Complex Numbers Exponential - Generalizations - in Abstract Algebra

In Abstract Algebra

Exponentiation for integer exponents can be defined for quite general structures in abstract algebra.

Let X be a set with a power-associative binary operation which is written multiplicatively. Then xn is defined for any element x of X and any nonzero natural number n as the product of n copies of x, which is recursively defined by

begin{align} x^1 &= x \ x^n &= x^{n-1}x quadhbox{for }n>1
end{align}

One has the following properties

begin{align} (x^i x^j) x^k &= x^i (x^j x^k) quadtext{(power-associative property)} \ x^{m+n} &= x^m x^n \ (x^m)^n &= x^{mn}
end{align}

If the operation has a two-sided identity element 1 (often denoted by e), then x0 is defined to be equal to 1 for any x.

begin{align} x1 &= 1x = x quadtext{(two-sided identity)} \ x^0 &= 1
end{align}

If the operation also has two-sided inverses, and multiplication is associative then the magma is a group. The inverse of x can be denoted by x−1 and follows all the usual rules for exponents.

begin{align} x x^{-1} &= x^{-1} x = 1 quadtext{(two-sided inverse)} \ (x y) z &= x (y z) quadtext{(associative)} \ x^{-n} &= left(x^{-1}right)^n \ x^{m-n} &= x^m x^{-n}
end{align}

If the multiplication operation is commutative (as for instance in abelian groups), then the following holds:

If the binary operation is written additively, as it often is for abelian groups, then "exponentiation is repeated multiplication" can be reinterpreted as "multiplication is repeated addition". Thus, each of the laws of exponentiation above has an analogue among laws of multiplication.

When one has several operations around, any of which might be repeated using exponentiation, it is common to indicate which operation is being repeated by placing its symbol in the superscript. Thus, xn is x ∗ ··· ∗ x, while x#n is x # ··· # x, whatever the operations ∗ and # might be.

Superscript notation is also used, especially in group theory, to indicate conjugation. That is, gh = h−1gh, where g and h are elements of some group. Although conjugation obeys some of the same laws as exponentiation, it is not an example of repeated multiplication in any sense. A quandle is an algebraic structure in which these laws of conjugation play a central role.

Read more about this topic:  Complex Numbers Exponential, Generalizations

Other articles related to "abstract, in abstract, abstracts, algebra":

History Of The Race And Intelligence Controversy - History - 1960-1980
... ("Level I" ability), they had difficulties with abstract conceptual reasoning ("Level II" ability) ... in the population according to SES level 1 and level 2, associative learning and abstract thinking (g), respectively ... of associative learning, but they fall behind on abstract thinking ...
Joseph Stanislaus Ostoja-Kotkowski - Biographical Summary of Works
... Ostoja introduced the new abstract expression of Europe both to lecturers and students at the Victorian School of Art, Melbourne ... Ostoja's work in abstract expression began to receive accolades ... the set for The Egg and a new, light/colour abstract presentation for two performances of the South Australian Ballet Theatre ...
List Of Academic Databases And Search Engines
... eJournal Multidisciplinary science (student based) Student driven research abstracts, posters, articles, science specific search engine, public forum Free APeJ ... Multidisciplinary Scholarly journals published in Africa Free abstracts Subscription full-text African Journals OnLine AgeLine Sociology, Gerontology ... Subscription Airiti Inc Analytical Abstracts Chemistry Subscription Royal Society of Chemistry Analytical sciences digital library Analytical sciences Free National Science ...
Timeline Of Category Theory And Related Mathematics - 1945–1970
... varying sets (a generalization of abstract sets). 1956 Henri Cartan–Samuel Eilenberg Influential book Homological Algebra, summarizing the state of the art in its topic at that time ... Grothendieck Abelian categories in homological algebra that combine exactness and linearity ...
Graham Priest - Selected Works - Articles
... http//onlinelibrary.wiley.com/doi/10.1111/j.0031-8094.2000.00187.x/abstract ... http//onlinelibrary.wiley.com/doi/10.1111/j.0066-7372.2003.00065.x/abstract ... http//www.doaj.org/doaj?func=abstract id=970235 recNo=6 toc=1 uiLanguage=en ...

Famous quotes containing the words algebra and/or abstract:

    Poetry has become the higher algebra of metaphors.
    José Ortega Y Gasset (1883–1955)

    We must trust infinitely to the beneficent necessity which shines through all laws. Human nature expresses itself in them as characteristically as in statues, or songs, or railroads, and an abstract of the codes of nations would be an abstract of the common conscience.
    Ralph Waldo Emerson (1803–1882)