TL;DR
This paper investigates equidecomposable magmas, characterizing their properties, their relation to free magmas, and methods to classify finite presented examples up to isomorphism.
Contribution
It provides an alternative characterization of freeness in equidecomposable magmas and methods to classify finite presented cases.
Findings
Equidecomposable magmas can be split into a free submagma and its complement.
An alternative characterization of freeness using a weaker condition.
Finite presented equidecomposable magmas can be classified up to isomorphism.
Abstract
A magma is called equidecomposable when the operation is injective, or, in other words, if implies that and . 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 tranformations 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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