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Starting from a familiar concept in group theory, of defining a group 'by generators and relations', we can say that in general a free object of a certain specific algebraic type will have 'generators and no relations'. Yet. If we want to use the method of generators and relations in generality, we split the approach up as
Therefore in these terms from universal algebra we need to understand free objects for step 1, and the nature of congruences for step 2.
An example would be free monoids. These are rather simpler than free groups: the free monoid on a set X, is the monoid of all finite strings using X as alphabet, with operation concatenation of strings. The identity is the empty string. (See also Kleene star.)
As that example suggests, free objects look like constructions from syntax; and we can 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). An example for that is the way a free magmaIn abstract algebra, a magma (also called a groupoid is a particularly basic kind of algebraic structure. Specifically, a magma consists of a set M equipped with a single binary operation M × M → M''. A binary operation is closed by definition, but n on X turns out to be the magma of binary treeTrees (structure) In computer science, a binary tree is an ordered tree data structure in which each node has at most two children. Typically the child nodes are called left and right''. One common use of binary trees is binary search trees; another is bis labelled at the leaves by X. That construction generalises (from a single binary operationThe word operation can mean any of several things: The method, act, process, or effect of using a device or system. See military operations, manufacturing operations, anomalous operation. The tactical shooting PC game Operation Flashpoint In medicine, a s to any collection of ' aritiesIn mathematics, the arity of a function or an operator is the number of arguments or operands it takes, A function or operator can thus be described as unary binary ternary etc. Terms such as 7-ary or n ary are also used. Sometimes, it is useful to consid') in a way the free object concept makes much more palatable. (See also Herbrand universeIn mathematical logic, for any formal language with a set of symbols (constants and functional symbols), the Herbrand universe recursively defines the set of all terms that can be composed by applying functional composition from the basic symbols. It is n.)
In general, the setting for a free object is like this: a category C of algebraic structureIn abstract algebra, an algebraic structure consists of a set together with one or more operations on the set which satisfy certain axioms. In case there are no ambiguities, we usually identify the set with the algebraic structure. For example, a group Gs (sets plus operations, obeying some laws) has a functorFor the usage in computer science, see the function object article. In category theory, a functor is a special type of mapping between categories. Functors can be thought of as morphisms in the category of all ( small) categories. Functors were first cons F to Sets, the category of setsCategory theory In mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. It is the most basic and the most commonly used category in mathematics. The category is usually denoted simply as Set . and functions, that simply ignores the operations. We call F a forgetful functor. Free objects are created by a left adjoint G to F: for a set X the free object on X as 'generators' is G(X). There are general existence theorems that apply.
Other types of forgetfulness also give rise to objects quite like free objects: for example the tensor algebra construction on a vector space as left adjoint to the functor on associative algebras that ignores the algebra structure. It is therefore often also called a free algebra.
For specific kinds of free objects see: