This paper investigates the algebraic properties of cyclic stamp folding, contrasting them with those of traditional  stamp folding. While the set of partially folded states for standard stamp folding forms a Heyting algebra, the set of partially folding states of the cyclic stamp folding under a partial order based on contacting faces forms only a join-semilattice, as a “meet” operation is not universally defined. This incomplete structure necessitates the search for a more descriptive algebraic framework.

To achieve this, we analyzed the combinatorial structure of the folded states and identified a minimal generating set composed of two fundamental patterns: “stars” (alternating mountain-valley folds) and “trees” (consecutive mountain or valley folds). Any valid folded state can be constructed from a combination of these basic components. Based on this generating set, we define a category for the set of all final folded states. We demonstrate that this category is a monoidal category, where the tensor product allows for the construction of complex folded states from simpler ones. This provides a novel and more comprehensive algebraic structure for cyclic stamp folding.

cyclic stamp folding, monoidal category, semilattice

Received: June 20, 2025; Accepted: July 21, 2025; Published: July 29, 2025

How to cite this article: Yiyang Jia and Jun Mitani, Combinatorics and algebra of cyclic stamp folding, JP Journal of Algebra, Number Theory and Applications 64(5) (2025), 509-529. https://doi.org/10.17654/0972555525026

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