Replica
of an algebraic system 
in a given class 
of algebraic systems of the same signature
An algebraic system 
from 
possessing the following properties: 1) there is a surjective homomorphism 
from 
onto 
; 2) if 
and if 
is a homomorphism from 
to 
, then 
for some homomorphism 
from the system 
to 
. The replica of the system 
in the class 
(if it exists) is uniquely defined up to an isomorphism. The class 
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).
References
| [1] | A.I. Mal'tsev, "Algebraic systems" , Springer (1973) (Translated from Russian) |
Comments
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 
. Let 
, where 
is a finite-dimensional vector space, and let 
be the smallest algebraic Lie subalgebra of 
that contains 
. The elements of 
are called the replicas of 
. One has that 
is nilpotent if and only if 
for all replicas 
of 
.
References
| [a1] | C. Chevalley, "Théorie des groupes de Lie" , 2 , Hermann (1951) pp. Chapt. II, §14 |
Replica. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Replica&oldid=48514