Is the intersection of any collection of equational theories itself an equational theory?

Question

An equational theory, under my definition, is the set of all equational consequences of a set of equations. For example, the set of equational consequences of the commutative property is an equational theory. Certainly, the union of two equational theories is not always itself an equational theory. Is the intersection of any collection of equational theories itself an equational theory?

Answer 1

Yes, this is true. Really nothing about equations is being used here: the key point is that in any reasonable context, the intersection of deductively closed sets is deductively closed. Note that $X$ is the set of (equational) consequences of some set of (equational) sentences iff $X$ is a deductively closed set of (equational) sentences. I think that second definition is easier to work with.