Varieties of infinitary universal algebras (Q802586)

Let \(\tau\) be a type with a proper class of operation symbols. We say that an equational theory E for algebras of type \(\tau\) is legitimate if algebras of type \(\tau\) satisfying E form a class. A subclass V of algebras satisfying a legitimate theory E is a variety if it is closed under products, subalgebras and homomorphic images. The author shows that a universe is weakly compact if and only if each variety can be described by means of equalities between implicit operations (i.e. a variety is a class of algebras of type \(\tau\) satisfying the theory E and some equations between implicit operations). Thus if a universe is weakly compact then we can generalize Reiterman's theorem for finite algebras to algebras of type \(\tau\) satisfying theory E.