Free algebra

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).

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