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Algebraic Properties of Codimension Series of PI-Algebras

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Silvia Boumova, Vesselin Drensky

Rings and Algebras · Submitted 1 Nov 2011

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

The authors' main claim We show that if c(A,t) and c(B,t) are rational functions, then c(R,t) is also rational.

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arXiv ID1111.0319 (version 1)
Pith IDOS5SLKHQ, Pith's permanent record of this paper and its review
CitationSilvia Boumova, Vesselin Drensky. "Algebraic Properties of Codimension Series of PI-Algebras." arXiv:1111.0319 (2011). https://pith.science/paper/OS5SLKHQ
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,338 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.22 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.06 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

For a PI-algebra R over a field of characteristic 0 let T(R) be the T-ideal of the polynomial identities of R and let c(R,t) be the codimension series of R (i.e., the generating function of the codimension sequence of R). Let A, B and R be PI-algebras such that T(R)=T(A)T(B). We show that if c(A,t) and c(B,t) are rational functions, then c(R,t) is also rational. If c(A,t) is rational and c(B,t) is algebraic, then c(R,t) is also algebraic. The proof is based on the fact that the product of two exponential generating functions behaves as the exponential generating function of the sequence of the degrees of the outer tensor products of two sequences of representations of symmetric groups.

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 For a PI-algebra R over a field of characteristic 0 let T(R) be the T-ideal of the polynomial identities of R and let c(R,t) be the codimension series of R (i.e., the generating function of the codimension sequence of R). background 1.00 0.10 0.02 asserted 0.78
s1 Let A, B and R be PI-algebras such that T(R)=T(A)T(B). background 0.89 0.09 0.02 asserted 0.56
s2 We show that if c(A,t) and c(B,t) are rational functions, then c(R,t) is also rational. result 1.00 0.94 0.53 asserted 1.00
s3 If c(A,t) is rational and c(B,t) is algebraic, then c(R,t) is also algebraic. result 1.00 0.90 0.20 asserted 1.00
s4 The proof is based on the fact that the product of two exponential generating functions behaves as the exponential generating function of the sequence of the degrees of the outer tensor products of two sequences of representations of symmetric groups. method 1.00 0.64 0.03 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

For a PI-algebra R over a field of characteristic 0 let T(R) be the T-ideal of the polynomial identities of R and let c(R,t) be the codimension series of R (i.e., the generating function of the codimension sequence of R). Let A, B and R be PI-algebras such that T(R)=T(A)T(B). We show that if c(A,t) and c(B,t) are rational functions, then c(R,t) is also rational. If c(A,t) is rational and c(B,t) is algebraic, then c(R,t) is also algebraic. The proof is based on the fact that the product of two exponential generating functions behaves as the exponential generating function of the sequence of the degrees of the outer tensor products of two sequences of representations of symmetric groups.

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1 arXiv paper cite this in 3 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Cites it as background 1 paper
  • Moreover, for binary symmetric operads with one or two generators (essentially, for quotients of the operad of associative algebras), such generating series has been extensively studied for decades under the name of codimension series of varieties of algebras, see [ref] .

    On generating series of finitely presented operads · Main results · 0.96

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