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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)?

  • 0
    It’s a proper class, not a set.2012-08-21
  • 0
    To add on Brian's comment, it is usually denoted by $\mathrm{Ord}$ or $\mathrm{On}$.2012-08-21

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