ok lets think about this, like Alan Turing, aged about 16. You like quantum mechanics math, and are fascinated by relativistic solutions to things like the Schrodinger equation.
So, you learn about continuous wave functions. But, its somehow easier to reason with discrete wave functions since you don’t needs such advanced calculus skills, now. You can model with rather simpler group theory (staying with algebra), or just fiddle with matrices.
Having got to representing quantum states (that steam room calculation environment, remember) using kets (ie column vectors), one quickly embraces the notion of basis. And, its not too hard to get to operators, that change basis. Even more interesting are those hermitian operators that force the steam room to stop steaming and come to a decision – when you take a slice subscape associated with a particular eigenvalue and project the stream onto that plane. If it projects just so, you know your wavefunction was an eigenfunction of the operator in question.
Now you get to operators that themselves as generics, as we would say today. That is, if A is the operator, then F(A) is a generic operator tuned up for F. one such F(A) is A*A… or do the operator twice in sequence. One we have the notion of sequence, then we ask: can we calulate the two parts in parallel (or not)? Commutators rear the head, answering yes or no.
Non-commutative operators are interesting to Turing at an early age. For within lies a yet more interesting notion, of uncertainty. And, he gets to play with quaternion groups, that showcase the non-commutativty while still facillitatin representaion in terms of cyclic groups. It is from the non-parallelism of the algebra plus the entirely cyclic nature of the representation that he gets to the notion that one can impose a “series of constraints” on a quantum system – representing the non-commutative points. These in some characterise their dual space, giving us what we would call the coding and dual-coding spaces today.
In the case of the enigma bombe design, with diagonalboard, we are in a world in which showcases both constraints and paralleilsm, with the constraints of the dual space controlling the calulations in the foreground space that are using tensor parallelism as much as possible.