Study of Algebraic Structures
Algebraic structures appear in most branches of mathematics, and students can encounter them in many different ways.
- Beginning study: In American universities, groups, vector spaces and fields are generally the first structures encountered in subjects such as linear algebra. They are usually introduced as sets with certain axioms.
- Advanced study:
- Abstract algebra studies properties of specific algebraic structures.
- Universal algebra studies algebraic structures abstractly, rather than specific types of structures.
- Varieties
- Category theory studies interrelationships between different structures, algebraic and non-algebraic. To study a non-algebraic object, it is often useful to use category theory to relate the object to an algebraic structure.
- Example: The fundamental group of a topological space gives information about the topological space.
Read more about this topic: Outline Of Algebraic Structures
Famous quotes containing the words study of, study, algebraic and/or structures:
“If usually the “present age” is no very long time, still, at our pleasure, or in the service of some such unity of meaning as the history of civilization, or the study of geology, may suggest, we may conceive the present as extending over many centuries, or over a hundred thousand years.”
—Josiah Royce (1855–1916)
“While one should always study the method of a great artist, one should never imitate his manner. The manner of an artist is essentially individual, the method of an artist is absolutely universal. The first is personality, which no one should copy; the second is perfection, which all should aim at.”
—Oscar Wilde (1854–1900)
“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 there are people who feel that God wants them to change the structures of society, that is something between them and their God. We must serve him in whatever way we are called. I am called to help the individual; to love each poor person. Not to deal with institutions. I am in no position to judge.”
—Mother Teresa (b. 1910)