DOMINIONS AND PRIMITIVE POSITIVE FUNCTIONS

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.

PRIMITIVE POSITIVE FORMULAS PREVENTING A FINITE BASIS OF QUASI-EQUATIONS

A finite unary algebra with a primitive positive formula that defines the graph of a finite group operation does not have a finite basis for its quasi-equations.

The syntax and semantics of entailment in duality theory

Both syntactic and semantic solutions are given for the entailment problem of duality theory. The test algebra theorem provides both a syntactic solution to the entailment problem in terms of primitive positive formula and a new derivation of the corresponding result in clone theory, viz. the syntactic description of Inv(Pol(R)) for a given set R of unitary relations on a finite set. The semantic solution to the entailment problem follows from the syntactic one, or can be given in the form of an algorithm. It shows, in the special case of a purely relational type, that duality-theoretic entailment is describable in terms of five constructs, namely trivial relations, intersection, repetition removal, product, and retractive projection. All except the last are concrete, in the sense that they are described by a quantifier-free formula. It is proved that if the finite algebra M generates a congruence-distributive variety and all subalgebras of M are subdirectly irreducible, then concrete constructs suffice to describe entailment. The concept of entailment appropriate to strong dualities is also introduced, and described in terms of coordinate projections, restriction of domains, and composition of partial functions.