Even-odd multiplicative algebraic structure with idempotency?

Question

What is the algebraic structure for the multiplications of even elements and odd elements?

Please notice that $o*o=o$, $e*e=e$ (idempotency) and $o\not = e$.

1st structure is such that

1. even times even is even.
2. even times odd is odd.
3. odd times odd is odd.

and its multiplicative matrix

*| o e 
--------- 
o| o o
e| o e

2nd structure is such that

1. even times even is even.
2. even times odd is Even.
3. odd times odd is odd.

and its multiplicative matrix

*| O E 
--------- 
O| O E
E| E E

Answer 1

The operation is associative, but you are confusing identity and idempotent. One has $oo = o$ and $ee = e$ and hence both $o$ and $e$ are idempotent. However $oe = eo = e$ and $oo = o$. Thus $o$ is an identity. The resulting structure is a monoid.

Answer 2

If you consider a field of two elements, $0,1$ with $0\neq 1$, we have the addition structure

$$0+0=0$$ $$ 0+1=1$$ $$1+0=1$$ $$1+1=0$$

the multiplication structure differs for fields; we don't consider the inclusion of $0$ in the structure because there are no zero divisors. And again, we have

$$0\cdot 0=0$$ $$0\cdot 1=0$$ $$1\cdot 0=0$$ $$1\cdot 1=1$$

Now $0\mapsto e$ and $1\mapsto o$.