations between confluence for such a system and associativity of the
multiplication. Moreover we develop a reduction strategy to ensure confluence
and which allows us to define a multiplication on normal forms associative up
to a given congruence of . Finally we show that this operation is
associative if, and only if, the rewriting system under consideration is
confluent
Thue systems for pattern recognition
This report presents a synoptic overview of Thue Systems. Thue Systems were introduced in the early 1900s by the Norwegian mathematician and logician Axel Thue. In this report the author suggests ways in which such systems can be used in pattern recognition.peer-reviewe
The Lamplighter Group
The Lamplighter Group is an Algebraic Group whose behavior models the dynamics of a geometric system. In this thesis, a survey paper following a set of notes as written by Professor Jennifer Taback of Bowdoin College, we define this geometric system and the connection to the Lamplighter Group. Several subsequent results are then proven regarding word length of elements in the group and behavior within its Cayley Graph. A preliminary section is included to introduce the reader to several topics including Free Groups, Group Presentation, and the properties of the Cayley Graph
A strong geometric hyperbolicity property for directed graphs and monoids
We introduce and study a strong "thin triangle"' condition for directed
graphs, which generalises the usual notion of hyperbolicity for a metric space.
We prove that finitely generated left cancellative monoids whose right Cayley
graphs satisfy this condition must be finitely presented with polynomial Dehn
functions, and hence word problems in NP. Under the additional assumption of
right cancellativity (or in some cases the weaker condition of bounded
indegree), they also admit algorithms for more fundamentally
semigroup-theoretic decision problems such as Green's relations L, R, J, D and
the corresponding pre-orders.
In contrast, we exhibit a right cancellative (but not left cancellative)
finitely generated monoid (in fact, an infinite class of them) whose Cayley
graph is a essentially a tree (hence hyperbolic in our sense and probably any
reasonable sense), but which is not even recursively presentable. This seems to
be strong evidence that no geometric notion of hyperbolicity will be strong
enough to yield much information about finitely generated monoids in absolute
generality.Comment: Exposition improved. Results unchange