Submodels
A useful tool for thinking about mathematical objects is the model. We tend to like stating what concepts we can use to analyze it (the language), and a set of statements from that langauge that describe it.
That is, a model is a set with specified structure.
A submodel is a subset, that keeps as much of that structure as is relevant.
That is, S is a submodel of M if for all statements in M that relate to elements in S, that statement is also true in S.
Here are some examples
Subgraphs
An edge in a graph is in its subgraph if both ends are vertices of the subgraph.
For example, a 5-cycle is a subgraph of the Petersen graph.
But actually there is a second 5-cycle that is a subgraph, from the inner vertices
Subgroups
If x and y are elements of a group, and * is the combiner, then statements of it are of the form x*y=z
To find a subgroup, if it has x and y, then it must also have z, so x*y=z is also true in the subgroup.
For example, the 2-element group is a subgroup of the Klein-4 Group:
Subrings
The even integers are a subring of the integers. That is, if you add and multiply integers, and then restrict that to even integers, everything is exactly the same.
Subfields
Similarly to the above examples, the real numbers are a subfield of the complex numbers, and the rational numbers are a subfield of the real numbers.
(How subfields relate is a really cool branch of abstract algebra called Galois theory)
Suborders
A partial order can, in the same way, have a suborder.
In fact, that suborder can be lineär.