TLDR: Is there a notion of Kleene Algebra for star-free regular expressions?
Kleene Algebras are algebraic structures that are equivalent to regular expressions. A Kleene Algebra is an idempotent semiring (non-commutative multiplication) with an extra unary operation $*$ that satisfies that following equations: \begin{align} &1+x\cdot x^* = x^* \\ &1+x^*\cdot x = x^* \\ &z+xy\le y \implies x^*\cdot z\le y \\ &z+yx\le y \implies z\cdot x^*\le y \end{align} where $x\le y$ means there exists $z$ such that $x+z=y$. Equivalence with regular expressions formally means, given regular expressions $r_1$ and $r_2$, $\mathcal{L}(r_1)=\mathcal{L}(r_2)$ iff we can prove $r_1=r_2$ using the axioms of Kleene Algebra.
Star-free regular expressions are regular expressions without the Kleene star but with the ability to take complements i.e. $$r_1,r_2::= \emptyset\mid a,b,c\in A\mid r_1\cup r_2\mid r_1\cap r_2\mid r_1\cdot r_2\mid (r_1)^c$$ where $A$ is the set of alphabets. Observe that complementation restores some Kleene star powers but not all: $A^*=(\emptyset)^c$ while $(aa)^*$ has no star-free form.
I was wondering how to tweak Kleene Algebra so that we capture exactly star-free expressions. While coming up with equation templates for the complementation operation the challenge is to get an equation for $(r_1\cdot r_2)^c$ because the rest of the interleaving of operations is taken care of by De Morgan's laws. Also, since Kleene Algebras do not explicitly talk about the alphabet set, I guess $a^c$ should be considered a normal form.
