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normalizing reduction (Definition)
Definition 1   Let $ X$ be a set and $ \rightarrow$ a reduction (binary relation) on $ X$. An element $ x \in X$ is said to be in normal form with respect to $ \rightarrow$ if $ x \nrightarrow y$ for all $ y \in X$, i.e., if there is no $ y \in X$ such that $ x \rightarrow y$. Equivalently, $ x$ is in normal form with respect to $ \to$ iff $ x\notin \operatorname{dom}(\to)$. To be irreducible is a common synonym for `to be in normal form'.

Denote by $ \stackrel{*}{\rightarrow}$ the reflexive transitive closure of $ \rightarrow$. An element $ y \in X$ is said to be a normal form of $ x \in X$ if $ y$ is in normal form and $ x \stackrel{*}{\rightarrow} y$.

A reduction $ \to$ on $ X$ is said to be normalizing if every element $ x\in X$ has a normal form.

Examples.

  • Let $ X$ be any set. Then no elements in $ X$ are in normal form with respect to any reduction that is either reflexive. If $ \to$ is a symmetric relation, then $ x\in X$ is in normal form with respect to $ \to$ iff $ x$ is not in the domain (or range) of $ \to$.
  • Let $ X=\lbrace a,b,c,d\rbrace$ and $ \to = \lbrace (a,a),(a,b),(a,c),(b,c),(a,d)\rbrace$. Then $ c$ and $ d$ are both in normal form. In addition, they are both normal forms of $ a$, while $ d$ is a normal form only for $ a$. However, $ X$ is not normalizing because neither $ c$ nor $ d$ have normal forms.
  • Let $ X$ be the set of all positive integers greater than $ 1$. Define the reduction $ \to$ on $ X$ as follows: $ a\to b$ if there is an element $ c\in X$ such that $ a=bc$. Then it is clear that every prime number is in normal form. Furthermore, every element $ x$ in $ X$ has $ n$ normal forms, where $ n$ is the number of prime divisors of $ x$. Clearly, $ n\ge 1$ for every $ x\in X$. As a result, $ X$ is normalizing.



"normalizing reduction" is owned by CWoo.
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See Also: terminating reduction, diamond lemma

Also defines:  normal form, irreducible, normalizing
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Cross-references: prime divisors, prime number, clear, integers, positive, addition, range, domain, symmetric relation, Reflexive, reflexive transitive closure, iff, binary relation, reduction
There are 22 references to this entry.

This is version 7 of normalizing reduction, born on 2008-03-31, modified 2008-04-02.
Object id is 10460, canonical name is NormalizingReduction.
Accessed 408 times total.

Classification:
AMS MSC: 68Q42 (Computer science :: Theory of computing :: Grammars and rewriting systems)
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