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Representability and local representability of algebraic theories. (English) Zbl 0936.18007
A finitary monosorted algebraic theory $$\mathcal T$$ is called locally representable (or representable) in a category $$\mathcal K$$ with finite products if each of its initial segments is the domain of a full faithful finite-products-preserving functor into $$\mathcal K$$ (or if $$\mathcal T$$ itself is the domain of such a functor).
The present paper investigates the question when local representability of an algebraic theory, that is, the existence of possibly separate representations of each of its segments, implies its representability.
Theories $$\mathcal T$$ that are representable in any category with finite products in which they are locally representable are completely characterized.
Reviewer: R.Halaš (Olomouc)

##### MSC:
 18C10 Theories (e.g., algebraic theories), structure, and semantics 18B15 Embedding theorems, universal categories
##### Keywords:
monosorted algebraic theory; representability
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