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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.2012-08-21
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    To add on Brian's comment, it is usually denoted by $\mathrm{Ord}$ or $\mathrm{On}$.2012-08-21

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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](http://en.wikipedia.org/wiki/Near-ring), contrary to the statement that you quoted, requires a near ring to be a group under addition.2012-08-21
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    @Brian: I did mention this below the quote...2012-08-21
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    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.2012-08-21
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    @Brian: Yes. I did made a change or two. I didn't see your comment until much later though...2012-08-21
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    I may not have updated the page before making it. (And being on dial-up sometimes has slightly odd consequences.)2012-08-21
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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.2012-08-21