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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