An arXiv preprint describes algorithms that decide whether a target element belongs to the subgroup generated by a finite set of elements in wreath products with a finite base group. The result is stated uniformly for every natural-number dimension.
Those wreath products are handled through shift-stable subgroups of finitely supported maps. In plain language, a finitely supported map involves only finitely many indexed positions, and the paper uses the word tile for one of these maps. Shift-stable means the subgroup is considered under the relevant shifts of those positions. The base group G can be any finite group in the stated framework.
The central idea is a finite test
At the center is the paper's definition of a standard basis. It requires the stable subgroup generated by every leading term in the subgroup under study to equal the stable subgroup generated by the leading terms of a finite list of basis elements. Here, a leading term is the part selected by a chosen monomial ordering.
That comparison becomes a membership test. The paper says a member has only trivial reductions, and that any trivial reduction certifies membership. The standard basis therefore turns the abstract question into a check based on reductions against a finite set of elements.
Building the basis
To produce the finite basis, the paper gives a Buchberger-inspired algorithm for the non-abelian case. Its criterion tests every associated S-tile, and the candidate set qualifies as a standard basis only when all of those reductions are trivial. The iterative procedure is stated to terminate after finitely many loop repetitions.
Several supporting routines fill out the construction. One computes finite generators for a specified saturation subgroup in the last canonical shift direction. Another, variable elimination, computes a finite generating set for the relevant shift-stable subgroup when the width parameter d is at least one.
From tiles to a group-wide decision
The next step is nested subgroup membership. An inductive algorithm decides whether a target tile belongs to a nested subgroup described by finite tile sets at each level, including positive and negative directions, when the base group and the tile sets are finite.
Subgroup Membership in a finite-base wreath product is then reduced to that nested tile problem. The input consists of a finite set S of wreath-product elements and a target h; the question is whether h belongs to the subgroup generated by S. The paper's main result says this question is uniformly decidable for finite G and every natural-number dimension. The word uniform means the statement covers each such dimension rather than only one fixed case.
The boundaries are part of the result
The result concerns subgroup membership, not every related membership problem. Rational Subset Membership is outside the result and is reported as undecidable for non-trivial G when the dimension n is at least two. Submonoid Membership remains an open problem for finite G when n is at least three.
The document is an arXiv preprint, identified as version 2 and dated 31 August 2026.
Paper data and sources
Original title: Standard bases for shift-stable groups and Subgroup Membership in wreath products
Authors: Ruiwen Dong
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text