# Free algebra

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in a class \$\mathfrak K\$ of universal algebras

An algebra \$F\$ in \$\mathfrak K\$ with a free generating system (or base) \$X\$, that is, with a set \$X\$ of generators such that every mapping of \$X\$ into any algebra \$A\$ from \$\mathfrak K\$ can be extended to a homomorphism of \$F\$ into \$A\$ (see Free algebraic system). Any non-empty class of algebras that is closed under subalgebras and direct products and that contains non-singleton algebras, has free algebras. In particular, free algebras always exist in non-trivial varieties and quasi-varieties of universal algebras (see Variety of universal algebras; Algebraic systems, quasi-variety of). A free algebra in the class of all algebras of a given signature \$\Lambda\$ is called absolutely free. An algebra \$A\$ of signature \$\Lambda\$ is a free algebra in some class of universal algebras of signature \$\Lambda\$ if and only if \$A\$ is intrinsically free, that is, if it has a generating set \$X\$ such that every mapping of \$X\$ into \$A\$ can be extended to an endomorphism of \$A\$. If a free algebra has an infinite base, then all its bases have the same cardinality (see Free Abelian group; Free algebra over a ring; Free associative algebra; Free Boolean algebra; Free group; Free semi-group; Free lattice; Free groupoid; Free module; and also Free product). Clearly, every element of a free algebra with a base \$X\$ can be written as a word over the alphabet \$X\$ in the signature of the class being considered. It is natural to ask: When are different words equal as elements of the free algebra? In certain cases the answer is almost trivial (semi-groups, rings, groups, associative algebras), while in others it is fairly complicated (Lie algebras, lattices, Boolean algebras), and sometimes it does not have a recursive solution (alternative rings).

Sometimes the phrases "free algebra" and "free algebraic system" are identified in meaning. See Free algebraic system.

How to Cite This Entry:
Free algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Free_algebra&oldid=31432
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article