  It is known that exactly eight varieties of Heyting algebras have a
model-completion, but no concrete axiomatisation of these model-completions
were known by now except for the trivial variety (reduced to the one-point
algebra) and the variety of Boolean algebras. For each of the six remaining
varieties we introduce two axioms and show that 1) these axioms are satisfied
by all the algebras in the model-completion, and 2) all the algebras in this
variety satisfying these two axioms have a certain embedding property. For four
of these six varieties (those which are locally finite) this actually provides
a new proof of the existence of a model-completion, this time with an explicit
and finite axiomatisation.
