Let binary operation $ \circ $ on set $X$ be function $\circ : X \times X \rightarrow X$.
Binary operation on set X is :
unitary if for some element $1 \in X$ and any $x \in X$ we've got $(1 \circ x) = x = (x \circ 1)$
alternate if for any $x,y \in X$ we've got $x \circ y = y \circ x$.
1) $x \circ (y \circ x) = y \implies (x \circ y) \circ x = y$
2) Let $\circ$ and $\star$ be binary operation on $X$. $\circ$ and $\star$ are unitary and for any $a,b,c,d \in X$ we've got $(a \circ b) \star (c \circ d) = (a \star c) \circ (b \star d)$. Show that $\star$ and $\circ$ are alternate and identical.
Thanks for help.