In mathematics, and more specifically abstract algebra, the term algebraic structure generally refers to an arbitrary set (called carrier set or underlying set) with one or more finitary operations defined on it.
Common examples of structures include groups, rings, fields and lattices. More complex algebraic structures can be defined by introducing multiple operations, different underlying sets, or by altering the defining axioms. Examples of more complex structures include vector spaces, modules and algebras.
The properties of specific algebraic structures are studied in the branch known as 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 " means that we have defined ring operations on the set . For another example, the group can be seen as a set that is equipped with an algebraic structure, namely the operation .
Read more about Algebraic Structure: Overview, Hybrid Structures, Universal Algebra, Category Theory
Famous quotes containing the words algebraic and/or structure:
“I have no scheme about it,—no designs on men at all; and, if I had, my mode would be to tempt them with the fruit, and not with the manure. To what end do I lead a simple life at all, pray? That I may teach others to simplify their lives?—and so all our lives be simplified merely, like an algebraic formula? Or not, rather, that I may make use of the ground I have cleared, to live more worthily and profitably?”
—Henry David Thoreau (1817–1862)
“If rightly made, a boat would be a sort of amphibious animal, a creature of two elements, related by one half its structure to some swift and shapely fish, and by the other to some strong-winged and graceful bird.”
—Henry David Thoreau (1817–1862)