I am wondering if given to varieties $V,W$ in the sense of Universal algebra once could describe free algebras over $V\cap W$ in some meaningful way? Now what I mean by meaningful is not even completely clear to me. I would like to use this description to say something regarding the subalgebras of $F_{V\cap W}(x_1,...,x_n)$ if I know something about subalgebras of $F_{V}(x_1,...,x_n)$ and $F_{W}(x_1,...,x_n)$. I am aware this isn't completely well posed question but maybe someone is able to help.
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I think the question is well-posed (+1). Now, from the top of my head, I don't see a solution. Surely one could say use the identities valid in $V$, which allow you to compute $F_V(\bar x)$ and apply them to $F_W(\bar x)$. But that is the same as computing $F_{V\cap W}(\bar x)$ from start. Perhaps there is another way... – amrsa Nov 09 '23 at 13:34