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 Bacsich, as follows: in any prevariety having at most $$\mathfrak {s}$$ s non-logical symbols and an axiomatization requiring at most $$\mathfrak {m}$$ m variables, if the epimorphisms into structures with at most $$\mathfrak {m}+\mathfrak {s}+\aleph _0$$ m + s + ℵ 0 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 $$\,\vdash $$ ⊢ with suitable infinitary definability properties of $$\,\vdash $$ ⊢ , while not making the standard but awkward assumption that $$\,\vdash $$ ⊢ comes furnished with a proper class of variables.

中文翻译：

---

Epimorphisms，可定义性和基数

我们在句法方面描述了类似一阶结构的任意类（与基本类相反）中的外同态的范围。这使我们能够加强 Bacsich 的结果，如下所示：在任何具有至多 $$\mathfrak {s}$$ s 个非逻辑符号和最多需要 $$\mathfrak {m}$$ m 个变量的公理化，如果到最多具有 $$\mathfrak {m}+\mathfrak {s}+\aleph _0$$ m + s + ℵ 0 个元素的结构的表同是满射的，那么所有的表同也是满射。使用这些事实，我们制定并证明了可管理的“桥接定理”，将逻辑 $$\,\vdash $$ ⊢ 的代数对应物中的所有表同态的满射性与合适的无限可定义属性 $$\,\vdash $$ 相匹配⊢ ，虽然没有做出标准但尴尬的假设 $$\,