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Erschienen in:
2015 | OriginalPaper | Buchkapitel
3. Algebraic Structures
verfasst von : Jörg Liesen, Volker Mehrmann
Erschienen in: Linear Algebra
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Abstract
An algebraic structure is a set with operations between its elements that follow certain rules. As an example of such a structure consider the integers and the operation ‘+.’ What are the properties of this addition? Already in elementary school one learns that the sum a + b of two integers a and b is another integer. Moreover, there is a number 0 such that 0 + a = a for every integer a, and for every integer a there exists an integer \(-a\) such that \((-a)\) + a = 0. The analysis of the properties of such concrete examples leads to definitions of abstract concepts that are built on a few simple axioms. For the integers and the operation addition, this leads to the algebraic structure of a group.