Principle of Induction applied to Interval of Naturally Ordered Semigroup

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Theorem

Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup.

Let $\left[{p \,.\,.\, q}\right]$ be a closed interval of $\left({S, \circ, \preceq}\right)$.

Let $T \subseteq \left[{p \,.\,.\, q}\right]$ such that the minimal element of $\left[{p \,.\,.\, q}\right]$ is in $T$.

Let:

$x \in T: x \prec q \implies x \circ 1 \in T$


Then:

$T = \left[{p \,.\,.\, q}\right]$


Proof

Let $T' = T \cup \left\{{x \in S: q \prec x}\right\}$.

Then $T'$ satisfies the conditions of the Principle of Mathematical Induction.

Therefore:

$T' = \left\{{x \in S: p \preceq x}\right\}$

Therefore:

$T = \left[{p \,.\,.\, q}\right]$

$\blacksquare$


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