3 General implications
In this chapter we record some general implications between equational laws.
The singleton law (Definition 2.2) implies all other laws.
This is clear from substitution.
Trivial.
Every law \(E\) has a dual \(E^{\mathrm{op}}\), formed by replacing the magma operation \(\circ \) with its opposite \(\circ ^{\mathrm{op}}:(x,y) \mapsto y \circ x\). For instance, the opposite of the law \(x \circ y = x \circ z\) is \(y \circ x = z \circ x\). A list of equations and their duals can be found here. Of the 4694 equations under consideration, 84 are self-dual, leaving 2305 pairs of dual equations.
The implication graph has a duality symmetry:
If \(E,F\) are equational laws, then \(E\) implies \(F\) if and only if \(E^{\mathrm{op}}\) implies \(F^{\mathrm{op}}\).
This is because a magma \(M\) obeys a law \(E\) if and only if the opposite magma \(M^{\mathrm{op}}\) obeys \(E^{\mathrm{op}}\).
Some equational laws can be “diagonalized”:
An equational law of the form
where \(x_1,\dots ,x_n\) and \(y_1,\dots ,y_m\) are distinct indeterminates, implies the diagonalized law
In particular, if \(G(y_1,\dots ,y_m)\) can be viewed as a specialization of \(F(x'_1,\dots ,x'_n)\), then these two laws are equivalent.
From two applications of 1 one has
and
whence the claim.
Thus for instance, Definition 2.7 is equivalent to Definition 2.2.