presentations of algebraic objects

Given an algebraic categoryPlanetmathPlanetmathPlanetmathPlanetmath with enough free objects one can use the general description of the free object to provide a precise description of all other objects in the categoryMathworldPlanetmath. The process is called a presentationMathworldPlanetmathPlanetmathPlanetmath.

Suppose A is an object generated by a subset X. Then if there exists a free object on X, F, then there exists a unique morphismMathworldPlanetmath f:FA which matches the embeddingPlanetmathPlanetmath of X in F to the embedding of X in A.

As we are in an algebraic category we have a fundamental homomorphism theoremMathworldPlanetmath (we take this as our definition of an algebraic category in this context). This means there is a notion of kernel K of f and quotientPlanetmathPlanetmath F/K such that F/K is isomorphicPlanetmathPlanetmathPlanetmath to A.

Now F is generted by X so every element of F is expressed as an informal word over X. [By informal word we mean whatever process encodes general elements as generated by X. For example, in groups and semigroupsPlanetmathPlanetmath these are actual formal words, but in algebrasMathworldPlanetmathPlanetmath these can be linear combinations of words or polynomialsMathworldPlanetmathPlanetmath with indeterminants in X, etc.] Hence a set of generatorsPlanetmathPlanetmathPlanetmath for the kernel K will be expressed as words over X.

Definition 1.

A presentation of an object A is a pair of sets X|R where X generates A and R is a set of informal words over X such that the free object F on X and the normal subobject K of F generated by R has the property F/KA.

Once again, normal refers to whatever property is required for subobject to allow quotients, so normal subgroupMathworldPlanetmath or ideals, etc.

Existence of presentations is dependent on the category being considered. The common categories: groups, rings, and modules all have presentations.

It is generally not possible to insist that a presentation is unique. First we have the variableMathworldPlanetmath choice of generators. Secondly, we may choose various relationsMathworldPlanetmathPlanetmath. Indeed, it is possible that the relations will generate different subobjects K such that F/KA. In practice, presentations are a highly compactified description of an object which can hide many essential features of the object. Indeed, in the extreem case are the theoremsMathworldPlanetmath of Boone which show that in the category of groups it is impossible to tell if an arbitrary presentation is a presentation of the trivial group. For a detailed account of these theorems refer to

Joseph Rotman, An Introduction to the Theory of Groups, Springer, New York, Fourth edition, 1995.

Title presentations of algebraic objects
Canonical name PresentationsOfAlgebraicObjects
Date of creation 2013-03-22 16:51:27
Last modified on 2013-03-22 16:51:27
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Definition
Classification msc 08B20
Related topic Presentationgroup
Defines presentation