Translation-invariant magmas

A translation-invariant magma is a magma whose carrier \(G\) is an abelian group \((G,+,-,0)\), and whose magma operation takes the form

\[ y \diamond x = x + f(x-y) \]

for some function \(f: G \to G\). Thus the translations on \(G\) become magma isomorphisms.

Example. A magma \(G\) satisfying the left (E4) or right (E5) absorption laws is translation-invariant. Equip the carrier \(G\) with an abelian group structure \((G,+,-,0)\) and define \(f: G \to G\) as either \(f(x) = -x\) or \(f(x) = 0\).

Example. Linear magmas \(x\diamond y = ax+by\) on a field \((\mathbb{F},+,-,\cdot,0,1)\) are translation-invariant if \(a + b = 1\), since \((\mathbb{F},+,-,0)\) forms an abelian group, and one can set \(f(x) = -ax\).

Note that if an example of the latter sort suffices to refute the implication between \(P\) and \(Q\) then by the Lefschetz principle one can construct a counterexample where the field \(\mathbb{F}\) is finite. Consequently, \(P\) and \(Q\) cannot constitute an Austin pair. However, these linear magmas can still serve as starting points for the modified translation-invariant models studied in A survey of examples.

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