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Abstract Systems

Thus far we assembled a sizable collection of concepts, terms, and notations. We have illustrated these with examples that are sometimes intuitive, and elementary. We are working towards a common language.

Not many years ago, algebra was considered to be merely symbolic arithmetic and this view is still not uncommon. In arithmetic calculations specific numbers were used while letters representing numbers were used in algebraic calculations. Eventually mathematicians became aware that some of the symbolic statements from this generalized arithmetic, which were true when symbols were replaced by numbers, were true when the symbols were replaced by other objects. Thus algebra evolved into the study of mathematical systems which have many of the properties of the "ordinary" number system.

Definition 1: A mathematical system S is a set S = {E,O,A} where E is a nonempty set of elements, O is a set of relations and operations on E, and A is a set of axioms concerning the elements of E and O. The elements of E are called the elements of the system.

Inasmuch as certain concepts have been assumed or defined for the sets of E and O, they are also taken to be part of S. We shall take the equality of elements in E as the basic equivalence relation of every mathematical system.

A mathematical system is analyzed by introducing definitions, new relations, new operations, etc. into the system and proving theorems concerning the various elements of components of the system. The resulting body is usually designated as the theory of the system.

A system S = {E,O A} is classified as abstract if the elements of the sets E and O are undefined except as their properties, which are set forth by A, define them. Otherwise it is thought of as a concrete system, a special case of an abstract system.

Systems which possess many of the properties of the systems of integers are called algebraic systems. There are also geometrical systems, topological systems, etc.