Note to Steve: This task needs a lot of work to get ready for publication
Note: This task may seem abstract, but upon playing around with the structures various particular examples will emerge.
Many mathematical theories can be broken into sets $X$ of 'elements' $\{ x_1, x_2, x_3, \dots \}$ which can be combined using two different mathematical operations, which we can abstractly call $*$ and $\dagger$
In this problem we require that $a*b=b*a$ and $ a\dagger b = b\dagger a$ and that $a*b$ and $a\dagger b$ are contained in the set $X$ for every $a$ and $b$.
Can you invent (i.e. specify precisely in some way what goes to what) the properties of $*$ and $\dagger$ in which you can combine elements of these sets of objects:
$X_1 = \{0, 1\}$
$X_6 = \{0, 1, 2, 3, 4, 5, 6\}$
$X_7 = \{0, 1, 2, 3, 4, 5, 6, 7\}$
$\mathbb{N} = \{1, 2, 3, \dots\}$
$2$ by $2$ matrices - you can choose what sorts of elements are in each matrix.
Words (i.e. strings of letters) made from the letters $a$ and $b$.
For each invention can you find an element $a$ such that $a*b=b$ or $a\dagger b = b$ for every element $b$?