48 research outputs found
Automatic structures for semigroup constructions
We survey results concerning automatic structures for semigroup
constructions, providing references and describing the corresponding automatic
structures. The constructions we consider are: free products, direct products,
Rees matrix semigroups, Bruck-Reilly extensions and wreath products.Comment: 22 page
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
Automatic presentations for semigroups
Special Issue: 2nd International Conference on Language and Automata Theory and Applications (LATA 2008)This paper applies the concept of FA-presentable structures to semigroups. We give a complete classification of the finitely generated FA-presentable cancellative semigroups: namely, a finitely generated cancellative semigroup is FA-presentable if and only if it is a subsemigroup of a virtually abelian group. We prove that all finitely generated commutative semigroups are FA-presentable. We give a complete list of FA-presentable one-relation semigroups and compare the classes of FA-presentable semigroups and automatic semigroups. (C) 2009 Elsevier Inc. All rights reserved.PostprintPeer reviewe
On finite complete rewriting systems, finite derivation type, and automaticity for homogeneous monoids
This paper investigates the class of finitely presented monoids defined by homogeneous (length-preserving) relations from a computational perspective. The properties of admitting a finite complete rewriting system, having finite derivation type, being automatic, and being biautomatic are investigated for this class of monoids. The first main result shows that for any consistent combination of these properties and their negations, there is a homogeneous monoid with exactly this combination of properties. We then introduce the new concept of abstract Rees-commensurability (an analogue of the notion of abstract commensurability for groups) in order to extend this result to show that the same statement holds even if one restricts attention to the class of n-ary homogeneous monoids (where every side of every relation has fixed length n). We then introduce a new encoding technique that allows us to extend the result partially to the class of n-ary multihomogenous monoids
Uniform decision problems in automatic semigroups
We consider various decision problems for automatic semigroups, which involve
the provision of an automatic structure as part of the problem instance. With
mild restrictions on the automatic structure, which seem to be necessary to
make the problem well-defined, the uniform word problem for semigroups
described by automatic structures is decidable. Under the same conditions, we
show that one can also decide whether the semigroup is completely simple or
completely zero-simple; in the case that it is, one can compute a Rees matrix
representation for the semigroup, in the form of a Rees matrix together with an
automatic structure for its maximal subgroup. On the other hand, we show that
it is undecidable in general whether a given element of a given automatic
monoid has a right inverse.Comment: 19 page
Finite Gr\"obner--Shirshov bases for Plactic algebras and biautomatic structures for Plactic monoids
This paper shows that every Plactic algebra of finite rank admits a finite
Gr\"obner--Shirshov basis. The result is proved by using the combinatorial
properties of Young tableaux to construct a finite complete rewriting system
for the corresponding Plactic monoid, which also yields the corollaries that
Plactic monoids of finite rank have finite derivation type and satisfy the
homological finiteness properties left and right . Also, answering a
question of Zelmanov, we apply this rewriting system and other techniques to
show that Plactic monoids of finite rank are biautomatic.Comment: 16 pages; 3 figures. Minor revision: typos fixed; figures redrawn;
references update
Crystal monoids & crystal bases: rewriting systems and biautomatic structures for plactic monoids of types An, Bn, Cn, Dn, and G2
The vertices of any (combinatorial) Kashiwara crystal graph carry a natural monoid structure given by identifying words labelling vertices that appear in the same position of isomorphic components of the crystal. Working on a purely combinatorial and monoid-theoretical level, we prove some foundational results for these crystal monoids, including the observation that they have decidable word problem when their weight monoid is a finite rank free abelian group. The problem of constructing finite complete rewriting systems, and biautomatic structures, for crystal monoids is then investigated. In the case of Kashiwara crystals of types An, Bn, Cn, Dn, and G2 (corresponding to the q-analogues of the Lie algebras of these types) these monoids are precisely the generalised plactic monoids investigated in work of Lecouvey. We construct presentations via finite complete rewriting systems for all of these types using a unified proof strategy that depends on Kashiwara's crystal bases and analogies of Young tableaux, and on Lecouvey's presentations for these monoids. As corollaries, we deduce that plactic monoids of these types have finite derivation type and satisfy the homological finiteness properties left and right FP∞. These rewriting systems are then applied to show that plactic monoids of these types are biautomatic and thus have word problem soluble in quadratic time
On finite complete rewriting systems, finite derivation type, and automaticity for homogeneous monoids
This paper investigates the class of finitely presented monoids defined by homogeneous (length-preserving) relations from a computational perspective. The properties of admitting a finite complete rewriting system, having finite derivation type, being automatic, and being biautomatic are investigated for this class of monoids. The first main result shows that for any consistent combination of these properties and their negations, there is a homogeneous monoid with exactly this combination of properties. We then introduce the new concept of abstract Rees-commensurability (an analogue of the notion of abstract commensurability for groups) in order to extend this result to show that the same statement holds even if one restricts attention to the class of n-ary homogeneous monoids (where every side of every relation has fixed length n). We then introduce a new encoding technique that allows us to extend the result partially to the class of n-ary multihomogenous monoids
Automatic semigroups : constructions and subsemigroups
In this thesis we start by considering conditions under which some standard semigroup constructions preserve automaticity. We first consider Rees matrix semigroups over a semigroup, which we call the base, and work on the following questions: (i) If the base is automatic is the Rees matrix semigroup automatic? (ii) If the Rees matrix semigroup is automatic must the base be automatic as well? We also consider similar questions for Bruck-Reilly extensions of monoids and wreath products of semigroups. Then we consider subsemigroups of free products of semigroups and we study conditions that guarantee them to be automatic. Finally we obtain a description of the subsemigroups of the bicyclic monoid that allow us to study some of their properties, which include finite generation, automaticity and finite presentability