A presentation of an algebraic theory is a theory with equality. Specified by:
- A set of variables
- A set of n-ary operations
- A set of axioms
The terms of the theory are defined inductivly:
term ::= variable | operation(term,...)
An axiom is an equality between terms
Presentation of a Group
We start with the 'presentation of a group' as a way of defining that group, we can then try to generalise this to other algebraic structures.
A group is a particulary simple example because:
- There is only one binary operation.
- The binary operation always has an inverse, this means that an equation can always be put in the form: term=1 (where '1' is the identity). So a relation is really just a term.
So it turns out that we can specify the group by its generators plus a set of relations among the generators (relations that produce the identity element are usually written without the equals sign).
This is known as the 'Presentation of a group'.We specify this as follows:
Γ< generators | relations >
Using the triangle example, from this page, we have:
Γ< a,b | aa,bb,bab=aba>
We could have set all relations to be equal to unity by replacing bab=aba with baba-1b-1a-1=1
Can we use this to generate all the entries in the table and thus completely define the structure?
Lets take an example of multiplying 'ab' by 'ab':
ab * ab = abab [associatively]
= aaba [since bab=aba]
= ba [since aa=1]
Doing this for all the other entries in the Cayley table shows that this completely defines the group. How can we work out the minimum set of relations that will define the group?