Free algebraic structures

ring of polynomials (with one or several variables) with coefficients in a given ring

free object in the category of groups; a group admitting a presentation without any relations

in algebra, a module that has a basis

commutative group whose elements are unique integer combinations of basis elements

operation that takes two groups G and H and constructs a new group G ∗ H

a “generic” algebraic structure over the given set, fulfilling no other equations except those given by the defining axioms

small category, whose objects are sets of natural numbers of the form {0,1,…,n}, and whose morphisms are nondecreasing functions

concept in mathematics

free object in the category of associative algebras

free object in the category of Lie algebras