Structure under Right Operation is Semigroup

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Theorem

Let $\struct {S, \to}$ be an algebraic structure in which the operation $\to$ is the right operation.


Then $\struct {S, \to}$ is a semigroup.


Proof

We need to verify the semigroup axioms:

A semigroup is an algebraic structure $\struct {S, \circ}$ which satisfies the following properties:

\((\text S 0)\)   $:$   Closure      \(\ds \forall a, b \in S:\) \(\ds a \circ b \in S \)      
\((\text S 1)\)   $:$   Associativity      \(\ds \forall a, b, c \in S:\) \(\ds a \circ \paren {b \circ c} = \paren {a \circ b} \circ c \)      


By the nature of the right operation, $\struct {S, \to}$ is closed:

$\forall x, y \in S: x \to y = y \in S$

whatever $S$ may be.

Hence Semigroup Axiom $\text S 0$: Closure holds.


From Right Operation is Associative, $\to$ is associative.

Hence Semigroup Axiom $\text S 1$: Associativity holds.


So $\struct {S, \to}$ is a semigroup.

$\blacksquare$


Also see


Sources