Free algebra

in a class of universal algebras

An algebra in with a free generating system (or base) , that is, with a set of generators such that every mapping of into any algebra from can be extended to a homomorphism of into (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 is called absolutely free. An algebra of signature is a free algebra in some class of universal algebras of signature if and only if is intrinsically free, that is, if it has a generating set such that every mapping of into can be extended to an endomorphism of . 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 can be written as a word over the alphabet 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).

Comments

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