Definition:Mathematical System

Definition
A mathematical system is a set $\mathcal S = \left({E, O, A}\right)$ where:


 * $E$ is a non-empty set of elements


 * $O$ is a set of relations and operations on the elements of $E$


 * $A$ is a set of axioms concerning the elements of $E$ and $O$.

Abstract System
A mathematical system $\mathcal S = \left({E, O, A}\right)$ is classed as abstract if the elements of $E$ and $O$ are defined only by their properties as specified in $A$.

Concrete System
A mathematical system $\mathcal S = \left({E, O, A}\right)$ is classed as concrete if the elements of $E$ and $O$ are understood as objects independently of their existence in $\mathcal S$ itself.

The distinction between abstract and concrete is of questionable value from a modern standpoint, as it is a moot point, for example, as to whether the natural numbers exist independently of Peano's axioms or are specifically defined by them.

Algebraic System
A mathematical system $\mathcal S = \left({E, O, A}\right)$ is classed as algebraic if it has many of the properties of the set of integers.

This is usually because such a system is itself an abstraction of certain properties of the integers.

The axioms are usually not considered as separate entities from the operations, as their nature is implicit in the operations themselves.

Specifically, an algebraic system can be defined as follows:

Also defined as
Some sources represent a mathematical system as a set, as opposed to an ordered tuple:
 * $\mathcal S = \left\{{E, O, A}\right\}$

but this approach has conceptual disadvantages.