scholarly journals An algebraic characterization of groups with soluble word problem

1974 ◽  
Vol 18 (1) ◽  
pp. 41-53 ◽  
Author(s):  
William W. Boone ◽  
Graham Higman
Keyword(s):  
Word Problem ◽  
Focal Point ◽  
Algebraic Characterization ◽  
Necessary And Sufficient Condition ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Finitely Presented Group ◽  
Necessary And Sufficient ◽  
Finitely Presented

The following theorem is the focal point of the present paper. It stipulates an algebraic condition equivalent, in any finitely generated group, to the solubility of the word problem.THEOREM I. A necessary and sufficient condition that a finitely generated group G have a soluble word problem is that there exist a simple group H, and a finitely presented group K, such that G is a subgroup of H, and H is a subgroup of K.

scholarly journals Calibrating word problems of groups via the complexity of equivalence relations

2016 ◽  
Vol 28 (3) ◽  
pp. 457-471 ◽  
Author(s):  
ANDRÉ NIES ◽  
ANDREA SORBI
Keyword(s):  
Word Problem ◽  
Equivalence Relation ◽  
Word Problems ◽  
Truth Table ◽  
Equivalence Relations ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Finitely Presented Group ◽  
Finitely Presented ◽  
Computably Enumerable

(1) There is a finitely presented group with a word problem which is a uniformly effectively inseparable equivalence relation. (2) There is a finitely generated group of computable permutations with a word problem which is a universal co-computably enumerable equivalence relation. (3) Each c.e. truth-table degree contains the word problem of a finitely generated group of computable permutations.

Context-sensitive languages and G-automata

2017 ◽  
Vol 27 (02) ◽  
pp. 237-249
Author(s):  
Rachel Bishop-Ross ◽  
Jon M. Corson ◽  
James Lance Ross
Keyword(s):  
Word Problem ◽  
Finitely Generated ◽  
Closure Properties ◽  
Finitely Generated Group ◽  
Context Sensitive ◽  
The Family

For a given finitely generated group [Formula: see text], the type of languages that are accepted by [Formula: see text]-automata is determined by the word problem of [Formula: see text] for most of the classical types of languages. We observe that the only exceptions are the families of context-sensitive and recursive languages. Thus, in general, to ensure that the language accepted by a [Formula: see text]-automaton is in the same classical family of languages as the word problem of [Formula: see text], some restriction must be imposed on the [Formula: see text]-automaton. We show that restricting to [Formula: see text]-automata without [Formula: see text]-transitions is sufficient for this purpose. We then define the pullback of two [Formula: see text]-automata and use this construction to study the closure properties of the family of languages accepted by [Formula: see text]-automata without [Formula: see text]-transitions. As a further consequence, when [Formula: see text] is the product of two groups, we give a characterization of the family of languages accepted by [Formula: see text]-automata in terms of the families of languages accepted by [Formula: see text]- and [Formula: see text]-automata. We also give a construction of a grammar for the language accepted by an arbitrary [Formula: see text]-automaton and show how to get a context-sensitive grammar when [Formula: see text] is finitely generated with a context-sensitive word problem and the [Formula: see text]-automaton is without [Formula: see text]-transitions.

scholarly journals ON GENERATORS AND PRESENTATIONS OF SEMIDIRECT PRODUCTS IN INVERSE SEMIGROUPS

2009 ◽  
Vol 79 (3) ◽  
pp. 353-365 ◽  
Author(s):  
E. R. DOMBI ◽  
N. RUŠKUC
Keyword(s):  
Semidirect Product ◽  
Inverse Semigroups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finitely Generated ◽  
Semidirect Products ◽  
Necessary And Sufficient ◽  
Finitely Presented

AbstractIn this paper we prove two main results. The first is a necessary and sufficient condition for a semidirect product of a semilattice by a group to be finitely generated. The second result is a necessary and sufficient condition for such a semidirect product to be finitely presented.

scholarly journals Bounded homomorphisms and finitely generated fiber products of lattices

2020 ◽  
Vol 30 (04) ◽  
pp. 693-710
Author(s):  
William DeMeo ◽  
Peter Mayr ◽  
Nik Ruškuc
Keyword(s):  
Word Problem ◽  
Time Algorithm ◽  
Finitely Generated ◽  
Bounded Lattice ◽  
Exponential Time ◽  
Unpublished Result ◽  
Lattice Homomorphisms ◽  
Finitely Presented ◽  
Exponential Time Algorithm

