I was reading a book containing a typo to the effect that they defined the distributive property as: $$ a \circ (b\times c) = (a \times b) \circ (a \times c) \tag{*}\label{*} $$
which is wrong of course. I will call the property (*) "reverse distributivity" for now. It got me wondering:
Are there any examples of structures with this "reverse distributivity"? What can we say about such a structure? And are there names for these things?
Some findings so far:
If we assume the existence of neutral elements, then things quickly degenerate. Assume that $(M, \circ, 1_\circ, \times, 1_\times)$ is an algebraic structure with two binary operators satisfying (*), and where $1_\circ$ and $1_\times$ are neutral elements. Then we have: $$ 1_\circ = (1_\circ \times 1_\times) \circ (1_\circ \times 1_\times) \stackrel{\eqref{*}} = 1_\circ \circ (1_\times \times 1_\times) = 1_\times $$ so the identity elements are in fact equal. Let $1 := 1_\times = 1_\circ$. Then, for any $a,b\in M$: $$ a \times b = 1 \circ (a \times b) \stackrel{\eqref{*}} = (1 \times a ) \circ (1 \times b) = a \circ b $$ so in fact the two compositions are the same. In this case (*) becomes $$ a \circ (b\circ c) = (a \circ b) \circ (a \circ c) $$ which seems to be known as self-distributivity and shows up in a number of places (e.g. group conjugation and logical implication).
But if we want two (different) compositions that satisfy (*), then this shows that they at least cannot both have neutral elements. (If we only assume the existence of $1_\times$, then we can show that $a \circ a = a \circ 1_\times$ for all $a$). I haven't gone much further than this.