concatenations

Concatenation is the operation of joining two sequences end to end to form a new sequence. It is a fundamental operation across strings, lists, arrays, and other sequence types. The result preserves the order of operands.

In formal language theory and computer science, concatenation of languages L1 and L2 is the set { xy | x in L1, y in L2 }. It is associative: (L1 L2) L3 = L1 (L2 L3). The empty string ε acts as the identity: L ε = ε L = L. In general, concatenation is not commutative: for example, "ab" concatenated with "c" differs from "c" concatenated with "ab".

In programming, concatenation is often performed with operators or functions, such as + or append. For strings, naive repeated concatenation can be inefficient in languages with immutable strings, leading to quadratic time behavior. Efficient approaches include using a string builder, a join operation, or buffering. For instance, in Python, ''.join(list_of_strings) is typically efficient; in Java, StringBuilder is a common choice.

Concatenation is used in text processing, data serialization, protocol construction, and parsing workflows. It also extends to other sequences, such as lists of items, where concatenation combines lists end-to-end. In formal settings, concatenation operators may be extended to more complex structures like regular expressions or finite automata, reflecting its role as a basic constructive operation across disciplines.