image (or range) of each homomorphism is equal to the kernel of the next: The sequence of groups and homomorphisms may be either finite or infinite. A similar definition can be made for certain other algebraic structures. For example, one could have an exact sequence of vector spaces and linear maps, or of modules and module homomorphisms. Spectral sequences In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a generalization of exact sequences, and since their introduction by , they have become an important research tool, particularly in homotopy theory. Set theory An ordinal-indexed sequence is a generalization of a sequence. If α is a limit ordinal and X is a set, an α-indexed sequence of elements of X is a function from α to X. In this terminology an ω-indexed sequence is an ordinary sequence. Computing In computer science, finite sequences are called lists. Potentially infinite sequences are called streams. Finite sequences of characters or digits are called strings. Streams Infinite sequences of digits (or characters) drawn from a finite alphabet are of particular interest in theoretical computer science. They are often referred to simply as sequences or streams, as opposed to finite strings. Infinite binary sequences, for instance, are infinite sequences of bits (characters drawn from the alphabet {0, 1}). The set C = {0, 1}∞ of all infinite binary sequences is sometimes called the Cantor space. An infinite binary sequence can represent a formal language (a set of strings) by setting the n th bit of the sequence to 1 if and only if the n th string (in shortlex order) is in the language. This representation is useful in the diagonalization method for proofs. See also Enumeration On-Line Encyclopedia of Integer Sequences Recurrence relation Sequence space Operations Cauchy product Examples Discrete-time signal Farey sequence Fibonacci sequence Look-and-say sequence Thue–Morse sequence List of integer sequences Types ±1-sequence Arithmetic progression Automatic sequence Cauchy sequence Constant-recursive sequence Geometric progression Harmonic progression Holonomic sequence Regular sequence Pseudorandom binary sequence Random sequence Related concepts List (computing) Net (topology) (a generalization of sequences) Ordinal-indexed sequence Recursion (computer science) Set (mathematics) Tuple Permutation Notes References External links The On-Line Encyclopedia of Integer Sequences Journal of Integer Sequences (free) Elementary mathematics