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In this version small mistakes are corrected and the exposition is changed assuggested by the referee (to appear in Canadian Journal of Mathematics).  The first main result of the paper is a criterion for a partially commutativegroup $\GG$ to be a domain. It allows us to reduce the study of algebraic setsover $\GG$ to the study of irreducible algebraic sets, and reduce theelementary theory of $\GG$ (of a coordinate group over $\GG$) to the elementarytheories of the direct factors of $\GG$ (to the elementary theory of coordinategroups of irreducible algebraic sets).  Then we establish normal forms for quantifier-free formulas over anon-abelian directly indecomposable partially commutative group $\HH$.Analogously to the case of free groups, we introduce the notion of ageneralised equation and prove that the positive theory of $\HH$ has quantifierelimination and that arbitrary first-order formulas lift from $\HH$ to $\HH\astF$, where $F$ is a free group of finite rank. As a consequence, the positivetheory of an arbitrary partially commutative group is decidable.

Elements of Algebraic Geometry and the Positive Theory of Partially Commutative Groups