Amalgam

A set $ M $ represented as the set-theoretic union of a family $ \{ {A _ {i} } : {i \in I } \} $ of algebraic systems (cf. Algebraic system) $ A _ {i} $ of a given class $ \mathfrak K $ with intersections $ U _ {ij} $, where for all $ i, j $ the intersection

$$ A _ {i} \cap A _ {j} = U _ {i j } = U _ {j i } $$

is non-empty and is a subsystem of each of the systems $ A _ {i} , A _ {j} $. If there exists a system $ B $ in the class $ \mathfrak K $ containing all $ A _ {i} $( $ i \in I $) as subsystems, then says one that the amalgam is imbeddable in the system $ B $.

An amalgam of two groups and, in general, any amalgam of groups $ \{ {A _ {i} } : {i \in I } \} $, in which all intersections $ U _ {ij} $( $ i \neq j $) coincide and are equal to $ U $, is always imbeddable in a group, e.g. in the free product of the groups $ A _ {i} $( $ i \in I $) with the common subgroup $ U $. 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 $ \mathfrak K $ be the class of all algebras over a given field $ F $ or the class of commutative, anti-commutative or Lie algebras over a field $ F $. An amalgam $ \{ {A _ {i} } : {i \in I } \} $ of $ \mathfrak K $- algebras $ A _ {i} $ with identical intersections $ U _ {ij} = U $( for all $ i \neq j $) is imbeddable in the $ \mathfrak K $- free product of these algebras with the common subalgebra $ U $[3]. It has been shown [4] that an amalgam $ \{ {T _ {i} } : {i \in I } \} $ of associative skew-fields $ T _ {i} $ with identical intersections $ T _ {ij} = T $( $ i \neq j $) 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

Comments

The reference to [1] in the article above should have been to H. Neumann's original papers [a1], [a2].

The amalgamation problem for a class of algebraic systems is generally understood to mean the problem of imbedding an amalgam of two systems, as defined above, in a system of the class. As mentioned above, the amalgamation problem for groups is solvable, but for other classes it can be an interesting and important question. E.g., it has been shown [a3], [a4] that the amalgamation problem for various classes of lattices is closely related to the interpolation problem in logic.

References

[a1]	H. Neumann, "Generalized free products with amalgamated subgroups" Amer. J. Math. , 70 (1948) pp. 590–625
[a2]	H. Neumann, "Generalized free products with amalgamated subgroups" Amer. J. Math. , 71 (1949) pp. 491–540
[a3]	L.L. Maksimova, "Craig's interpolation problem and amalgamable varieties" Soviet Math. Dokl. , 18 (1977) pp. 1550–1553
[a4]	A.M. Pitts, "Amalgamation and interpolation in the category of Heyting algebras" J. Pure Appl. Alg. , 29 (1983) pp. 155–165