We investigate when fiber products of lattices are finitely generated and obtain a new characterization of bounded lattice homomorphisms onto lattices satisfying a property we call Dean’s condition (D) which arises from Dean’s solution to the word problem for finitely presented lattices. In particular, all finitely presented lattices and those satisfying Whitman’s condition satisfy (D). For lattice epimorphisms [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text] are finitely generated and [Formula: see text] satisfies (D), we show the following: If [Formula: see text] and [Formula: see text] are bounded, then their fiber product (pullback) [Formula: see text] is finitely generated. While the converse is not true in general, it does hold when [Formula: see text] and [Formula: see text] are free. As a consequence, we obtain an (exponential time) algorithm to decide boundedness for finitely presented lattices and their finitely generated sublattices satisfying (D). This generalizes an unpublished result of Freese and Nation.

Whitehead Graphs and the Σ1-Invariants of Infinite Groups

1998 ◽  
Vol 08 (01) ◽  
pp. 23-34 ◽  
Author(s):  
Susan Garner Garille ◽  
John Meier
Keyword(s):  
Local Structure ◽  
Universal Cover ◽  
Finitely Generated ◽  
Infinite Groups ◽  
Finitely Generated Group ◽  
Finite Presentation ◽  
Finitely Presented Group ◽  
Finitely Presented

Let G be a finitely generated group. The Bieri–Neumann–Strebel invariant Σ1(G) of G determines, among other things, the distribution of finitely generated subgroups N◃G with G/N abelian. This invariant can be quite difficult to compute. Given a finite presentation 〈S:R〉 for G, there is an algorithm, introduced by Brown and extended by Bieri and Strebel, which determines a space Σ(R) that is always contained in, and is sometimes equal to, Σ1(G). We refine this algorithm to one which involves the local structure of the universal cover of the standard 2-complex of a given presentation. Let Ψ(R) denote the space determined by this algorithm. We show that Σ(R) ⊆ Ψ ⊆ Σ1(G) for any finitely presented group G, and if G admits a staggered presentation, then Ψ = Σ1(G). By casting this algorithm in terms of connectivity properties of graphs, it is shown to be computationally feasible.

scholarly journals Groups with a quotient that contains the original group as a direct factor

1992 ◽  
Vol 45 (3) ◽  
pp. 513-520 ◽  
Author(s):  
Ron Hirshon ◽  
David Meier
Keyword(s):  
Free Group ◽  
Normal Subgroups ◽  
Direct Factor ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Finitely Presented Group ◽  
Finitely Generated Groups ◽  
Original Group ◽  
Hopfian Group ◽  
Finitely Presented

We prove that given a finitely generated group G with a homomorphism of G onto G × H, H non-trivial, or a finitely generated group G with a homomorphism of G onto G × G, we can always find normal subgroups N ≠ G such that G/N ≅ G/N × H or G/N ≅ G/N × G/N respectively. We also show that given a finitely presented non-Hopfian group U and a homomorphism φ of U onto U, which is not an isomorphism, we can always find a finitely presented group H ⊇ U and a finitely generated free group F such that φ induces a homomorphism of U * F onto (U * F) × H. Together with the results above this allows the construction of many examples of finitely generated groups G with G ≅ G × H where H is finitely presented. A finitely presented group G with a homomorphism of G onto G × G was first constructed by Baumslag and Miller. We use a slight generalisation of their method to obtain more examples of such groups.

Time-Complexity of the Word Problem for Semigroups and the Higman Embedding Theorem

1998 ◽  
Vol 08 (02) ◽  
pp. 235-294 ◽  
Author(s):  
Jean-Camille Birget
Keyword(s):  
Word Problem ◽  
Time Complexity ◽  
Linear Time ◽  
Algorithm Design ◽  
Embedding Theorem ◽  
Algebraic Characterization ◽  
Algorithmic Problem ◽  
Finitely Generated ◽  
Isoperimetric Function ◽  
Finitely Presented

