Let $S$ be a non-empty set and $*$ be a binary operation on $S$ i.e. let $(S,*)$ be a magma such that
$a*(b*c)=(c*a)*b \space;\forall a,b,c\in S $ , then is it consistent that $(S,*)$ has neither any left identity
nor any right identity ?
( I can prove that it has a left identity iff it has a right identity and then $(S,*)$ also becomes commutative )
It is possible that $S$ does not have a right or a left identity. Indeed, let $|S|>1$ and let $s \in S$. Define $a \ast b$ to be equal to $s$ for all $a,b \in S$. Let $t$ be an element of $S$ different from $s$. We have $s \ast t = t \ast s = s$, hence $s$ is neither a left nor a right identity. Since for each $u \in S$ with $u \neq s$ we have $u \ast u = s$, also elements of $S$ not equal to $s$ are not a left nor a right identity. We conclude that $S$ has no left identity or right identity.