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The strings over an alphabet, with the concatenation operation, form an associative algebraic structure with identity element the null string—a free monoid. Sets of strings with concatenation and alternation form a semiring, with concatenation (*) distributing over alternation (+); 0 is the empty set and 1 the set consisting of just the null ...
Concatenation theory, also called string theory, character-string theory, or theoretical syntax, studies character strings over finite alphabets of characters, signs, symbols, or marks. String theory is foundational for formal linguistics , computer science, logic, and metamathematics especially proof theory. [ 1 ]
String homomorphisms are monoid morphisms on the free monoid, preserving the empty string and the binary operation of string concatenation. Given a language , the set () is called the homomorphic image of . The inverse homomorphic image of a string is defined as
If is a set of strings, then is defined as the smallest superset of that contains the empty string and is closed under the string concatenation operation. If V {\\displaystyle V} is a set of symbols or characters, then V ∗ {\\displaystyle V^{*}} is the set of all strings over symbols in V {\\displaystyle V} , including the empty string ε ...
String concatenation is an associative, but non-commutative operation. The empty string ε serves as the identity element; for any string s, ...
COBOL uses the STRING statement to concatenate string variables. MATLAB and Octave use the syntax "[x y]" to concatenate x and y. Visual Basic and Visual Basic .NET can also use the "+" sign but at the risk of ambiguity if a string representing a number and a number are together. Microsoft Excel allows both "&" and the function "=CONCATENATE(X,Y)".
Various operations, such as copying, concatenation, tokenization and searching are supported. For character strings, the standard library uses the convention that strings are null-terminated: a string of n characters is represented as an array of n + 1 elements, the last of which is a "NUL character" with numeric value 0.
In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero elements, often called the empty string and denoted by ε or λ, as the identity element.