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Give an example of a set with two binary operations, addition and multiplication, in which the left distributive law holds but the right distributive law does not hold. I.e.: $$a(b+c)=ab+ac\text{, but }(b+c)a=ba+ca.$$ If this is not possible, then prove that one implies the other.

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The multiplication of ordinal numbers is only left-distributive, not right-distributive, see here, and also here. A further discussion is given here.

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