Editing Model theory (section)

===Structures and types===
While not every type is realised in every structure, every structure realises its isolated types.
If the only types over the empty set that are realised in a structure are the isolated types, then the structure is called ''atomic''.

On the other hand, no structure realises every type over every parameter set; if one takes all of <math>\mathcal{M}</math> as the parameter set, then every 1-type over <math>\mathcal{M}</math> realised in <math>\mathcal{M}</math> is isolated by a formula of the form ''a = x'' for an <math>a \in \mathcal{M}</math>. However, any proper elementary extension of <math>\mathcal{M}</math> contains an element that is ''not'' in <math>\mathcal{M}</math>. Therefore, a weaker notion has been introduced that captures the idea of a structure realising all types it could be expected to realise.
A structure is called ''saturated'' if it realises every type over a parameter set <math>A \subset \mathcal{M}</math>  that is of smaller cardinality than <math>\mathcal{M}</math> itself.

{{Anchor|homogeneous}}While an automorphism that is constant on ''A'' will always preserve types over ''A'', it is generally not true that any two sequences <math>a_1, \dots, a_n</math> and <math>b_1, \dots, b_n</math> that satisfy the same type over ''A'' can be mapped to each other by such an automorphism. A structure <math>\mathcal{M}</math> in which this converse does hold for all ''A'' of smaller cardinality than <math>\mathcal{M}</math> is called '''homogeneous'''.

The real number line is atomic in the language that contains only the order <math><</math>, since all ''n''-types over the empty set realised by  <math>a_1, \dots, a_n</math> in  <math>\mathbb{R}</math> are isolated by the order relations between the  <math>a_1, \dots, a_n</math>. It is not saturated, however, since it does not realise any 1-type over the countable set <math>\mathbb{Z}</math> that implies ''x'' to be larger than any integer.
The rational number line  <math>\mathbb{Q}</math> is saturated, in contrast, since  <math>\mathbb{Q}</math> is itself countable and therefore only has to realise types over finite subsets to be saturated.{{sfn|Marker|2002|pp=125-155}}