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What do you call boolean functions which are identical accurate to argument order?

EDIT1

I meant not symmetric function.

I mean, for example, implication function with truth table

00=1 01=1 10=0 11=1

and the function

00=1 01=0 10=1 11=1

i.e. giving the same if arguments exchanged.

What do you call the relation of these functions to each other?

I.e.

$$f_1(x_1,x_2,...,x_k,...,x_l,...x_n)=f_2(x_1,x_2,...,x_l,...,x_k,...x_n)$$

i.e. the positions of arguments $k$ and $l$ was exchanged, which gave different function.

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If we have a boolean function $f$, or any other kind from $A^n$ to $B$, and a particular permutation $\pi$, you can talk about the function induced by $\pi$. I do not know of a special name in the Boolean case. –  André Nicolas Jun 18 '12 at 20:30

1 Answer 1

There are different equivalence relations.

In your case, $f_1$ and $f_2$ are said to be P-Equivalent or Permutation Equivalent.

P-Equivalence is a subset of NPN-Equivalence.

Furthermore, you can check Spectral Equivalence and Affine Equivalence. They are even more generalized forms of NPN-Equivalence.

If you want to learn more, Progress in Applications of Boolean Functions 1 is a good resource.

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