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