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Homenormalizing reduction
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normalizing reduction
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 $X$ 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, $n\geq 1$ for every $x\in X$. As a result, $X$ is normalizing.
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