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arXiv · 2606.17695

On monoids up to symmetry

Dinh Van Le·Tim Römer·Nguyen Thi Vien
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Abstract

We study monoids in an infinite-dimensional setting that are invariant under the action of the infinite symmetric group Sym. Our main result establishes a local--global principle characterizing equivariant finite generation for arbitrary Sym-invariant monoids, extending earlier results that required additional assumptions. We further analyze local--global phenomena for other fundamental properties, including positivity, normality, seminormality, and simplicity. In addition, we obtain structural results for symmetric monoids, including characterizations of positivity and non-positivity, a description of their groups of units, and explicit formulas for the ranks of local symmetric monoids and stabilizing Sym-invariant chains.

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Dinh Van Le, Tim Römer, Nguyen Thi Vien. 2026-06-16. On monoids up to symmetry. https://doi.org/10.1016/j.jpaa.2026.108243

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