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|
all morphisms are isomorphisms
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
We can now go on and look at specific structures in more detail: