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The Word Problem Versus the Isomorphism Problem
Author(s) -
Glass A. M. W.,
Madden James J.
Publication year - 1984
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-30.1.53
Subject(s) - word problem (mathematics education) , isomorphism (crystallography) , word (group theory) , linguistics , mathematics , computer science , arithmetic , philosophy , chemistry , crystallography , crystal structure
In previously studied cases in varieties of algebras, the word and isomorphism problems have had the same solution. For abelian groups and loops, for example, both are soluble [15, Section 3.3; 9; 10]; for groups, semigroups and lattice-ordered groups, for example, both are insoluble [7, 19, 1, 18, 12, 11]. In contrast, Miller [17, p. 77], shows that there are recursive classes of finitely presented groups with uniformly soluble word problem but with insoluble isomorphism problem. Our contribution is to provide a variety of algebras defined by a finite number of laws where this dichotomy also occurs. Specifically, we have the following.