Dominions and Primitive Positive Functions

Abstract

For A ≤ B first order structures in a class K, say that A is an epic substructure of B in K if for every C ∈ K and all homomorphisms g, g : B → C, if g and g agree on A, then g = g . We prove that A is an epic substructure of B in a class K closed under ultraproducts if and only if A generates B via operations definable in K with primitive positive formulas. Applying this result we show that a quasivariety of algebras Q with an n-ary near-unanimity term has surjective epimorphisms if and only if SPnPu(Q RSI ) has surjective epimorphisms. It follows that if F is a finite set of finite algebras with a common near-unanimity term, then it is decidable whether the (quasi)variety generated by F has surjective epimorphisms.

Epimorphisms, Definability and Cardinalities

“…The connection between such 'implicitly defined' constructs and epimorphisms was remarked upon in the algebraic literature long ago (e.g., see Freyd [17, p. 93] and Isbell [27]), but it is characterized in a syntactically sharper manner in Theorem 3 of Campercholi's recent paper [11] (see Remark 3.3 below). There, however, it is confined to classes closed under ultraproducts.…”

“…and c ∈ B, and (1) holds. Restricting to the case where K is closed under P U (whence each Σ can be chosen finite, by Remark 3.2), we obtain a more elementary proof of Campercholi's result[11, Thm. 3] 2.…”

Epimorphisms in varieties of subidempotent residuated structures

“…Theorem 2. (Campercholi [9,Thm. 22]) If a congruence permutable variety K with EDPM lacks the ES property, then some FSI member of K has a Kepic proper subalgebra.…”

“…Of these properties, only (9) and (10) rely on the square-increasing law. It follows from (3) that • is isotone in both arguments, and from (4) and ( 5) that → is isotone in its second argument and antitone in its first.…”

“…It is called a deductive filter of A if, moreover, e ∈ F . In that case, F is a submonoid of A; •, e , by (9), and whenever b ∈ A and a, a → b ∈ F , then b ∈ F (as a • (a → b) b, by (2)).…”