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Equidecomposable magmas

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Abstract

A magma is called equidecomposable when the operation is injective, or, in other words, if \(x+y=x'+y'\) implies that \(x=x'\) and \(y=y'\). A magma is free iff it is equidecomposable and graded, hence the notion of equidecomposability is very related to the notion of freeness although it is not sufficient. We study main properties of such magmas. In particular, an alternative characterization of freeness, which uses a weaker condition, is proved. We show how equidecomposable magmas can be split into two disjoint submagmas, one of which is free. Certain transformations on finite presentations permit to obtain a reduced form which allows us identify all the finite presented equidecomposable magmas up to isomorphisms.

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Acknowledgements

Many thanks to the anonymous reviewer for so exhaustive and accurate revision, specially for shortening the proof in the Example 3.7 and the reference of the article of Jónsson and Tarski.

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  1. Departament de Ciències de la Computació, Universitat Politècnica de Catalunya, Campus Nord, Edifici Omega, Jordi Girona Salgado 1-3, 08034, Barcelona, Catalonia, Spain

    Carles Cardó

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  1. Carles Cardó
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Correspondence to Carles Cardó.

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Presented by J. B. Nation.

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This research was supported the recognition 2017SGR-856 (MACDA) from AGAUR (Generalitat de Catalunya).

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Cardó, C. Equidecomposable magmas. Algebra Univers. 81, 57 (2020). https://doi.org/10.1007/s00012-020-00685-3

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  • DOI: https://doi.org/10.1007/s00012-020-00685-3

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