# Mal'tsev local theorems

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Theorems on transferring properties of local parts of a model (in logic) to the whole model, established by A.I. Mal'tsev. A system $\{ M_i : I \in I \}$ of subsets of a set is called a local covering of it if each element of the set is contained in some $M_i$ and any two $M_i$, $M_j$ are contained in a third subset $M_k$. Examples of local coverings are: the system of all finite subsets of a set, and the system of all finitely-generated subgroups of a given group. A model $M$ locally has a property $\sigma$ if there is a local covering of $M$ consisting of submodels with the property $\sigma$. A local theorem holds for a property $\sigma$ (and a corresponding class of models) if every model locally having property $\sigma$ has the property in the large.
A source of a great variety of local theorems is the following fundamental local theorem of Mal'tsev (or the compactness theorem of the restricted predicate calculus) : If each finite subsystem of an infinite system of axioms of the restricted predicate calculus is consistent, then the whole system is consistent. Mal'tsev  gave a general method for obtaining concrete local theorems in group theory with the help of the fundamental local theorem, thus making a major contribution to model theory. Later, by improving the method itself, he proved  a local theorem for any property described by so-called quasi-universal axioms. The question of the validity of a local theorem for a property $\sigma$, which had previously been investigated separately for each $\sigma$, has thus been reduced to a common and quite "grammatical" question: Is it possible to describe $\sigma$ by universal axioms?