A generalization and a new proof of Plotkin's reduction theorem

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It is known that Plotkin's reduction theorem is very important for his theory of universal algebraic geometry [arXiv:math. GM/0210187], [arXiv:math. GM/0210194]. It turns out that this theorem can be generalized to arbitrary categories containing two special objects and in this case its proof becomes considerable more simple. This new proof and applications are the subject of the present paper.