Epimorphisms, Definability and Cardinalities

Abstract

Generalizing a theorem of Campercholi, we characterize, in syntactic terms, the ranges of epimorphisms in an arbitrary class of similar first-order structures (as opposed to an elementary class). This allows us to strengthen a result of Isbell, as follows: in any prevariety having at most s non-logical symbols and an axiomatization requiring at most m variables, if the epimorphisms into structures with at most m + s + ℵ0 elements are surjective, then so are all of the epimorphisms. Using these facts, we formulate and prove manageable 'bridge theorems', matching the surjectivity of all epimorphisms in the algebraic counterpart of a logic ⊢ with suitable infinitary definability properties of ⊢, while not making the standard but awkward assumption that ⊢ comes furnished with a proper class of variables.

Epimorphisms in varieties of subidempotent residuated structures

“…The weak ES property for a variety K rules out non-surjective K-epimorphisms h : A − → B in all cases where B is generated by the union of h[A] and a finite set. By [36,Thm. 5.4], it is equivalent to the demand that no finitely generated member of K has a K-epic proper subalgebra.…”

“…In particular, when a logic L is algebraized, in the sense of [8], by a variety K of algebras, then the ES property for K-i.e., the demand that all epimorphisms in K be surjective-amounts to the so-called infinite Beth definability property for L. The most general version of this 'bridge theorem' was formulated and proved by Blok and Hoogland [6,Thms. 3.12,3.17] (also see [36,Thm. 7.6] and the antecedents cited in both papers).…”

Profiniteness and representability of spectra of Heyting algebras

“…First, an intermediate logic is said to be hereditarily structurally complete if all its finitary extensions [31] are structurally complete in the sense that their admissible rules are derivable (see for instance [60]). On the other hand, an intermediate logic L has the infinite Beth definability property if implicit definitions can be turned explicit in L (we refer to [16,52] for the technical details). Proof.…”

The algebraic significance of weak excluded middle laws

“…The main result of the present paper generalizes this characterization of $$\mathsf {KC}$$ to a signature‐independent framework. It is in the spirit of the ‘bridge theorems’ of abstract algebraic logic [13, 17] that correlate, for instance, syntactic interpolation or definability properties with model‐theoretic amalgamation or epimorphism‐surjectivity demands [2, 14, 34], and deduction‐like theorems with congruence extensibility properties [4, 6, 13, 37].…”