difficult, depending on the particular algebraic object in question. For example, the free group in two generators is easily described. By contrast, little or nothing is known about the structure of free Heyting algebras in more than one generator. The problem of determining if two different strings belong to the same equivalence class is known as the word problem. As the examples suggest, free objects look like constructions from syntax; one may reverse that to some extent by saying that major uses of syntax can be explained and characterised as free objects, in a way that makes apparently heavy 'punctuation' explicable (and more memorable). Free universal algebras Let be any set, and let be an algebraic structure of type generated by . Let the underlying set of this algebraic structure , sometimes called its universe, be , and let be a function. We say that (or informally just ) is a free algebra (of type ) on the set of free generators if, for every algebra of type and every function , where is a universe of , there exists a unique homomorphism such that Free functor The most general setting for a free object is in category theory, where one defines a functor, the free functor, that is the left adjoint to the forgetful functor. Consider a category C of algebraic structures; the objects can be thought of as sets plus operations, obeying some laws. This category has a functor, , the forgetful functor, which maps objects and functions in C to Set, the category of sets. The forgetful functor is very simple: it just ignores all of the operations. The free functor F, when it exists, is the left adjoint to U. That is, takes sets X in Set to their corresponding free objects F(X) in the category C. The set X can be thought of as the set of "generators" of the free object F(X). For the free functor to be a left adjoint, one must also have a Set-morphism . More explicitly, F is, up to isomorphisms in C, characterized by the following universal property: Whenever A is an algebra in C, and is a function (a morphism in the category of sets), then there is a unique C-morphism such that . Concretely, this sends a set into the free object on that set; it is the "inclusion of a basis". Abusing notation, (this abuses notation