Infinite magma constructions

The need to construct infinite magmas primarily arises in the context of Austin laws and Austin pairs. An Austin law admits infinite models but no nontrivial finite ones, while an Austin pair consists of laws \(P\) and \(Q\) such that every finite model satisfying the law \(P\) also satisfies \(Q\), but some infinite magma satisfies \(P\) without also satisfying \(Q\). Examples are given in Finite implications and infinite models.

Here we survey techniques for constructing infinite magmas that serve as counterexamples for implications between laws. Many of the techniques presented trace their origins to the first analysis of the Asterix equation (E65 \(({x = y \diamond (x \diamond (y \diamond x))})\)), reviewed in The Asterix equation.

• Translation-invariant magmas
• Greedy algorithm constructions
• A survey of examples
• The Dupont equation

The dependency explorer needs JavaScript and its graph data.

Download the complete graph data.