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SESSION 16

Idempotents, involutions, and graphs

1.  Solving exercises on idempotents and involutions

In Article III are some exercises about automorphisms, involutions, and idempotents. Exercise 4 asks whether the endomap α of the integer numbers, defined by assigning to each integer its negative: α(x) =x, is an involution or an idempotent, and what its fixed points are. What is an involution?

OMER: An endomap that composed with itself gives the identity.

Right. This means that for each element x its image is mapped back to x, hence it goes in a cycle of length 2 unless it is a fixed point. The internal diagram of an involution consists of some cycles of length 2 and some fixed points:

On the other hand, an idempotent endomap is one that applied ...

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