Closed Interval of Naturally Ordered Semigroup with Successor equals Union with Successor
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Theorem
Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.
Then:
- $\forall m, n \in \struct {S, \circ, \preceq}: m \preceq n \implies \closedint m {n \circ 1} = \closedint m n \cup \set {n \circ 1}$
where $\closedint m n$ is the closed interval between $m$ and $n$.
Proof
Let $m \preceq n$. Then:
\(\ds \) | \(\) | \(\ds x \in \closedint m {n \circ 1}\) | ||||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds m \preceq x \land x \preceq \paren {n \circ 1}\) | Definition of Closed Interval | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds m \preceq x \land \paren {x \prec n \circ 1 \lor x = n \circ 1}\) | Definition of Strictly Precede |
\(\ds \) | \(\) | \(\ds x \in \closedint m n \cup \set {n \circ 1}\) | ||||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds m \preceq x \land \paren {x \preceq n \lor x = n \circ 1}\) | Definition of Closed Interval and Definition of Set Union | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds m \preceq x \land \paren {x \prec n \lor x = n \lor x = n \circ 1}\) | Definition of Strictly Precede | |||||||||||
\(\ds \) | \(\leadstoandfrom\) | \(\ds m \preceq x \land \paren {x \prec n \circ 1 \lor x = n \circ 1}\) | Definition of Strictly Precede |
Thus:
- $\closedint m {n \circ 1} = \closedint m n \cup \set {n \circ 1}$
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Corollary $16.4.2$