Thanks, Brian. Meanwhile I see from Exercise 4.48(18) of MMT that every
HA with a gap at the top (1 covering a penultimate b) is SI. So
presumably the other half of the argument (that Var preserves gaps
between finite linearly ordered HAs) is that the SIs of Var(n) are
exactly the chains of cardinality between 2 and n. Hence if there were
a V intermediate between V(n) and V(n+1) it would have to contain an SI
in Var(n+1) not in Var(n), impossible since all but one of the SIs of
Var(n+1) are in Var(n) and that missing one is n+1 itself making V =
V(n+1).

Looking beyond finite chains, is it the case that any SI of Var(A) is a
subalgebra of A when A is SI?

And what can be said of gaps A < B when B is SI? Is Var(A) < Var(B)
necessarily a gap in that case?

Vaughan