# 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 ^{}*.

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 morphism^{}
$f:F\to A$ which matches the embedding^{} of $X$ in $F$ to the embedding
of $X$ in $A$.

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 $K$ of $f$ and quotient^{} $F/K$ such that $F/K$ is isomorphic^{} 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 semigroups^{} these are actual formal words, but in algebras^{} these can be linear combinations
of words or polynomials^{} with indeterminants in $X$, etc.] Hence a set of generators^{} for the kernel $K$ will be expressed as words over $X$.

###### Definition 1.

A presentation of an object $A$ is a pair of sets $\mathrm{\u27e8}X\mathrm{|}R\mathrm{\u27e9}$ 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 $F\mathrm{/}K\mathrm{\cong}A$.

Once again, normal refers to whatever property is required for subobject to allow quotients, so normal subgroup^{} 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 variable^{} choice of generators. Secondly, we may choose various relations^{}. Indeed, it is possible that the relations will generate different subobjects $K$ such that $F/K\cong A$. 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 |
---|---|

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 |