Defining relationships

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defining relations, of a universal algebra $ G $ with respect to a system $ \{ {g _ {i} } : {i \in I } \} $ of generators for it

Relations of the form

$$ u _ {j} ( g _ {i1}, \dots, g _ {in} ) = \ v _ {j} ( g _ {i1}, \dots, g _ {in} ),\ \ j \in J, $$

between generators (where $ u _ {j} , v _ {j} $ are terms in the signature of the algebra in question), such that all remaining relations of this form are consequences of the given ones and the identities of the variety in which $ G $ is being studied. Normally when one speaks of a presentation of an algebra by generators and defining relations, one is thinking of a quotient algebra of the free algebra of the variety with the same generators by the congruence defined by all pairs $ ( u _ {j} , v _ {j} ) $, $ j \in J $. In the case of multi-operator groups (especially, groups, algebras, rings, and modules) the form of the defining relations may be simplified: they can be written either as $ w _ {j} = 0 $ or as $ w _ {j} = 1 $ (in groups).

Defining relations are not uniquely determined even for the same system of generators. For example, the cyclic group of order two with generator $ a $ can be given by one defining relation $ a ^ {2} = 1 $, as well as by the two defining relations $ a ^ {6} = 1 $ and $ a ^ {4} = 1 $. Special transformations (Tietze transformations in groups (see [2]) and their analogues in various varieties of algebras) exist which allow one to convert from one presentation of an algebra by generators and defining relations into another presentation of the same algebra. In these conditions, for finitely-presented groups (or algebras), i.e. ones given by finite systems of generators and defining relations, it is possible, using a finite number of Tietze transformations, to pass from any such presentation to any other (finite) presentation by generators and defining relations. If an algebra is finitely generated, then from any system of generators for it it is possible to select a finite subsystem of generators; if an algebra in some finite system of generators is given by a finite number of defining relations, then given any other finite system of generators it is possible to choose a finite subsystem of defining relations from any system of defining relations.

The study of finitely-presented algebras has generated a whole series of algorithmic problems, such as the problem of equality (identity), the isomorphism problem and others (see Algorithmic problem). A series of results has been obtained for algebras with one defining relation. For example, in groups with one defining relation, the problem of equality is solvable, and the elements of finite order, the centre, and all subgroups with non-trivial identities have been described (see also Group calculus).


[1] P.M. Cohn, "Universal algebra" , Reidel (1981)
[2] A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)
[3] W. Magnus, A. Karrass, B. Solitar, "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience) (1966)
How to Cite This Entry:
Defining relationships. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.L. Shmel'kin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article