Which algebraic structure captures the ordinal arithmetic? 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)?

 A: With only addition, the ordinals form a monoid.
The ordinal numbers with both addition and multiplication form a non-commutative semiring.
To quote from Wikipedia's page about semirings:

A near-ring does not require addition to be commutative, nor does it require right-distributivity. Just as cardinal numbers form a semiring, so do ordinal numbers form a near-ring.

(Although in the page of near-rings it is required that addition has inverse; so perhaps ordinals just form a non-commutative semiring).
A: In 1956 Tarski published a book titled Ordinal Algebras (table of contents and review in Journal of Symbolic Logic and review in Mathematical Gazette and review, in German, in Philosophische Rundschau) that I think is relevant to what you're asking. I've paged through this book many times in university libraries (and I've also often paged through Tarski's 1949 book Cardinal Algebras—Preface and review in Bulletin of the American Mathematical Society and review in Mathematical Gazette and review in Journal of Symbolic Logic and review in Science Progress), but I don't know very much about it. Maybe some of the set theory specialists in StackExchange can say more about these two books by Tarski.
