Outline Of Algebraic Structures
In mathematics, there are many types of algebraic structures which are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a collection of axioms.
Another branch of mathematics known as universal algebra studies algebraic structures in general. From the universal algebra viewpoint, most structures can be divided into varieties and quasivarieties depending on the axioms used. Some axiomatic formal systems that are neither varieties nor quasivarieties, called nonvarieties, are sometimes included among the algebraic structures by tradition.
Concrete examples of each structure will be found in the linked Wikipedia article.
Algebraic structures are so numerous today that this article will inevitably be incomplete. In addition to this, there are sometimes multiple names for the same structure, and sometimes one name will be defined by disagreeing axioms by different authors. Most structures appearing on this page will be common ones which most authors agree on. Other web lists of algebraic structures, organized more or less alphabetically, include Jipsen and PlanetMath. These lists mention many structures not included below, and may present more information about some structures than is presented here.
Read more about Outline Of Algebraic Structures: Study of Algebraic Structures, Types of Algebraic Structures, Algebraic Structures With Additional Non-algebraic Structure, Algebraic Structures in Different Disciplines
Famous quotes containing the words outline of, outline, algebraic and/or structures:
“The beginning of an acquaintance whether with persons or things is to get a definite outline of our ignorance.”
—George Eliot [Mary Ann (or Marian)
“One by one objects are defined—
It quickens: clarity, outline of leaf
But now the stark dignity of
entrance—Still, the profound change
has come upon them: rooted, they
grip down and begin to awaken”
—William Carlos Williams (1883–1963)
“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)
“The American who has been confined, in his own country, to the sight of buildings designed after foreign models, is surprised on entering York Minster or St. Peter’s at Rome, by the feeling that these structures are imitations also,—faint copies of an invisible archetype.”
—Ralph Waldo Emerson (1803–1882)