Let $(X,*)$ be a semigroup. Furthermore, for every $a$ and $b$ in $X$, if $a\ne b$, then $a*b\ne b*a$.

Based on the defined semigroup, choose the correct equalities from the options given below.

A. For every $a$ in $X,\ a*a=a$

B. For every $a,b$ in $X,\ a*b*a=a$

C. For every $a,b,c$ in $X,\ a*b*c=a*c$

Given condition means if two elements commute, then they must be equal.

For A:

$a*(a*a)=(a*a)*a$ by associativity.

So, $a$ and $a*a$ commute.

Hence, $a=a*a$.

So, A is true.

For B:

$a*(a*b*a)=(a*a)*b*a=a*b*a$

$(a*b*a)*a=a*b*(a*a)=a*b*a$

So, $a$ and $a*b*a$ commute.

Hence, $a*b*a=a$.

So, B is true.

For C:

Using B, $c*a*c=c$ and $a*c*a=a$.

Now,

$(a*b*c)(a*c)=a*b*(c*a*c)=a*b*c$

$(a*c)(a*b*c)=(a*c*a)*b*c=a*b*c$

So, $a*b*c$ and $a*c$ commute.

Hence, $a*b*c=a*c$.

So, C is also true.