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Published online by Cambridge University Press: 12 March 2014
By a variety we mean a class of algebras in a language 
, containing only function symbols, which is closed under homomorphisms, submodels, and products. A variety 
is said to be strongly abelian if for any term 
in , the quasi-identity
holds in .
In [1] it was proved that if a strongly abelian variety has less than the maximal possible uncountable spectrum, then it is equivalent to a multisorted unary variety. Using Shelah's Main Gap theorem one can conclude that if is a classifiable (superstable without DOP or OTOP and shallow) strongly abelian variety then is a multisorted unary variety. In fact, it was known that this conclusion followed from the assumption of superstable without DOP alone.
This paper is devoted to the proof that the superstability assumption is enough to obtain the same structure result. This fulfills a promise made in [2]. Namely, we will prove the following
Theorem 0.1. If is a superstable strongly abelian variety, then it is multisorted unary.
