Replica

of an algebraic system $ A $ in a given class $ \mathfrak K $ of algebraic systems of the same signature

An algebraic system $ K _ {0} $ from $ \mathfrak K $ possessing the following properties: 1) there is a surjective homomorphism $ \phi _ {0} $ from $ A $ onto $ K _ {0} $; 2) if $ K \in \mathfrak K $ and if $ \phi $ is a homomorphism from $ A $ to $ K $, then $ \phi = \phi _ {0} \psi $ for some homomorphism $ \psi $ from the system $ K _ {0} $ to $ K $. The replica of the system $ A $ in the class $ \mathfrak K $( if it exists) is uniquely defined up to an isomorphism. The class $ \mathfrak K $ is called replica full if it contains a replica for any algebraic system of the same signature. A class of algebraic systems of a fixed signature is replica full if and only if it contains a one-element system and is closed with respect to taking subsystems and direct products. The axiomatizable replica-full classes (and only these) are quasi-varieties (cf. Quasi-variety).

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The concept of a replica is closely related to that of a universal problem (cf. Universal problems).

A second notion of replica occurs in the theory of algebraic Lie algebras, the Lie algebras of algebraic subgroups of $ \mathop{\rm GL} ( V) $. Let $ X \in \mathop{\rm End} ( V) $, where $ V $ is a finite-dimensional vector space, and let $ \mathfrak g ( X) $ be the smallest algebraic Lie subalgebra of $ \mathfrak g \mathfrak l ( V) $ that contains $ X $. The elements of $ \mathfrak g ( X) $ are called the replicas of $ X $. One has that $ X $ is nilpotent if and only if $ \mathop{\rm Tr} ( XX ^ \prime ) = 0 $ for all replicas $ X ^ \prime $ of $ X $.

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