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We investigate closure operators and describe their properties for$E$-combinations and $P$-combinations of structures and their theories. Weprove, for $E$-combinations, that the existence of a minimal generating set oftheories is equivalent to the existence of the least generating set, andcharacterize syntactically and semantically the property of the existence ofthe least generating set. For the class of linearly ordered language uniformtheories we solve the problem of the existence of least generating set withrespect to $E$-combinations and characterize that existence in terms of orders.

Closures and generating sets related to combinations of structures