Border (String)

Definition

Border (String)

A string is a border of a string if is both a prefix and a suffix of .

Trivial

Definition

Trivial String Border

Given a string , then , the empty string, and , the string itself, are called trivial string borders.

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Chain

Definition

Chain (Border, String)

Of two borders of the same string, the shorter is a border of the longer. For strings ,

Thus the borders of form a nested chain: ordering them by length also orders them by the border relation. Two borders of equal length are the same string, since both are the prefix of of that length.

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Transitivity

Definition

Transitivity (Border, String)

A border of a border is a border of the original string. For strings ,

This is transitivity of the border relation. The diagram shows one arrangement; the two occurrences of in need not be disjoint.

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Examples

The borders are , , and .

The borders are and .

The borders are , , , and . The full string itself is not a border.