﻿ Successor ordinal

Successor ordinal

In mathematics, are the terms successor and predecessor, the mental concepts of " descent or succession to " and " counting " formalized and generalized.

Successor and predecessor in scoring and in orders

When counting the successor of an integer is intuitively the next larger number: So about 2 the successor of 1, 3, the successor of 2, etc. When counting down to get from 9 to its predecessor 8, etc. This naive in itself discovery, the children always understand back in the game, one can formalize a mathematical characterization of the natural numbers, which was developed by Giuseppe Peano and his honor called Peano axiom system.

During the up - down counting one finds that it does not depend on the significance of the number words, but only on their order. This finding suggests a generalization of the " Zählnachbarn " predecessor and successor on graphs and ordered sets to:

Definitions

Let ( M, < ) is a strictly ordered set. Then say

• C successor of b if b < c and no smaller element than c exists with this property
• A predecessor of b if a

For a strict total ordering this definition ensures at the same time that the predecessor and successor (if any) are uniquely determined. In general, however, an element several, not mutually comparable predecessors and / or successors. This general approach will be taken to graph theory further. It thus comes close to vormathematischen Ethnicity.

In order theory is defined as:

• C is the successor of b if b < c and every other element is greater with this property
• Is a predecessor of b if a

This predecessors and successors are (if any) in non- totally ordered sets clearly. Thus, the counting process is rather shown.

Example

The graph shown illustrates the divider relation in the set of divisors of the number 12 The abstract relation is represented here by arrows and has the meaning " 3 divides 6", " 1 divides 4" etc. The order is not total, because there are elements, which can not be compared with each other, for example, is not a divisor of 2 3 or vice versa. But the purpose of the second order theoretical definition, the two no successor has a predecessor, in terms of the first, more general definition, the 2 has a predecessor and two successors.

Applications

• In a well-ordered set ( ordinal number ) of each element has a unique successor, unless it is the maximum of well-ordered set. Elements without predecessor hot here Limes elements or limit ordinals.
• The existence of predecessors and successors in minor amounts can also be studied with topological means. See order topology.
• The concept of predecessors and successors in directed graphs is explained (graph theory ) in the article neighborhood.
• Order Theory
• Number
590155
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