An addendum to "The theory of implicit operations"

Abstract

In this addendum to [4], we provide a pair of counterexamples relevant to the theory of implicit operations. More precisely, we exhibit a pp expansion of a variety that fails to be a variety (although it is a quasivariety). Furthermore, we construct a sequence of varieties possessing a congruence preserving Beth companion, but no simple Beth companion.

arXiv commentAddendum to arXiv:2512.14326