The subalgebra systems of direct powers

A simple characterization of the subalgebra systems of direct powers of finitary universal algebras on a fixed infinite setA is given. For |I|≥|A| such subalgebra system of anI-power is precisely an algebraic closure systemS onA ^I closed under mutations ofI (which encompass both the precomposition by permutations ofI and allowing the values at specified elements ofI to become unrestricted) and such that each function in the intersection ofS is constant. For |I|<|A| the subalgebra systems ofI-powers are obtained as the restrictions toI of such closure systems on someA ^J withJ⊇I and |J|=|A|. Presented by J. D. Monk.