Definition For Linearly Ordered Groups
Let x and y be positive elements of a linearly ordered group G. Then x is infinitesimal with respect to y (or equivalently, y is infinite with respect to x) if, for every natural number n, the multiple nx is less than y, that is, the following inequality holds:
The group G is Archimedean if there is no pair x,y such that x is infinitesimal with respect to y.
Additionally, if K is an algebraic structure with a unit (1) — for example, a ring — a similar definition applies to K. If x is infinitesimal with respect to 1, then x is an infinitesimal element. Likewise, if y is infinite with respect to 1, then y is an infinite element. The algebraic structure K is Archimedean if it has no infinite elements and no infinitesimal elements.
Read more about this topic: Archimedean Property
Famous quotes containing the words definition, ordered and/or groups:
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“Twenty-four-hour room service generally refers to the length of time that it takes for the club sandwich to arrive. This is indeed disheartening, particularly when you’ve ordered scrambled eggs.”
—Fran Lebowitz (b. 1950)
“Some of the greatest and most lasting effects of genuine oratory have gone forth from secluded lecture desks into the hearts of quiet groups of students.”
—Woodrow Wilson (1856–1924)