Subsemigroup of Ordered Semigroup is Ordered
Theorem
Let $\struct {S, \circ, \preceq}$ be an ordered semigroup.
Let $\struct {T, \circ_T}$ be a subsemigroup of $\struct {S, \circ}$.
Then the ordered structure $\struct {T, \circ_T, \preceq_T}$ is also an ordered semigroup.
In the above:
- $\circ_T$ denotes the operation induced on $T$ by $\circ$
- $\preceq_T$ denotes the restriction of $\preceq$ to $T \times T$.
Proof
It is necessary to ascertain that $\struct {T, \circ {\restriction_T} }$ fulfils the ordered semigroup axioms:
An ordered semigroup is an algebraic system $\struct {S, \circ, \preceq}$ which satisfies the following properties:
\((\text {OS} 0)\) | $:$ | Closure | \(\ds \forall a, b \in S:\) | \(\ds a \circ b \in S \) | |||||
\((\text {OS} 1)\) | $:$ | Associativity | \(\ds \forall a, b, c \in S:\) | \(\ds a \circ \paren {b \circ c} = \paren {a \circ b} \circ c \) | |||||
\((\text {OS} 2)\) | $:$ | Compatibility of $\preceq$ with $\circ$ | \(\ds \forall a, b, c \in S:\) | \(\ds a \preceq b \implies \paren {a \circ c} \preceq \paren {b \circ c} \) | |||||
where $\preceq$ is an ordering | \(\ds a \preceq b \implies \paren {c \circ a} \preceq \paren {c \circ b} \) |
In this context, we see that $\text {OS} 0$ and $\text {OS} 1$ are fulfilled a fortiori by dint of $\struct {T, \circ {\restriction_T} }$ being a subsemigroup of $\struct {S, \circ}$.
We have that $\struct {S, \circ, \preceq}$ is an ordered semigroup.
From Restriction of Ordering is Ordering, we have that $\preceq_T$ is an ordering.
Hence:
\(\ds \forall a, b \in S: \, \) | \(\ds a \preceq b\) | \(\implies\) | \(\ds \paren {a \circ c} \preceq \paren {b \circ c}\) | Ordered Semigroup Axiom $\text {OS} 2$ on $\struct {S, \circ, \preceq}$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall a, b \in T: \, \) | \(\ds a \preceq_T b\) | \(\implies\) | \(\ds \paren {a \circ_T c} \preceq_T \paren {b \circ_T c}\) |
and:
\(\ds \forall a, b \in S: \, \) | \(\ds a \preceq b\) | \(\implies\) | \(\ds \paren {c \circ a} \preceq \paren {c \circ b}\) | Ordered Semigroup Axiom $\text {OS} 2$ on $\struct {S, \circ, \preceq}$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall a, b \in T: \, \) | \(\ds a \preceq_T b\) | \(\implies\) | \(\ds \paren {c \circ_T a} \preceq_T \paren {c \circ_T b}\) |
Hence $\preceq_T$ fulfils Ordered Semigroup Axiom $\text {OS} 2$ on $\struct {T, \circ_T, \preceq_T}$.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 15$: Ordered Semigroups