Magic Wands

Abstract

A typical way of constructing an algebra B with a given congruence lattice C is to construct an algebra A with a much larger congruence lattice and then “collapsing” sufficiently many pairs of congruences of the form con(a, b) and con(c, d) in B, so that the congruence lattice “shrinks” to C. To do this we need a “magic wand” that will make a ≡ b equivalent to c ≡ d. Such a magic wand may be a pair of partial operations f and g such that f(a) = c, f(b) = d, and g(c) = a, g(d) = b. This is the start of the Congruence Lattice Characterization Theorem of Universal Algebras of Grätzer and Schmidt (see [58], and also [22] and [25].