Why do we need another definition?
We have the definition for generator being an element of a group.
But a sturdy element is instead an element of a set.
We could widen the definition for generator to be an element of a set.
Would that create problems?
Should we say super generator instead of sturdy element?
But this might require or imply a generator could be an element of a set.
There is nothing wrong with describing the results of using a sturdy element modulo enclosing set as being a cyclic group.
However to call those results a cyclic subgroup implies a parent group and that is not always the case.
We could widen the definition of subgroup to include a parent set also.
Would that create problems?
A new definition “super element” avoids having to widen existing definitions.
Perhaps because a debate about widening generator or subgroup has not yet happened shows that these elements and their behaviour have not yet been studied more widely.
“They are just generators”
But they do not belong to a group [in all cases]
“They are super generators”
We need to be careful we do not imply group membership [in all cases] by reusing the term generator
“They generate cyclic subgroups”
We need to be careful we do not imply group membership [in all cases] by the current definition of a subgroup
But wouldn’t having a sturdy element require a new notation?
If we reuse the angle brackets that we use for generator typically then we could risk confusion.
How about double angle brackets
<<-1+2^t>> for the sturdy element -1+2^t
<<1+2^t>> for the sturdy element 1+2^t
<<-1+3^t>> for the sturdy element -1+3^t
<<1+3^t>> for the sturdy element 1+3^t
Each of those sturdy elements has a context set or enclosing set so should be quoted in more complete form as shown next.
