More complex structures can be defined by introducing multiple operations, different underlying sets, or by altering the defining axioms. Examples of more complex algebraic structures include vector spaces, modules, and algebras.
The properties of specific algebraic structures are studied in abstract algebra. The general theory of algebraic structures has been formalized in universal algebra. Category theory is used to study the relationships between two or more classes of algebraic structures, often of different kinds. For example, Galois theory studies the connection between certain fields and groups, algebraic structures of two different kinds.
In a slight abuse of notation, the word "structure" can also refer only to the operations on a structure, and not the underlying set itself. For example, a phrase "we have defined a ring structure (a structure of ring) on the set A" means that we have defined ring operations on the set A. For another example, the group (\mathbb Z, +) can be seen as a set \mathbb Z that is equipped with an algebraic structure, namely the operation +.
- Garrett Birkhoff and Saunders MacLane, 1999 (1967). Algebra, 2nd ed. Chelsea.
- Michel, Anthony N., and Herget, Charles J., 1993 (1981). Applied Algebra and Functional Analysis. Dover.
A monograph available free online:
- Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. A Course in Universal Algebra. Springer-Verlag. ISBN 3-540-90578-2.
- Mac Lane, Saunders (1998) Categories for the Working Mathematician. 2nd ed. (Graduate Texts in Mathematics 5). Springer-Verlag.
- Taylor, Paul, 1999. Practical Foundations of Mathematics. Cambridge University Press.
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