Axiomatizable class

In mathematics, an axiomatizable class is a class of mathematical structures which are all models of a fixed set of sentences in formal (typically first order) logic.

For example, the axiomatic sentences of a multiplicative group are:

$\forall xyz \, \, (xy)z = x(yz)$

$\forall x\,\, x \cdot 1 = x$

$\forall x\,\, x \cdot x^{-1} = 1.$

The axioms of a left R-module are the axioms of a multiplicative group, together with the additional sentences

$\forall xy \,\, r(x+y)=r(x)+r(y)$ for all $r\in R$

$\forall x \,\, (r+s)(x)=r(x)+s(x)$ for all $r,s\in R$

$\forall x \,\, (rs)(x)=r(s(x))$ for all $r,s\in R$

$\forall x \,\, 1(x)=x.$

Many of the common classes of mathematics are easily axiomatizable, including the rings, fields, lattices, boolean algebras and the like.

References

Wilfrid Hodges (1997). A shorter model theory. Cambridge University Press. ISBN 0-521-58713-1.