Does anyone know the name of this type of map

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The discussion centers around a specific algebraic structure involving a map denoted as (_)*, characterized by the property that (x)*** = (x)*. It is noted that this map is not an involution in general. Furthermore, the elements e satisfying e** = e do not form a substructure due to their lack of closure under addition. A participant suggests the term 'hyperinvolution' as a potential name for this type of map.

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I have been exploring an algebraic structure with a map (_)* such that
(x)*** = (x)*
but in general it is not an involution. Also, the set of elements e such that
e** = e
do not form a substructure because they are not closed to addition.

Has anyone seen such maps before, or know/can suggest a name?

Thank you!
 
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How about calling this a 'hyperinvolution'?
 

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