Strict Lower Closure of Sum with One
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Theorem
Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.
Then:
- $\forall n \in \struct {S, \circ, \preceq}: \paren {n \circ 1}^\prec = n^\prec \cup \set n$
where $n^\prec$ is defined as the strict lower closure of $n$, that is, the set of elements strictly preceding $n$.
Proof
First note that as $\struct {S, \circ, \preceq}$ is well-ordered and hence totally ordered, the Trichotomy Law applies.
Thus:
\(\ds \forall m \in S: \, \) | \(\ds \) | \(\) | \(\ds m \notin n^\prec\) | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds \neg \ m \prec n\) | Definition of Strict Lower Closure of Element | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds m = n \lor n \prec m\) | Trichotomy Law | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds n \preceq m\) | Definition of Strictly Precede |
So:
\(\ds \forall p \in S: \, \) | \(\ds \) | \(\) | \(\ds p \notin \paren {n \circ 1}^\prec\) | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds n \circ 1 \preceq p\) | from the above | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds n \prec p\) | Sum with One is Immediate Successor in Naturally Ordered Semigroup |
Similarly:
\(\ds \forall p \in S: \, \) | \(\ds \) | \(\) | \(\ds p \notin n^\prec \cup \set n\) | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds n \preceq p \land n \ne p\) | from the above | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds n \prec p\) | Definition of Strictly Precede |
So:
- $p \notin n^\prec \cup \set n \iff p \notin \paren {n \circ 1}^\prec$
Thus:
- $\relcomp S {\paren {n \circ 1}^\prec} = \relcomp S {n^\prec \cup \set n}$
from the definition of relative complement.
So:
\(\ds \paren {n \circ 1}^\prec\) | \(=\) | \(\ds \relcomp S {\relcomp S {\paren {n \circ 1}^\prec} }\) | Relative Complement of Relative Complement | |||||||||||
\(\ds \) | \(=\) | \(\ds \relcomp S {\relcomp S {n^\prec \cup \set n} }\) | from above | |||||||||||
\(\ds \) | \(=\) | \(\ds n^\prec \cup \set n\) | Relative Complement of Relative Complement |
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Corollary $16.4.1$