Inverse limit

Given a mapping A from a lattice to a set of algebraic structures (sets, groups, algebras, rings etc.), and a mapping from each pair of lattice elements i, j such that i>j to a morphism f_i,j: A_i -> A_j such that f_i,k = f_i,j o f_j,k, we define the inverse limit, A, as the set of all mappings a_i from the lattice to the union of all A_j such that for every i a_i is a member of A_i and such that for every i > j, f_i,j(a<sub>i) = a_j.

The inverse limit A together with the functions f_i({a_j}) = a_i (the projections) has the universal property that for every structure B and a set of morphisms g_i: B -> A_i such that for every i>j, g_j = g_i o f_i,j there is a unique morphism g: B -> A such that for every i, g_i = g o f_i.

Note that A is often the same kind of algebraic structure as the A_i, with the operations defined element-wise: this holds for rings, algebras, fields, groups and vector spaces, amongst other. If every structure is finite, we can give A the product topology of discrete spaces. Since the rules describing an inverse limit are closed, A will be compact and Haussdorf in this case.

Examples:

• Z_p, the ring of p-adic integers, is the inverse limit of the rings Z/pⁿ (see modular arithmetic) with the lattice being the natural numbers with the usual order, and the functions being "take remainder". The natural topology on the p-adic integers is the same as the one described here.