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binary operation, (a ⋆ b) ⋆ b= a , a ⋆ (a ⋆ b)= b

Source: Czech and Slovak Match 1996 P2

October 1, 2017
Binary operationalgebra

Problem Statement

Let ⋆ be a binary operation on a nonempty set MM. That is, every pair (a,b)∈M(a,b) \in M is assigned an element aa ⋆b b in MM. Suppose that ⋆ has the additional property that (a(a ⋆ b)b) ⋆b=a b= a and aa ⋆ (a(a ⋆b)=b b)= b for all a,b∈Ma,b \in M. (a) Show that aa ⋆ b=bb = b ⋆ aa for all a,b∈Ma,b \in M. (b) On which finite sets MM does such a binary operation exist?
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