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I-automorphism-invariance satisfies intermediate subalgebra condition

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Statement

Suppose \mathcal{V} is a variety of algebras, A is an algebra of \mathcal{V} and B is a subalgebra of A such that for every I-automorphism \varphi of A, \varphi(B) \subseteq B. Then, if C is a subalgebra of A containing B, every I-automorphism of C also sends B to itself.

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