# presentations of algebraic objects

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  $f:F\rightarrow A$ which matches the embedding  of $X$ in $F$ to the embedding of $X$ in $A$.

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 $\langle X|R\rangle$ 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 $F/K\cong A$.

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 PresentationsOfAlgebraicObjects 2013-03-22 16:51:27 2013-03-22 16:51:27 CWoo (3771) CWoo (3771) 5 CWoo (3771) Definition msc 08B20 Presentationgroup presentation