Translation Monoids and Recursive Evaluation in Finite Binary Algebras

Let $A=(A,\star)$ be a finite binary algebra, not necessarily associative. For each $n\geq 1$, every full binary bracketing on $x_1,\dots,x_n$ determines an $n$-ary term operation on $A$, and hence an evaluation word obtained by listing its values on $A^n$ in lexicographic order. This produces an $m^n\times C_{n-1}$ array, where $m=|A|$ and $C_{n-1}$ is the $(n-1)$st Catalan number. We show that the recursive structure of these arrays is governed by the translation monoid $$T(A)=\langle L_a,R_a:a\in A\rangle\leq A^A, \qquad L_a(x)=a\star x,\quad R_a(x)=x\star a.$$ More precisely, context maps arising from subterms are exactly the elements of $T(A)$, so every element of the translation monoid occurs as a recursive block map. We also prove that rank defines a natural chain of two-sided ideals in $T(A)$, that the minimum-rank elements form a minimal nonempty two-sided ideal, and that Green's $\mathcal J$-classes are contained in rank layers. Finally, we show by example that equal rank does not determine the $\mathcal J$-class in general.