In mathematics, there are many methods for translating a difficult problem into
an easier problem in a completely different setting. For example:
• a logarithm function translates the difficult problem of multiplying two positive
numbers into the easy problem of adding two different numbers;
• a Galois connection translates the problem of solving an equation over an
infinite field into the problem of finding all the subgroups of a finite group.
A natural duality provides another method for translating difficult problems into
easier ones. A natural duality can take a difficult, hard-to-visuahse problem in
a class of algebras and translate it into an easier, pictorial problem in a different
class of mathematical structures. For example, Priestley's duality [58] can
be used to translate problems about distributive lattices into problems about
ordered topological spaces. Over the past two decades, the theory of natural
dualities has developed into a practical tool for studying algebras.