# Maths - Structure

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.

### 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 Groupoids 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 ab and bc, then it also has an arrow ac
• A structure with reflexive law always has identity element, arrow from 'a' to 'a'.

Note: the above bullets correspond to equivalence relations.

## Set + Structure

In its purest form the only information contained in a set is the number of elements it contains. All other information, such as labeling the elements, can be added using functions between sets.

A total function maps every element in the domain to an element in the codomain. Although not every element in the codomain may necessarily be mapped too. So, in this way functions between sets are not necessarily symmetrical.

From this mathematical structure is genererated.

 Bijective functions are invertible. See: Surjective functions are not invertible in that we can't reverse the function and get back to where we started, for instance if we have a surjective mapping from set to set we can't get back to the same set, however we can go back to a 'set of sets'. This gives rise to interesting structure, see: Injective functions gives a subobject structure. If we have an injective mapping from set to set we can't get back to the same set, however we can have a mapping from set to Bool which tells us which element are in the subset, this is known as a characteristic function in sets or 'subobject classifier' more generally in category theory. This subobject structure is interesting see:

## Further Topics

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

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

 The Princeton Companion to Mathematics - This is a big book that attempts to give a wide overview of the whole of mathematics, inevitably there are many things missing, but it gives a good insight into the history, concepts, branches, theorems and wider perspective of mathematics. It is well written and, if you are interested in maths, this is the type of book where you can open a page at random and find something interesting to read. To some extent it can be used as a reference book, although it doesn't have tables of formula for trig functions and so on, but where it is most useful is when you want to read about various topics to find out which topics are interesting and relevant to you.