Preservation Theorems for AE-Sentences

Abstract

An AE-sentence is a sentence of the form ∀x1 ... xn∃! z1 ... zm α(x, z), where α is a finiteconjunction of equations. Given an arbitrary quasivariety Q, we give a purely semantical characterizationof when a classK ⊆ Qwith S(K) = Q is axiomatizable relative to Q by AE-sentences. Along the way,we also characterize axiomatizability by generalized AE-sentences, which are of the form described above,except that both the number of existential quantifiers and of equations in α are allowed to be infinite. Thearticle concludes with an analysis of how the main theorems can be improved in the context of finitelygenerated quasivarieties.