Finite Complete Rewriting System Research Articles

Finite Complete Rewriting Systems for the Monoids M, ρ, and M/ρ

Let 𝑀 be a monoid and 𝜌 be an equivalence relation on 𝑀 such that 𝜌 is a congruence. So, 𝜌 is a submonoid of the direct product of monoids 𝑀×𝑀, and 𝑀/𝜌={𝑥𝜌:𝑥∈𝑀} is a monoid with the operation (𝑥𝜌)(𝑦𝜌)=(𝑥𝑦)𝜌. First, an introductory lemma is proposed, proved and a relevant example is given. Then, it is shown that if 𝜌 can be presented by a finite complete rewriting system, then so can 𝑀. As the final part of the main result, it is proved that if 𝜌 can be presented by a finite complete rewriting system, then so can 𝑀/𝜌.

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.

Tame filling invariants for groups

A new pair of asymptotic invariants for finitely presented groups, called intrinsic and extrinsic tame filling functions, is introduced. These filling functions are quasi-isometry invariants that strengthen the notions of intrinsic and extrinsic diameter functions for finitely presented groups. We show that the existence of a (finite-valued) tame filling functions implies that the group is tame combable. Bounds on both intrinsic and extrinsic tame filling functions are discussed for stackable groups, including groups with a finite complete rewriting system, Thompson's group F, and almost convex groups.

Finite Gröbner–Shirshov bases for Plactic algebras and biautomatic structures for Plactic monoids

This paper shows that every Plactic algebra of finite rank admits a finite Gröbner–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 FP∞. Also, answering a question of Zelmanov, we apply this rewriting system and other techniques to show that Plactic monoids of finite rank are biautomatic.

Algorithms and topology of Cayley graphs for groups

Autostackability for finitely generated groups is defined via a topological property of the associated Cayley graph which can be encoded in a finite state automaton. Autostackable groups have solvable word problem and an effective inductive procedure for constructing van Kampen diagrams with respect to a canonical finite presentation. A comparison with automatic groups is given. Another characterization of autostackability is given in terms of prefix-rewriting systems. Every group which admits a finite complete rewriting system or an asynchronously automatic structure with respect to a prefix-closed set of normal forms is also autostackable. As a consequence, the fundamental group of every closed 3-manifold with any of the eight possible uniform geometries is autostackable.

On finite complete rewriting systems and large subsemigroups

Let S be a semigroup and T be a subsemigroup of finite index in S (that is, the set S ∖ T is finite). The subsemigroup T is also called a large subsemigroup of S. It is well known that if T has a finite complete rewriting system, then so does S. In this paper, we will prove the converse, that is, if S has a finite complete rewriting system, then so does T. Our proof is purely combinatorial and also constructive.

On properties not inherited by monoids from their Schützenberger groups

We give an example of a monoid with finitely many left and right ideals, all of whose Schützenberger groups are presentable by finite complete rewriting systems, and so each have finite derivation type, but such that the monoid itself does not have finite derivation type, and therefore does not admit a presentation by a finite complete rewriting system. The example also serves as a counterexample to several other natural questions regarding complete rewriting systems and finite derivation type. Specifically it allows us to construct two finitely generated monoids M and N with isometric Cayley graphs, where N has finite derivation type (respectively, admits a presentation by a finite complete rewriting system) but M does not. This contrasts with the case of finitely generated groups for which finite derivation type is known to be a quasi-isometry invariant. The same example is also used to show that neither of these two properties is preserved under finite Green index extensions.

Finite complete rewriting systems for regular semigroups

It is proved that, given a (von Neumann) regular semigroup with finitely many left and right ideals, if every maximal subgroup is presentable by a finite complete rewriting system, then so is the semigroup. To achieve this, the following two results are proved: the property of being defined by a finite complete rewriting system is preserved when taking an ideal extension by a semigroup defined by a finite complete rewriting system; a completely 0-simple semigroup with finitely many left and right ideals admits a presentation by a finite complete rewriting system provided all of its maximal subgroups do.

FINITENESS CONDITIONS FOR REWRITING SYSTEMS

Monoids that can be presented by a finite complete rewriting system have both finite derivation type and finite homological type. This paper introduces a higher dimensional analogue of each of these invariants, and relates them to homological finiteness conditions.

Subgroups of finite index in groups with finite complete rewriting systems

AbstractWe show that if a group G has a finite complete rewriting system, and if H is a subgroup of G with |G : H| = n, then H * Fn–1 also has a finite complete rewriting system (where Fn–1 is the free group of rank n – 1).

Finite Complete Rewriting Systems and Finite Derivation Type for Small Extensions of Monoids

LetSbe a monoid and letTbe a submonoid of finite index inS. The main results in this article state thatScan be presented by a finite complete rewriting system ifTcan, andShas finite derivation type ifThas.

Constructing finitely presented monoids which have no finite complete presentation

Squier (1987) showed that if a monoid is defined by a finite complete rewriting system, then it satisfies the homological finiteness condition FP3, and using this fact he gave monoids (groups) which have solvable word problems but cannot be presented by finite complete systems. In the present paper we show that a monoid cannot have a finite complete presentation if it contains certain special elements. This observation enables us to construct monoids without finite complete presentation in a direct and elementary way. We give a finitely presented monoid which has (1) a word problem solvable in linear time and (2) linear growth but (3) no finite complete presentation. We also give a finitely presented monoid which has (1) a word problem solvable in linear time, (2) finite derivation type in the sense of Squier and (3) the property FP∞, but (4) no finite complete presentation.

