Subsemigroup of Ordered Semigroup is Ordered

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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

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