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Examples

Let σ = { +, ×, −, 0, 1 } {\displaystyle \sigma =\{+,\times,-,0,1\}} be again the standard signature for fields. When regarded as σ {\displaystyle \sigma } -structures in the natural way, the rational numbers form a substructure of the real numbers, and the real numbers form a substructure of the complex numbers. The rational numbers are the smallest substructure of the real (or complex) numbers that also satisfies the field axioms.

The set of integers gives an even smaller substructure of the real numbers which is not a field. Indeed, the integers are the substructure of the real numbers generated by the empty set, using this signature. The notion in abstract algebra that corresponds to a substructure of a field, in this signature, is that of a subring, rather than that of a subfield.

The most obvious way to define a graph is a structure with a signature σ {\displaystyle \sigma } consisting of a single binary relation symbol E. {\displaystyle E.} The vertices of the graph form the domain of the structure, and for two vertices a {\displaystyle a} and b, {\displaystyle b,} (a, b) ∈ E {\displaystyle (a,b)\!\in {\text{E}}} means that a {\displaystyle a} and b {\displaystyle b} are connected by an edge. In this encoding, the notion of induced substructure is more restrictive than the notion of subgraph. For example, let G {\displaystyle G} be a graph consisting of two vertices connected by an edge, and let H {\displaystyle H} be the graph consisting of the same vertices but no edges. H {\displaystyle H} is a subgraph of G, {\displaystyle G,} but not an induced substructure. The notion in graph theory that corresponds to induced substructures is that of induced subgraphs.