What are the main relationships between exclusive OR / logical biconditional? Let $\mathbb{B} = \{0,1\}$ denote the Boolean domain.
Its well known that both exclusive OR and logical biconditional make $\mathbb{B}$ into an Abelian group (in the former case the identity is $0$, in the latter the identity is $1$).
Furthermore, I was playing around and noticed that these two operations 'associate' over each other, in the sense that $(x \leftrightarrow y) \oplus z$ is equivalent to $x \leftrightarrow (y \oplus z).$
This is easily seen via the following chain of equivalences.


*

*$(x \leftrightarrow y) \oplus z$

*$(x \leftrightarrow y) \leftrightarrow \neg z$

*$x \leftrightarrow (y \leftrightarrow \neg z)$

*$x \leftrightarrow (y \oplus z)$


Anyway, my question is, what are the major connections between the operations of negation, biconditional, and exclusive OR? Furthermore, does $(\mathbb{B},\leftrightarrow,\oplus,\neg)$ form any familiar structure? I know that the binary operations don't distribute over each other, so its not a ring.

You probably already know this, but the immediate connection between them is $(x\oplus y) \leftrightarrow \neg(x \leftrightarrow y)$. Then the exclusive OR reduces trivially to the biconditional, and vice versa.
