Left finite set

A left finite set is a subset of the rational numbers by which each only a finite number of elements with contains. Left-finite sets are needed to define the Levi-Civita field . ${\displaystyle k\in \mathbb {Q} }$ ${\displaystyle x}$ ${\displaystyle x<k}$

definition

A set is called final if and only if holds. It denotes the power of a crowd. Equivalent to this: is either finite or order isomorphic to the natural numbers. ${\displaystyle M\subset \mathbb {Q} }$ ${\displaystyle \forall k\in \mathbb {Q} :|\{x\in M:x<k\}|<\infty }$ ${\displaystyle |\cdot |}$ ${\displaystyle M}$

Examples

• Every finite set is left finite.
• the set of natural numbers is left finite, although it contains an infinite number of elements.

Counterexamples

• The set of integers is not left finite.

properties

• The elements of a left-finite set can be arranged in ascending order with regard to their order relation.
• Every non-empty left-finite set (i.e. is left-finite and ) has a minimum . ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle M\neq \emptyset }$
• The union and the intersection of two left-fining sets are again left-fining sets.
• A subset of a left-finite set is also left-finite.
• If and are two left-finite sets, then the set is also left-finite. ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle M+N:=\{m+n|m\in M,n\in N\}}$

swell

• Martin Berz: Calculus and Numerics on Levi-Civita Fields . In: Martin Berz, Christian Bischof, George Corliss, Andreas Griewank (eds.): Computational differentiation. Techniques, applications, and tools. Proceedings of the 2nd International Workshop held in Santa Fe, NM, February 12-14, 1996 . 1996, ISBN 0-89871-385-4 , chap. 2 (English, online [PDF; accessed June 6, 2013]).