# Combinators that form a 1-symbol basis of alternating associative combinatory logic

It is known that the combinators $$S,K$$ form a basis for lambda calculus. It's also known that the iota combinator $$\lambda x.((x S) K)$$ is a basis.

Chris Barker found that the iota combinator allows a 2-symbol basis by encoding the alternating associativity. This is described by the word translation function $$\tau:\{0,1\}^*\to\mathrm{COMB}$$, where $$\mathrm{COMB}$$ is an expression of combinators with normal application notation: \begin{align*} \tau(\varepsilon)&= I\\ \tau(x0)&=((\tau(x)S)K)\\ \tau(x1)&=(S(K\tau(x))) \end{align*} I now wondered what happens if you add the restriction of "alternating associativity" for a 1-symbol basis, specifically: $$\tau_{l,r}:\{1\}^*\to\mathrm{COMB}$$: \begin{align*} \tau_{l\mid r}(\varepsilon)&= E\\ \tau_l(x1)&=(L\tau_r(x))\\ \tau_r(x1)&=(\tau_l(x)R) \end{align*} The translation would start with $$\tau_l$$ if the length of the word is even and $$\tau_r$$ if the length is odd; such that $$E$$ will only appear on the left side of the innermost application.

With $$E,L,R$$ being arbitrary (preferably common) combinators, does such a basis exist?