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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)?

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It’s a proper class, not a set. – Brian M. Scott Aug 21 '12 at 17:42
To add on Brian's comment, it is usually denoted by $\mathrm{Ord}$ or $\mathrm{On}$. – Asaf Karagila Aug 21 '12 at 17:44

2 Answers 2

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).

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Ordinal addition isn’t commutative; the addition in a semiring is required to be. And ordinal addition isn’t a group operation, but this page, contrary to the statement that you quoted, requires a near ring to be a group under addition. – Brian M. Scott Aug 21 '12 at 18:04
@Brian: I did mention this below the quote... – Asaf Karagila Aug 21 '12 at 19:14
Didn’t see it. Did you by any chance do a really quick edit that didn’t show up as an edit? I could swear that I originally saw something slightly different. – Brian M. Scott Aug 21 '12 at 19:16
@Brian: Yes. I did made a change or two. I didn't see your comment until much later though... – Asaf Karagila Aug 21 '12 at 19:17
I may not have updated the page before making it. (And being on dial-up sometimes has slightly odd consequences.) – Brian M. Scott Aug 21 '12 at 19:19

In 1956 Tarski published a book titled Ordinal Algebras that I think is relevant to what you're asking about. I've glanced at it many times in university libraries (as well as Tarski's 1949 book Cardinal Algebras), but I don't really know much about it. Maybe some of the set theory specialists in StackExchange can say more about Tarski's book.

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I actually have Cardinal Algabras. I found it second hand for pennies. However it is related to the concept of Jonsson Cardinals which is not algebra in the modern sense of the word. – Asaf Karagila Aug 21 '12 at 19:16

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