## The Uniform Word Problem for Groups and Finite Rees Quotients of E-Unitary Inverse Semigroups (2001)

Citations: | 3 - 1 self |

### BibTeX

@MISC{Steinberg01theuniform,

author = {Benjamin Steinberg},

title = {The Uniform Word Problem for Groups and Finite Rees Quotients of E-Unitary Inverse Semigroups},

year = {2001}

}

### OpenURL

### Abstract

If C is a class of groups closed under taking subgroups, we show that the decidability of the uniform word problem for C is implied by the decidability of the membership problem for the class of nite Rees quotients of E-unitary inverse semigroups with maximal group image in C. The converse is shown if C is a pseudovariety. When C is a pseudovariety, the above problems are shown to be equivalent to the problem of embedding a nite labeled graph in the Cayley graph of a group in C. This latter problem is shown to be equivalent to deciding whether a nite labeled graph is a Schutzenberger graph of an E-unitary inverse semigroup with maximal group image in C. 1.

### Citations

71 |
Inverse Semigroups: the Theory of Partial Symmetries
- Lawson
- 1998
(Show Context)
Citation Context ... group (a map between semigroups is said to be idempotent pure if the inverse image of each idempotent consists of idempotents). The structure of such semigroups was completely described by McAlister =-=[15, 11-=-]. Date: May 17, 2001. 1991 Mathematics Subject Classication. 20F32, 20M18, 20F10, 20M07, 20M35. Key words and phrases. Uniform word problem, 0-E-unitary inverse semigroups, undecidability, Cayley gra... |

35 | Algorithmic problems in varieties
- Kharlampovich, Sapir
- 1995
(Show Context)
Citation Context ...variety of groups containing Z(N 2 Ab). Then results of Kharlampovich [7, 8] imply that the uniform word problem is undecidable for the following pseudovarieties: N; G \N; G \V; N\V; and G \N\V; see [=-=6, 9]-=- for more. On the other hand, the uniform word problem is solvable in any variety V of groups in which everysnitely generated group is residuallysnite, as well as in the pseudovariety ofsnite groups f... |

33 | Closed subgroups in the pro-V topologies, and the extension problem for inverse automata
- Margolis, Sapir, et al.
- 2001
(Show Context)
Citation Context ... [12]. Since N is a directed union of the N k , it follows that asnite inverse semigroup has an E-unitary cover over N if and only if it does over N\G. The latter problem was shown to be decidable in =-=[14]-=-. The author does not know of a published example of a decidable pseudovarietysC ofsnite groups such that the pseudovariety ofsnite inverse semigroups with E-unitary covers over C is undecidable. Howe... |

24 |
semilattices and inverse semigroups
- McAlister, Groups
- 1974
(Show Context)
Citation Context ... group (a map between semigroups is said to be idempotent pure if the inverse image of each idempotent consists of idempotents). The structure of such semigroups was completely described by McAlister =-=[15, 11-=-]. Date: May 17, 2001. 1991 Mathematics Subject Classication. 20F32, 20M18, 20F10, 20M07, 20M35. Key words and phrases. Uniform word problem, 0-E-unitary inverse semigroups, undecidability, Cayley gra... |

22 |
The word problem for abstract algebras
- Evans
- 1951
(Show Context)
Citation Context ...thers. It is known [6] that the uniform word problem for a pseudovariety of groups C is equivalent to the decidability of its universal theory. A result of THE UNIFORM WORD PROBLEM FOR GROUPS 3 Evans =-=[4]-=- shows that it is also equivalent to the decidability of the problem of embedding a partial group into a group in C. In Theorem 1.1, if C is just assumed to be closed under S then (2) and (3) are equi... |

21 |
Topology of graphs
- Stallings
- 1983
(Show Context)
Citation Context ...itions and properties concerning labeled graphs. The transition monoid of an inverse graph is an inverse monoid denoted M(). One can view the labeling as giving an immersion in the sense of Stallings =-=[21]-=- from to the inverse graph BA consisting of a single vertex and edges labeled in bijection with A [ A 1 . The following result generalizes [17, Theorem 7.1] (which proves the result for the case where... |

19 |
Presentations of inverse monoids
- Stephen
- 1990
(Show Context)
Citation Context ... (BA )), so there is a unique morphisms: (; i) ! (CGA (G); 1). To show thatsis an embedding, it suces to show it is injective on vertices (this is a well-known property of morphisms of inverse graphs =-=[24, 23, 19]-=-). Supposes(v 1 ) =s(v 2 ). Let v 1 = i[w 1 ] M and v 2 = i[w 2 ] M (note: [w 1 ] M ; [w 2 ] M 6= 0). Then [w 1 ] G = [w 2 ] G whence '([w 1 2 w 1 ] M ) = 1. We conclude [w 1 2 w 1 ] M is idempotent w... |

