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A noncommutative theory of Penrose tilings

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Christopher J. Mulvey, Pedro Resende

Category Theory · Submitted 25 Jun 2003

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We study in general the representations of involutive quantales on those of binary relations, and show that in the case of Pen the algebraically irreducible representations provide a complete classification of the set of Penrose tilings from which its representation as a quotient of Cantor space is recovered.

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arXiv IDmath/0306361 (version 2)
Pith IDZ7JC6PIY, Pith's permanent record of this paper and its review
CitationChristopher J. Mulvey, Pedro Resende. "A noncommutative theory of Penrose tilings." arXiv:math/0306361 (2003). https://pith.science/paper/Z7JC6PIY
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,367 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.09 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

Considering quantales as generalised noncommutative spaces, we address as an example a quantale Pen based on the Penrose tilings of the plane. We study in general the representations of involutive quantales on those of binary relations, and show that in the case of Pen the algebraically irreducible representations provide a complete classification of the set of Penrose tilings from which its representation as a quotient of Cantor space is recovered.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Considering quantales as generalised noncommutative spaces, we address as an example a quantale Pen based on the Penrose tilings of the plane. contribution 0.93 0.80 0.08 asserted 0.87
s1 We study in general the representations of involutive quantales on those of binary relations, and show that in the case of Pen the algebraically irreducible representations provide a complete classification of the set of Penrose tilings from which its representation as a quotient of Cantor space is recovered. result 0.68 0.93 0.74 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Considering quantales as generalised noncommutative spaces, we address as an example a quantale Pen based on the Penrose tilings of the plane. We study in general the representations of involutive quantales on those of binary relations, and show that in the case of Pen the algebraically irreducible representations provide a complete classification of the set of Penrose tilings from which its representation as a quotient of Cantor space is recovered.

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