Abstract An algebraically expandable (AE) class is a class of algebraic structures axiomatizable by sentences of the form $\forall \exists ! \mathop{\boldsymbol {\bigwedge }}\limits p = q$ . For a logic L algebraized by a quasivariety $\mathcal {Q}$ we show that the AE-subclasses of $\mathcal {Q}$ correspond to certain natural expansions of L, which we call algebraic expansions. These turn out to be a special case of the expansions by implicit connectives studied by X. Caicedo. We proceed to characterize all the AE-subclasses of abelian $\ell $ -groups and perfect MV-algebras, thus fully describing the algebraic expansions of their associated logics.

Abstract LetA≤Bbe structures, and ${\cal K}$ a class of structures. An elementb∈BisdominatedbyArelative to ${\cal K}$ if for all ${\bf{C}} \in {\cal K}$ and all homomorphismsg,g':B → Csuch thatgandg'agree onA, we havegb=g'b. Our main theorem states that if ${\cal K}$ is closed under ultraproducts, thenAdominatesbrelative to ${\cal K}$ if and only if there is a partial functionFdefinable by a primitive positive formula in ${\cal K}$ such thatFB(a1,…,an) =bfor somea1,…,an∈A. Applying this result we show that a quasivariety of algebras ${\cal Q}$ with ann-ary near-unanimity term has surjective epimorphisms if and only if $\mathbb{S}\mathbb{P}_n \mathbb{P}_u \left( {\mathcal{Q}_{{\text{RSI}}} } \right)$ has surjective epimorphisms. It follows that if ${\cal F}$ is a finite set of finite algebras with a common near-unanimity term, then it is decidable whether the (quasi)variety generated by ${\cal F}$ has surjective epimorphisms.

Let F = F, R be a finite Kripke frame.A congruence of F is a bisimulation of F that is also an equivalence relation on F. The set of all congruences of F is a lattice under the inclusion ordering.In this article we investigate this lattice in the case that F is a finite line frame.We give concrete descriptions of the join and meet of two congruences with a nontrivial upper bound.Through these descriptions we show that for every nontrivial congruence ρ, the interval [Id F , ρ] embeds into the lattice of divisors of a suitable positive integer.We also prove that any two congruences with a nontrivial upper bound permute.