Definition:Immediate Predecessor Element

Let $$\left({S; \le}\right)$$ be a poset.

Let $$a, b \in S$$.

Then $$a$$ is the predecessor to $$b$$ iff:
 * 1) $$a < b$$;
 * 2) $$\lnot \exists c \in S: a < c < b$$.

We say that "$$a$$ immediately precedes $$b$$".