Consider the set class $\mathrm{Ord}$ of all (finite and infinite) ordinal numbers, equipped with ordinal arithmetic operations: addition, multiplication, and exponentiation. It is closed under these operations. Addition is non-commutative and there are no additive or multiplicative inverses.
Is $(\mathrm{Ord}, +)$ a magma? What algebraic structure does $\mathrm{Ord}$ posses (under either/both $+, \times$ operations)?