Dice Question Streamline Icon: https://streamlinehq.com

Is A_gen always an ASL?

Determine whether the generic algebra of leading terms A_gen associated to a pseudo-ASL (A,≼) is always an ASL, even when A itself is not an ASL.

Information Square Streamline Icon: https://streamlinehq.com

Background

The discrete algebra of leading terms A_disc is always an ASL, regardless of whether the original algebra A is an ASL. It is unknown whether the analogous statement holds for the generic algebra of leading terms A_gen, which would strengthen the structural theory.

References

We also note that $disc$ is always an ASL, even if $A$ is not an ASL; we do not know whether this holds for $A_{gen}$.

— Gröbner Bases Native to Term-ordered Commutative Algebras, with Application to the Hodge Algebra of Minors (2510.11212 - Grochow et al., 13 Oct 2025) in Section 3 Algebras of leading terms, after Proposition 3.x (Proposition \ref{prop:agen-adisc})