Tomsekvensen

Tomsekvensen (the empty sequence) is the unique sequence containing zero elements. It has length zero and serves as the neutral element for the operation of concatenating sequences: for any sequence s, the concatenation s followed by the empty sequence, or the empty sequence followed by s, yields s.

Formal definition and notation: A finite sequence over a set S is a function from {1, 2, ..., n} to S for some nonnegative integer n. When n = 0 the function has an empty domain, producing the empty sequence. In many texts the empty sequence is denoted by ε, by angle brackets with nothing inside (for example ⟨⟩), or by an empty list symbol in programming contexts (such as []).

Key properties: The empty sequence has length 0 and is unique. It is a subsequence of any sequence, and it is simultaneously a prefix and a suffix of any sequence. It acts as the identity element for sequence concatenation: s concatenated with ε equals ε concatenated with s equals s.

Contexts and usage: The concept appears across mathematics, computer science, and formal language theory. It is commonly used as a base case in proofs by induction, in constructions involving concatenation, and in automata and grammar formalisms where the empty word may be part of a language or accepted by an automaton. In programming, the empty list often plays a similar role as a neutral element in recursive definitions or iterative algorithms.

Example: Over the alphabet {a, b}, the empty sequence is simply the sequence containing no symbols. See also empty word, empty string, empty list, and the identity element under concatenation.