# Amalgam

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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 ; for imbeddability of amalgams of semi-groups in a semi-group see ). 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 $U$. It has been shown  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.

How to Cite This Entry:
Amalgam. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Amalgam&oldid=45094
This article was adapted from an original article by L.A. Bokut'D.M. Smirnov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article