I have to describe with a formula in semigroups language $\{e,\cdot\}$ the traspositions of $S_4$. It does not work a formula to say that they are the elements of order 2 because the product of two disjointed traspositions still have order 2.
So I thought that the subset composed of the neutral element and all the products of two disjointed traspositions is a group isomorphic to $D_2$, therefore the traspositions can be described like:
$$(\neg (x=e)) \land (x \cdot x =e) \land \Bigl( \neg (\exists y_1 \exists y_2 (\neg(y_1=e)) \land (\neg(y_2=e)) \land (\neg(y_1 \cdot y_2 = e)) \land (y_1 \cdot y_1 = e) \land (y_2 \cdot y_2 = e) \land (y_1 \cdot y_2 \cdot y_1 \cdot y_2 = e) \land (x=y_1 \cdot y_2) )\Bigl)$$
Is it correct? I would appreciate confirmation.
