presentations of algebraic objects
Given an algebraic category 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 category. The process is called a presentation.
As we are in an algebraic category we have a fundamental homomorphism theorem (we take this as our definition of an algebraic category in this context). This means there is a notion of kernel of and quotient such that is isomorphic to .
Now is generted by so every element of is expressed as an informal word over . [By informal word we mean whatever process encodes general elements as generated by . For example, in groups and semigroups these are actual formal words, but in algebras these can be linear combinations of words or polynomials with indeterminants in , etc.] Hence a set of generators for the kernel will be expressed as words over .
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 variable choice of generators. Secondly, we may choose various relations. Indeed, it is possible that the relations will generate different subobjects such that . 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 theorems 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|
|Date of creation||2013-03-22 16:51:27|
|Last modified on||2013-03-22 16:51:27|
|Last modified by||CWoo (3771)|