We investigate closure operators and describe their properties for E-combinations and P-combinations of structures and their theories including the negation of finite character and the exchange property. It is shown that closure operators for E-combinations correspond to the closures with respect to the ultraproduct operator forming Hausdorff topological spaces. It is also shown that closure operators for disjoint P-combinations form topological T₀-spaces, which can be not Hausdorff. Thus topologies for E-combinations and P-combinations are rather different. We prove, for E-combinations, that the existence of a minimal generating set of theories is equivalent to the existence of the least generating set, and characterize syntactically and semantically the property of the existence of the least generating set: it is shown that elements of the least generating set are isolated and dense in its E-closure.

MSC

03C30, 03C15, 03C50, 54A05

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