Twisting a weak central groupoid

Occasionally, an equational law is preserved under a “twist” operation in which one replaces the magma operation \(x \diamond y\) by \(x \diamond' y := Tx \diamond Uy\) for some automorphisms \(T,U\) of the magma \(G\) that satisfy additional relations. In the case of the weak central groupoid law (1), we see that

\[ (y \diamond' x) \diamond' (x \diamond' (z \diamond' y))= (T^2 y \diamond TUx) \diamond (UTx \diamond (UTUz \diamond U^3y)) \]

so if \(T\) is an automorphism of order \(5\) and \(U = T^{-1}\) (so that \(T^2 = U^3\)), we conclude that this twisted magma is also a weak central groupoid. This can be used to generate further counterexamples. For instance, let \(M_2\) be the order two weak central groupoid with carrier \({\mathbb F}_2\) and with the NAND operation \(x \diamond y := 1 - xy\); this can easily be verified to be a weak central groupoid. It does not have any nontrivial automorphisms, but its fifth power \(M_2^{\otimes 5}\) has a cyclic shift \(T\) of order \(5\): \(T( (x_i)_{i\in \mathbb{Z}/5\mathbb{Z}} ) = (x_{i+1})_{i \in \mathbb{Z}/5\mathbb{Z}}\). If we twist \(M_2^{\otimes 5}\) by \(T\) and \(T^{-1}\), we obtain a weak central groupoid \(M\) with carrier \({\mathbb F}_2^5\) and magma operation

\[ (x_i)_{i\in \mathbb{Z}/5\mathbb{Z}} \diamond (y_i)_{i\in \mathbb{Z}/5\mathbb{Z}} = (1 - x_{i+1} y_{i-1})_{i \in \mathbb{Z}/5\mathbb{Z}}. \]

In particular, if \(x = (x_i)_{i\in \mathbb{Z}/5\mathbb{Z}}\), then

\[ x \diamond x = (1 - x_{i+1} x_{i-1})_{i \in \mathbb{Z}/5\mathbb{Z}} \]

and

\[ (x \diamond x) \diamond (x \diamond x) = (1 - (1 - x_{i+2} x_{i}) (1 - x_i x_{i-2}) )_{i \in \mathbb{Z}/5\mathbb{Z}} \]

which by the laws of boolean algebra simplify to

\[ (x \diamond x) \diamond (x \diamond x) = (x_i (x_{i-2} + x_i + x_{i+2}))_{i \in \mathbb{Z}/5\mathbb{Z}} \]

from which one can easily refute equation 151,

\[ x = (x \diamond x) \diamond (x \diamond x). \]

Informally, the reason for this is that equation 151 has a different semigroup twist symmetries: \(T^2 = 1, T = U^{-1}\) instead of \(T^5 = 1, T = U^{-1}\).

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