Question:
There are $2^4=16$ distinct binary boolean operators. Two operators are regarded the same if one can be obtained from the other by exchanging the operands (input). It is easy to see only $12$ operators remain. How many distinct subsets of the $12$ are closed under composition?
I think the $12$ operators are {$T,F$, Identity, Not, And, Nand, Or, Nor, Implies, Non-implies, Equivalent, Xor (Non-equivalent)} respectively, where $T$ and $F$ are $0$-ary, Identity and Not are unary and the rest are truly binary. It's not hard to apply brute force as the number of cases is sufficiently small ($2^{11}=2048$ cases, since the identity must be included, which is the compositions of $0$ operators). But I would like to see a more elegant argument, one that may provide some insight into why the closed subsets are what they are and why there are no others. Many thanks.
Bonus: It will be much more appreciated if additional interpretations of the restricted range of logic represented by each closed subset are provided.