The following algebraic characterization of the computational complexity of the word problem for finitely generated semigroups is proved, in the form of a refinement of the Higman Embedding Theorem: Let S be a finitely generated semigroup whose word problem has nondeterministic time complexity T (where T is a function on the positive integers which is superadditive, i.e. T(n+m) ≥T(n)+T(m)). Then S can be embedded in a finitely presented semigroup H in which the derivation distance between any two equivalent words x and y (and hence the isoperimetric function) is O (T(∣x∣+∣y∣)2). Moreover, there is a conjunctive linear-time reduction from the word problem of H to the word problem of S, so the word problems of S and H have the same nondeterministic time complexity (and also the same deterministic time complexity). Thus, a finitely generated semigroup S has a word problem in NTIME(T) (or in DTIME(To)) iff S is embeddable into a finitely presented semigroup H whose word problem is in NTIME(T) (respectively in DTIME(To)). In the other direction, if a finitely generated semigroup S is embeddable in a finitely presented semigroup H with isoperimetric function ≤ D (where D(n) ≥ n), then the word problem of S has nondeterministic time complexity O(D). The word problem of a finitely generated semigroup S is in NP (or more generally, in NTIME((T) O(1) )) iff S can be embedded in a finitely presented semigroup H with polynomial (respectively (T) O(1) ) isoperimetric function. An algorithmic problem L is in NP (or more generally, in NTIME((T) O(1) )) iff L is reducible (via a linear-time one-to-one reduction) to the word problem of a finitely presented semigroup with polynomial (respectively (T) O(1) ) isoperimetric function. In essence, this shows: (1) Finding embeddings into finitely presented semigroups or groups is an algebraic analogue of nondeterministic algorithm design; (2) the isoperimetric function is an algebraic analogue of nondeterministic time complexity.

scholarly journals The computability of group constructions II

1973 ◽  
Vol 8 (1) ◽  
pp. 27-60 ◽  
Author(s):  
R.W. Gatterdam
Keyword(s):  
Characteristic Function ◽  
Word Problem ◽  
Elementary Functions ◽  
Recursively Enumerable Set ◽  
Finitely Presented Groups ◽  
Finitely Generated Group ◽  
Finitely Presented Group ◽  
Recursively Enumerable ◽  
Finitely Presented ◽  
Computational Level

Finitely presented groups having word, problem solvable by functions in the relativized Grzegorczyk hierarchy, {En(A)| n ε N, A ⊂ N (N the natural numbers)} are studied. Basically the class E3 consists of the elementary functions of Kalmar and En+1 is obtained from En by unbounded recursion. The relativization En(A) is obtained by adjoining the characteristic function of A to the class En.It is shown that the Higman construction embedding, a finitely generated group with a recursively enumerable set of relations into a finitely presented group, preserves the computational level of the word problem with respect to the relativized Grzegorczyk hierarchy. As a corollary it is shown that for every n ≥ 4 and A ⊂ N recursively enumerable there exists a finitely presented group with word problem solvable at level En(A) but not En-1(A). In particular, there exist finitely presented groups with word problem solvable at level En but not En-1 for n ≥ 4, answering a question of Cannonito.

CIRCUITS, THE GROUPS OF RICHARD THOMPSON, AND coNP-COMPLETENESS

2006 ◽  
Vol 16 (01) ◽  
pp. 35-90 ◽  
Author(s):  
JEAN-CAMILLE BIRGET
Keyword(s):  
Simple Group ◽  
Word Problem ◽  
Partial Transformation ◽  
Transformation Groups ◽  
Combinational Circuits ◽  
Finitely Generated ◽  
Finitely Presented Group ◽  
Thompson Groups ◽  
Higman Group ◽  
Finitely Presented

We construct a finitely presented group with coNP-complete word problem, and a finitely generated simple group with coNP-complete word problem. These groups are represented as Thompson groups, hence as partial transformation groups of strings. The proof provides a simulation of combinational circuits by elements of the Thompson–Higman group G3,1.

scholarly journals Subgroups of finitely presented metabelian groups

1973 ◽  
Vol 16 (1) ◽  
pp. 98-110 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Metabelian Group ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Finitely Generated Group ◽  
Finitely Presented Group ◽  
Finitely Presented

In 1961 Graham Higman [1] proved that a finitely generated group is a subgroup of a finitely presented group if, and only if, it is recursively presented. Therefore a finitely generated metabelian group can be embedded in a finitely presented group.

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