Subsequente

Subsequente is a concept used in sequence analysis to describe a relation between two finite sequences. It concerns the occurrence of one sequence within another in the same order but not necessarily contiguously. In many contexts Subsequente is treated as equivalent to the standard subsequence relation, though some authors use the term to emphasize formal order preservation.

Formal definition: Let A = (a1, a2, ..., an) and B = (b1, b2, ..., bm) be sequences over a common alphabet. A is Subsequente of B if there exist indices 1 ≤ i1 < i2 < ... < in ≤ m such that a1 = b_{i1}, a2 = b_{i2}, ..., an = b_{in}. When this condition holds, A is said to be a Subsequente of B; equivalently, A appears as a subsequence of B.

Properties: The relation is reflexive (every sequence is a Subsequente of itself, via i_k = k), transitive (if A is Subsequente of B and B of C, then A is Subsequente of C), and antisymmetric up to equality. Thus it induces a partial order on the set of finite sequences over a fixed alphabet.

Example: A = (1, 3, 4) is Subsequente of B = (1, 2, 3, 4, 5) because one can choose i1=1, i2=3, i3=4, yielding a1=b1, a2=b3, a3=b4.

Applications: The Subsequente relation appears in text processing, motif matching in bioinformatics, and time-series analysis where preserving the order of elements is important. It also provides a formal basis for algorithms that test whether one sequence is contained within another, with approaches analogous to those used for standard subsequences.

See also: subsequence, order theory, pattern matching.