Nice Infinitary Logics

The first part of the introduction tries to explain the aim to a general mathematical audience, so may be skipped by a knowledgeable reader; naturally we should start by explaining what is first order logic and a general (abstract) logic from a model-theoretic perspective. We may consider classes of rings and classes of groups but usually we do not consider a class containing structures of both kinds. Formally a ring is a structure (or model) M consistent with its universe, a set of elements called |M | (but we may write a ∈ M) and interpretations + ,× and 0 of the binary function symbols +,× and the individual constant symbol (= zero place function symbol) 0. We also write M for {(a0, . . . , an−1) : a0, . . . , an−1 are elements of M}. Generally we have a so-called vocabulary τ consisting of relation symbols (= predicates) and function symbol x, each with a given number of places (= arity). For a ring M we consider many times the set of n-tuples satisfying some equations. Model theorists usually look at a wider class of such sets, which start with the family {{ā ∈ M : ā satisfies an equation φ} : n ∈ N and φ an equation} and close it under intersection of two (with the same n), complements inside the relevant (M) and projection (from M to M). So a first order formula for the vocabulary τ, φ = φ(x0, . . . , xn−1) is a scheme giving for a τ -structure M a subset φ(M) of M as above. If n = 0, φ(M) ∈ {{〈〉}, ∅}, then we call φ a sentence and