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Fix a commutative ring $\mathbf{k}$, two elements $\beta,\alpha\in\mathbf{k}$and a positive integer $n$. Let $\mathcal{X}$ be the polynomial ring over$\mathbf{k}$ in the $n(n-1)/2$ indeterminates $x_{i,j}$ for all $1\leq i<j\leqn$. Consider the ideal $\mathcal{J}$ of $\mathcal{X}$ generated by allpolynomials of the form $x_{i,j}x_{j,k}-x_{i,k}(x_{i,j}+x_{j,k}+\beta)-\alpha$for $1\leq i<j<k\leq n$. The quotient algebra $\mathcal{X}/\mathcal{J}$ (atleast for a certain choice of $\mathbf{k}$, $\beta$ and $\alpha$) has beenintroduced by Karola M\'esz\'aros as a commutative analogue of AnatolKirillov's quasi-classical Yang-Baxter algebra. A monomial in $\mathcal{X}$ issaid to be pathless if it has no divisors of the form $x_{i,j}x_{j,k}$ with$1\leq i<j<k\leq n$. The residue classes of these pathless monomials span the$\mathbf{k}$-module $\mathcal{X}/\mathcal{J}$, but (in general) are$\mathbf{k}$-linearly dependent. Recently, the study of Grothendieckpolynomials has led Laura Escobar and Karola M\'esz\'aros to defining a$\mathbf{k}$-algebra homomorphism $D$ from $\mathcal{X}$ into the polynomialring $\mathbf{k}[t_{1},t_{2},\ldots,t_{n-1}]$ that sends each $x_{i,j}$ to$t_{i}$. We show the following fact (generalizing a conjecture ofM\'esz\'aros): If $p\in\mathcal{X}$, and if $q\in\mathcal{X}$ is a$\mathbf{k}$-linear combination of pathless monomials satisfying $p\equivq\operatorname{mod}\mathcal{J}$, then $D(q)$ does not depend on $q$ (as long as$\beta$, $\alpha$ and $p$ are fixed). Thus, reducing a $p\in\mathcal{X}$ modulo$\mathcal{J}$ may lead to different results depending on the choices made inthe reduction process, but all of them become identical once $D$ is applied. Wealso find an actual basis of the $\mathbf{k}$-module $\mathcal{X}/\mathcal{J}$,using what we call forkless monomials.

$t$-Unique Reductions for M\'esz\'aros's Subdivision Algebra