We sometimes think of an algebra as a set plus some form of 'structure' (in a similar way that in the computer programming world we look at objects as data+functions) on these pages we look at this structure element of mathematical entities like algebras.

This structure may be defined in terms of:

- Functions such as Binary (or unary or 'n'ary) operations: (element,element) -> element
- Binary (or unary or 'n'ary) relations: (element,element) -> boolean
- mappings (or morphisms or arrows) between objects.

On this page we look at maps or morphisms between objects.

On this page we look at graphs.

### Structures with binary operation and identity element

A common type of structure is one with a binary operation and identity element. These structures may have:

- Single objects or multiple objects
- Composition law or not.
- Invertible or not.

As indicated in the following table:

single objects | multiple objects -> | |||

composition law | no composition law | |||

invertible | (permutation) Groups |
all morphisms are isomorphisms |
Graph | |

non-invertible | Monoids | Categories | Directed Graph |

These structures can all be represented by directed graphs with various restrictions. For instance,

- An invertible structure wont have an arrow from 'a' to 'b' unless there is also an arrow from 'b' to 'a'. (symmetry)
- A structure with composition law, if it has arrows a→b and b→c, then it also has an arrow a→c
- A structure with reflexive law always has identity element, arrow from 'a' to 'a'.

Note: the above bullets correspond to equivalence relations.

## Set + Structure

## Further Topics

We can now go on and look at specific structures in more detail: