A class \(K\) of algebras has few models if for some uncountable \(\kappa\) the number of algebras in \(K\) of cardinality \(\kappa\) is less than \(2^ \kappa\). A complete description of varieties with few models is given by B. Hart, S. Starchenko and M. Valeriote [“Vaught’s conjecture for varieties”, Trans. Am. Math. Soc. (to appear)]. The corresponding problem of characterizing varieties with few existentially closed algebras is open. Although it is not hard to note that these varieties are abelian, the problem is open even in the strongly abelian case. This means that for every term \(u(x,y)\) and tuples \(a,b,c,d\) and \(e\) from an algebra \(A\) the condition \(u(a,b)=u(c,d)\) implies \(u(a,e)=u(c,e)\).
The following theorems are proved in the present paper.
Theorem 1. Let a strongly abelian locally finite variety \(V\) have superstable model completion. Then there is a number \(n\) such that every term of \(V\) depends at most on \(n\) variables.
Theorem 2. If some extension of the model completion of a unary variety \(V\) has not the dimensional order property, then \(V\) is linear: for all nonconstant terms \(t(x)\), \(s(x)\) there is a term \(w(x)\) such that either the equation \(t(x)=w(s(x))\) or the equation \(s(x)=w(t(x))\) holds in \(V\).