18 | Finite state automata: A geometric approach
- Steinberg
(Show Context)
Citation Context ...alled inverse if it is connected, deterministic, co-deterministic, and if whenever there is an edge e from p to q with label a, there is an edge from q to p with label a 1 . The reader is referred to =-=[23-=-] for basic denitions and properties concerning labeled graphs. The transition monoid of an inverse graph is an inverse monoid denoted M(). One can view the labeling as giving an immersion in the sens... |

11 |
Inverse monoids, trees and context-free languages
- Margolis, Meakin
- 1993
(Show Context)
Citation Context ...with maximal group image in C if and only if it embeds in the Cayley graph of a group in C. Moreover, if C consists ofsnite groups and issnite, then M can be taken to besnite. Proof. It is well known =-=[13-=-] that any Schutzenberger graph of an E-unitary inverse monoid embeds in the Cayley graph of its maximal group image, so one direction is clear. For the other direction, we can use Corollary 3.3 and P... |

8 |
Inverse semigroups with zero: covers and their structure
- Bulman-Fleming, Fountain, et al.
- 1999
(Show Context)
Citation Context ... for being a Rees quotient of an Eunitary inverse semigroup is to have the property: x e = e 2 6= 0 implies x = x 2 ; inverse semigroups satisfying this property are called 0-E-unitary. However, in [=-=1-=-], an example is given of a Cliord semigroup which is 0-Eunitary, but which is not a Rees quotient of an E-unitary inverse semigroup. We show that it is undecidable whether asnite inverse semigroup is... |

8 |
automata and pseudovarieties of semigroups, Int
- Rhodes, Undecidability
- 1999
(Show Context)
Citation Context ...iety of all groups) and is of interest in its own right. The argument here is an adaptation of an unpublished proof of that theorem due to the author and J. McCammond; see also [23, Theorem 5.21] and =-=[18]-=- for similar proof techniques. We now prove (2) implies (1) in Theorem 1.1. Theorem 3.1. Let C be a class of groups closed under S. If it is decidable whether asnite inverse graph embeds in the Cayley... |

6 | Combinatorial group theory, inverse monoids, automata, and global semigroup theory
- Delgado, Margolis, et al.
(Show Context)
Citation Context ...lar covering (or even asnite regular covering). The following lemma is key in the rest of what follows; it gives an inverse semigroup theoretic characterization of the subgraphs of Cayley graphs; see =-=[3]-=- for more examples of connections between embedding problems for inverse graphs into covers and inverse semigroups. If I is an inverse monoid generated by A, we use [w] I for the image in I of a word ... |

4 |
The word problem for groups and Lie algebras, doctor’s thesis (in Russian
- Kharlampovich
- 1990
(Show Context)
Citation Context ... by Abelian groups, Z(N 2 Ab) the class of all groups G such that G=Z(G) 2 N 2 Ab (where Z(G) is the center of G), and V any variety of groups containing Z(N 2 Ab). Then results of Kharlampovich [7,=-= 8]-=- imply that the uniform word problem is undecidable for the following pseudovarieties: N; G \N; G \V; N\V; and G \N\V; see [6, 9] for more. On the other hand, the uniform word problem is solvable in a... |

4 |
On algorithmic unsolvability of the problem of identity, Dokl
- Novikov
(Show Context)
Citation Context ...t under switching between inverse monoids and inverse semigroups and so we do not worry about such distinctions. The uniform word problem for the variety of all groups is well known to be undecidable =-=[16]-=-. For the pseudovariety of allsnite groups, it was proved undecidable by Slobodskoii [20]. Let G be the pseudovariety of allsnite groups, N be the pseudovariety of all nilpotent groups, N k the variet... |

4 | Fundamental groups, inverse Schützenberger automata, and monoid presentations
- Steinberg
- 2000
(Show Context)
Citation Context ...phen shows [24] that L(SchA (m)) = fw 2 (A [ A 1 ) jw m (in M)g: There is a natural homomorphism from M onto M() called the Schutzenbergersrepresentation. The following is a folklore result; see [2,=-= 17, 19, 2-=-2]. Proposition 4.1. An inverse graph is a Schutzenberger graph if and only if it is a Schutzenberger graph of a D-class of a (necessarily unique) 0-minimal ideal of M(). Proof. Let M = M() and suppos... |

