Strolling through common meadows

Bruno Dinis; Jo\~ao Dias

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Strolling through common meadows

João Dias, Bruno Dinis

Rings and Algebras · Submitted 9 Nov 2023

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

The authors' main claim We show that common meadows are unions of rings which are ordered by a partial order that defines a lattice.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,199 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.19 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.13 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

Meadows are a sort of commutative rings with a multiplicative identity element and a total multiplicative inverse operation. In this paper we study algebraic properties of common meadows, which are meadows that introduce, as the inverse of zero, an error term $\abf$ which is absorbent for addition. We show that common meadows are unions of rings which are ordered by a partial order that defines a lattice. These results allow us to generalize some classical algebraic constructions to the setting of common meadows. We also briefly consider common meadows from a categorical perspective.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Meadows are a sort of commutative rings with a multiplicative identity element and a total multiplicative inverse operation.	background 1.00	0.10	0.04	asserted 1.00
s1	In this paper we study algebraic properties of common meadows, which are meadows that introduce, as the inverse of zero, an error term $\abf$ which is absorbent for addition.	contribution 0.78	0.26	0.06	asserted 0.96
s2	We show that common meadows are unions of rings which are ordered by a partial order that defines a lattice.	result 1.00	0.94	0.48	asserted 1.00
s3	These results allow us to generalize some classical algebraic constructions to the setting of common meadows.	interpretation 0.82	0.88	0.37	asserted 0.97
s4	We also briefly consider common meadows from a categorical perspective.	contribution 0.95	0.81	0.03	asserted 0.79

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

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arXiv listing abstract v2

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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

Meadows are a sort of commutative rings with a multiplicative identity element and a total multiplicative inverse operation. In this paper we study algebraic properties of common meadows, which are meadows that introduce, as the inverse of zero, an error term $\abf$ which is absorbent for addition. We show that common meadows are unions of rings which are ordered by a partial order that defines a lattice. These results allow us to generalize some classical algebraic constructions to the setting of common meadows. We also briefly consider common meadows from a categorical perspective.

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How the literature uses it

2 arXiv papers cite this in 10 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

Papers citing it · 5 sentences without a confident label

Uses its method or tool 1 paper

A model of $E_𝖿𝗍𝖼-𝖼𝗆+ 𝖠𝖵𝖫$ in which $ϕ_2,3,5$ fails can be found following the structure theory for GCM's as set out in [ref] .

Rings with common division, common meadows and their conditional equational theories · AVL and conditional equations · 0.81

Cites it as background 2 papers

In [ref] one finds results concerning for precisely which WCR's have an expansion to a model of $E_𝖿𝗍𝖼-𝖼𝗆$ (i.e., to a generalised common meadow).

Rings with common division, common meadows and their conditional equational theories · Weak commutative rings with $$ · 0.80
We have developed algebraic and logical theories of these structures in [ref] , and for the structure theory of the class of common meadows we refer to [ref] .

On Ambiguity: The case of fraction, its meanings and roles · Fractions and division by zero in school · 0.89

What it cites

Compares with 1 paper

We would like to point out that our notion of pre-meadow with $$ is related with the notion of weak commutative ring with $$ in [ref] .

A Complete Finite Axiomatisation of the Equational Theory of Common Meadows · Pre-meadows · 1.00

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