Topic
Word problem for groups
About: Word problem for groups is a research topic. Over the lifetime, 351 publications have been published within this topic receiving 7575 citations.
Papers published on a yearly basis
1 / 2
Papers
TL;DR: The main theorem of as discussed by the authors states that a finitely generated group can be embedded in a finite presented group if and only if it has a recursively enumerable set of defining relations.
Abstract: The main theorem of this paper states that a finitely generated group can be embedded in a finitely presented group if and only if it has a recursively enumerable set of defining relations. It follows that every countable A belian group, and every countable locally finite group can be so embedded; and that there exists a finitely presented group which simultaneously embeds all finitely presented groups. A nother corollary of the theorem is the known fact that there exist finitely presented groups with recursively insoluble word problem . A by-product of the proof is a genetic characterization of the recursively enumerable subsets of a suitable effectively enumerable set.
323 citations
1 Jul 1987
TL;DR: It is shown that even if some equations are not orientable, the Knuth-Bendix procedure can still be modified into a reasonably efficient semi-decision procedure for word problems in equational theories.
Abstract: The Knuth-Bendix procedure for word problems in universal algebra is known to be very effective when it is applicable. However, the procedure will fail if it generates equations which cannot be oriented into rules (i.e., the system is not noetherian), or if it generates infinitely many rules (i.e., the system is not confluent). In 1981 Huet showed that even if the system is not confluent, the Knuth-Bendix procedure still yields a semi-decision procedure for word problems, provided that every equation can be oriented. In this paper we show that even if some equations are not orientable, the Knuth-Bendix procedure can still be modified into a reasonably efficient semi-decision procedure for word problems in equational theories. Thus, we have lifted the noetherian requirement in the Knuth-Bendix procedure. Several confluence results, extensions, and experiments are given. So are some comparisons with related work.
181 citations
Book•
1 Jan 1986
TL;DR: This paper is an overview of rewriting systems as a tool to solve word problems in usual algebras by identifying the mechanisms that induce the existence of a completion procedure for the finite presentations in this variety.
Abstract: This paper is an overview of rewriting systems as a tool to solve word problems in usual algebras. A successful completion of an equational theory, defining a variety of algebras, induces the existence of a completion procedure for the finite presentations in this variety.
157 citations
1 Jan 2009
TL;DR: In this article, it was shown that the word problem is undecidable for associative algebras, in which one does not assume that the algebra satisfies any identical relation (for instance, associativity).
Abstract: The word problem1, stated relative to one or another algebraic system, has attracted the attention of many mathematicians. In the works of A.A. Markov [1] and E. Post [3] it was proved for the first time that there exist algebraic systems (semigroups) with undecidable word problem. The most significant achievement in this direction is the result of P.S. Novikov [2] that establishes undecidability of the word problem for groups. In 1950, A.I. Zhukov [5], while studying free nonassociative algebras, established that in the case in which one does not assume that the algebra satisfies any identical relation (for instance, associativity) the word problem (as well as some other algorithmic problems) is decidable. From the results obtained for semigroups, it easily follows that the word problem is undecidable for associative algebras.
124 citations
IBM1
TL;DR: The main result of this paper solves the complementary problem of generating the set of all solutions and generates, given a word equation, a minimal and complete set of unifiers.
Abstract: The fundamental satisfiability problem for word equations has been solved recently by Makanin. However, this algorithm is purely a decision algorithm. The main result of this paper solves the complementary problem of generating the set of all solutions. Specifically, the algorithm in this paper generates, given a word equation, a minimal and complete set of unifiers. It stops if this set is finite.
121 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2020 | 1 |
| 2018 | 2 |
| 2017 | 6 |
| 2016 | 5 |
| 2015 | 9 |
| 2014 | 6 |