Tame combings, almost convexity and rewriting systems for groups

A finite complete rewriting system for a group is a finite presentation which gives an algorithmic solution to the word problem. Finite complete rewriting systems have proven to be useful in geometric group theory, yet little is known about the geometry of groups admitting such rewriting systems. Here we indicate some of the geometry that is implicit in groups with various types of finite rewriting systems. For example, we show that any group admitting a finite complete rewriting system is tame 1-combable, and if the rewriting system is geodesic, then the group is almost convex. Several properties for finitely presented groups have been defined which can be used to show that a closed P2-irreducible three-manifold has universal cover homeomorphic to R3. For example, work of Poenaru [P] shows that if the fundamental group is infinite and satisfies Cannon’s almost convexity property, then the universal cover is simply connected at infinity, and hence is R3 (see [BT]). Casson later discovered the property C2, which regrettably is presentation dependent, but which also implies that the universal cover is R3 [S-G]. Brick and Mihalik generalized the condition C2 to the quasi-simply-filtered condition [B-M], which is independent of presentation and also implies the covering property. Later Mihalik and Tschantz [M-T] defined the notion of a tame 1-combing for a finitely presented group, which implies the quasi-simply-filtered condition, and showed that asynchronously automatic groups and semihyperbolic groups are tame 1-combable. Using a fairly geometric argument we show that groups with finite complete rewriting systems have tame 1-combings. Using similar techniques, we obtain

A new finiteness condition for monoids presented by complete rewriting systems (after Craig C. Squier)

Recently, Craig Squier introduced the notion of finite derivation type to show that some finitely presentable monoid has no presentation by means of a finite complete rewriting system. A similar result was already obtained by the same author using homology, but the new method is more direct and more powerful. Here, we present Squier's argument with a bit of categorical machinery, making proofs shorter and easier. In addition we prove that if a monoid has finite derivation type, then its third homology group is of finite type.

Rewriting systems for Coxeter groups

A finite complete rewriting system for a group is a finite presentation which gives a solution to the word problem and a regular language of normal forms for the group. In this paper it is shown that the fundamental group of an orientable closed surface of genus g has a finite complete rewriting system, using the usual generators a 1,…, a g, b 1,…, b g along with generators representing their inverses. Constructions of finite complete rewriting systems are also given for any Coxeter group G satisfying one of the following hypotheses. (1) G has three or fewer generators. (2) G does not contain a special subgroup of the form 〈 s i , s j , s k | s 2 i = s 2 j = s 2 k = ( s i s j ) 2 = ( s i s k ) m = ( s j s k ) n = 〈 with m and n both finite and not both equal to two.

Soluble groups with a finite rewriting system

We describe a class of soluble groups with a finite complete rewriting system which includes all the soluble groups known to have such a system. It is an open question, related to deep questions in the theory of groups, whether it includes all soluble groups with such a system.

The problems of cyclic equality and conjugacy for finite complete rewriting systems

It is known that the left-conjugacy problem and the conjugacy problem are decidable for each finite, length-reducing, and complete rewriting system. This does not hold for the problem of cyclic equality as is shown here by presenting a finite, length-reducing, and complete rewriting system with an undecidable problem of cyclic equality. Further, a finite complete rewriting system T is constructed such that the left-conjugacy problem and the conjugacy problem are undecidable for T.

N-level rewriting systems

It is well known that a monoid which is presented by a finite complete rewriting system has a decidable work problem. On the other hand, the problem whether each finitely presentable recursive monoid can be presented by a finite complete rewriting system is still unsolved. It is conjectured that this is not always possible. By generalising the rewriting systems to the n-level rewriting systems it is shown that exactly the finitely presentable recursive monoids can be presented by finite complete 2-level rewriting systems.

Finite complete rewriting systems and the complexity of the word problem

It is well known that the word problem for a finite complete rewriting system is decidable. Here it is shown that in general this result cannot be improved. This is done by proving that each sufficiently rich complexity class can be realized by the word problem for a finite complete rewriting system. Further, there is a gap between the complexity of the word problem for a finite complete rewriting system and the complexity of the least upper bound for the lengths of the chains generated by this rewriting system, and this gap can get arbitrarily large. Thus, the lengths of these chains do not give any information about the complexity of the word problem. Finally, it is shown that the property of allowing a finite complete rewriting system is not an invariant of finite monoid presentations.

Finite complete rewriting systems for the Jantzen monoid and the greendlinger group

It is shown that for the presentation ( a, b; abbaab = λ) of the Jantzen monoid J no finite complete rewriting system exist that is based on a Knuth-Bendix ordering. However, a finite complete rewriting system is given for a different presentation of J that has four generators. Further, a finite complete rewriting system is given for the presentation 〈 a, b, c; abc = cba〉 of the Greendlinger group G. This system induces a polynomial-time algorithm for the word problem for G.