# Amalgam

A set represented as the set-theoretic union of a family of algebraic systems (cf. Algebraic system) of a given class with intersections , where for all the intersection

is non-empty and is a subsystem of each of the systems . If there exists a system in the class containing all () as subsystems, then says one that the amalgam is imbeddable in the system .

An amalgam of two groups and, in general, any amalgam of groups , in which all intersections () coincide and are equal to , is always imbeddable in a group, e.g. in the free product of the groups () with the common subgroup . There are, however, amalgams of groups which are not imbeddable in a group. (For conditions for imbeddability of amalgams of groups in a group see [1]; for imbeddability of amalgams of semi-groups in a semi-group see [2]). See also Amalgam of groups.

Let be the class of all algebras over a given field or the class of commutative, anti-commutative or Lie algebras over a field . An amalgam of -algebras with identical intersections (for all ) is imbeddable in the -free product of these algebras with the common subalgebra [3]. It has been shown [4] that an amalgam of associative skew-fields with identical intersections () is imbeddable in an associative skew-field.

#### References

 [1] A.G. Kurosh, "The theory of groups" , 2 , Chelsea (1960) pp. 33 (Translated from Russian) [2] A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 2 , Amer. Math. Soc. (1967) [3] A.I. Shirshov, "On a hypothesis in the theory of Lie algebras" Sibirsk. Mat. Zh. , 3 : 2 (1962) pp. 297–301 (In Russian) [4] P.M. Cohn, "The embedding of firs in skew fields" Proc. London Math. Soc. (3) , 23 (1971) pp. 193–213