Call A < B a subalgebra gap when A is a maximal proper subalgebra of B.
The respective varieties generated by A and B need not form a variety
gap V(A) < V(B), witness the signature 1-1 algebra B = ({0,1,2,3},f,g)
for which f(3) = 1, g(3) = (2), and f(x) = g(x) = 0 otherwise. (Writing
the arguments as 2-bit binary numerals 00, 01, 10, and 11, f clears the
leftmost 1 if any and g the rightmost.) This has one maximal proper
subalgebra, namely A = {0,1,2}. Both f(x) = g(x) and f(f(x)) = f(x)
hold in A but not B, yet adding the first equation to the theory of B
does not entail the second.

Sometimes however the generated varieties do preserve gaps, witness the
three smallest Heyting algebras 1 < 2 < 3, all chains. As for any
signature, V(1) is axiomatized by x = y (the inconsistent theory). V(2)
is Boolean algebras, and in Boolean algebra every nontautology famously
entails x = y, that is, V(1) < V(2) is a gap. The less well known gap
V(2) < V(3) can be shown using any refutation of excluded middle x v x'
= 1 in any non-Boolean Heyting algebra H. Let that refutation have
left-hand-side a v a' = b; then {0,b,1} is a subalgebra of H isomorphic
to 3, whence every proper extension of V(2) extends V(3). The one
delicate bit here is to show b' (= b --> 0) is in {0,b,1}, which follows
from 0 = b&b' = (a v a')&b' = a&b' v a'&b', whence a&b' = 0. But a' is
the greatest x such that a&x = 0 whence b' <= a'. So b' <= b whence b'
= 0.

1. What other subalgebra gaps between Heyting algebras are preserved by
their respective varieties? In particular what about the gaps 3 < 4 < 5
< ...?

2. With an eye towards a one-line abstract-nonsense proof of gaphood of
V(2) < V(3), what natural conditions are sufficient for the preservation
of subalgebra gaps under generation of their respective varieties?

This line of thought was prompted by a recent talk at Stanford by
Vladimir Lifshitz on stable models for logic programming, at which one
member of the audience expressed confidence that adding any equation to
the theory of the Heyting algebra 3 would axiomatize Boolean algebra.
("How could there be anything in between?") His intuition turned out to
be correct, though the argument he had in mind would seem to depend on a
suitable answer to question 2.

Analogous questions arise for quotient gaps.

Vaughan Pratt