4 |
More on Burnside’s Problem, Combinatorial and geometric group theory. Proceedings of a workshop held at Heriot-Watt University, Edinborough GB Spring of 93
- Zel’manov
- 1995
(Show Context)
Citation Context ...ofsnite groups from V; see [9]. In particular, the uniform word problem is decidable for the variety N k and the pseudovariety G\N k . Zel'manov's positive solution of the restricted Burnside problem =-=[25]-=- implies the solvability of the uniform word problem for any pseudovariety ofsnite groups withsxed exponent; see [9, 6]. 2. Strongly 0-E-unitary inverse semigroups If S and T are inverse semigroups wi... |

3 |
Pseudovarieties of Inverse Monoids
- Ruyle
- 1997
(Show Context)
Citation Context ... (BA )), so there is a unique morphisms: (; i) ! (CGA (G); 1). To show thatsis an embedding, it suces to show it is injective on vertices (this is a well-known property of morphisms of inverse graphs =-=[24, 23, 19]-=-). Supposes(v 1 ) =s(v 2 ). Let v 1 = i[w 1 ] M and v 2 = i[w 2 ] M (note: [w 1 ] M ; [w 2 ] M 6= 0). Then [w 1 ] G = [w 2 ] G whence '([w 1 2 w 1 ] M ) = 1. We conclude [w 1 2 w 1 ] M is idempotent w... |

2 |
Homomorphisms onto groups, Ivanov gosudarst. ped. Inst. ucenye Zap. Nauk 18
- Mal'cev
- 1958
(Show Context)
Citation Context ...moreover, this is decidable if V has asnite basis. This is the case for N k , assnitely generated nilpotent groups THE UNIFORM WORD PROBLEM FOR GROUPS 11 are polycyclic and polycyclic groups are LERF =-=[12]-=-. Since N is a directed union of the N k , it follows that asnite inverse semigroup has an E-unitary cover over N if and only if it does over N\G. The latter problem was shown to be decidable in [14].... |

1 |
Characterizations of Schutzenberger graphs in terms of their automorphism groups and fundamental groups
- Cowan, Reilly
- 1993
(Show Context)
Citation Context ...phen shows [24] that L(SchA (m)) = fw 2 (A [ A 1 ) jw m (in M)g: There is a natural homomorphism from M onto M() called the Schutzenbergersrepresentation. The following is a folklore result; see [2,=-= 17, 19, 2-=-2]. Proposition 4.1. An inverse graph is a Schutzenberger graph if and only if it is a Schutzenberger graph of a D-class of a (necessarily unique) 0-minimal ideal of M(). Proof. Let M = M() and suppos... |

1 |
Algorithmic problems for groups and 0-simple semigroups
- Hall, Kublanovskii, et al.
- 1997
(Show Context)
Citation Context ...roblem has been shown to be equivalent to the problem of determining whether a semigroup embeds in a 0-simple semigroup with structure group in C, or even a Brandt semigroup with structure group in C =-=[6]-=-. Here we relate the uniform word problem to the membership problem for the class ofsnite Rees quotients of E-unitary inverse semigroups with maximal group image in C. See [10] for more connections of... |

1 | The universal theory of the class of nilpotent groups is undecidable, Mat. Zemetki 33 - Kharlampovich - 1983 |

1 |
Potential divisibility in semigroups is undecidable
- Kublanovskii, Sapir
(Show Context)
Citation Context ...ith structure group in C [6]. Here we relate the uniform word problem to the membership problem for the class ofsnite Rees quotients of E-unitary inverse semigroups with maximal group image in C. See =-=[10]-=- for more connections of this problem with semigroup theory. We use S, H , P, P fin to denote the operators on collections of groups of closing under: subgroups, homomorphic images, arbitrary products... |

1 |
Inverse automata and monoids and the undecidability of the Cayley subgraph problem for groups
- Oliveira, Silva
(Show Context)
Citation Context ...migroup is a Rees quotient of an E-unitary inverse semigroup. A related problem is determining whether asnite inverse graph is a Schutzenberger graph for an E-unitary inverse monoid. It was shown in [=-=17]-=- that this problem is undecidable in general. This paper considers a vast generalization. The uniform word problem for a class C of groups asks for an algorithm to determine, given asnite set of relat... |

1 |
Undecidability of the universal theory of groups, Algebra i Logika 20
- Slobodskoii
- 1981
(Show Context)
Citation Context ...bout such distinctions. The uniform word problem for the variety of all groups is well known to be undecidable [16]. For the pseudovariety of allsnite groups, it was proved undecidable by Slobodskoii =-=[20-=-]. Let G be the pseudovariety of allsnite groups, N be the pseudovariety of all nilpotent groups, N k the variety of all nilpotent groups of step at most k, N 2 Ab the variety of all extensions of gr... |