In some contexts, the chains that are considered are order isomorphic to the natural numbers with their usual order or its opposite order. In this case, a chain can be identified with a monotone sequence, and is called an ascending chain or a descending chain, depending whether the sequence is increasing or decreasing. A partially ordered set has the descending chain condition if every descending chain eventually stabilizes. For example, an order is well founded if it has the descending chain condition. Similarly, the ascending chain condition means that every ascending chain eventually stabilizes. For example, a Noetherian ring is a ring whose ideals satisfy the ascending chain condition. In other contexts, only chains that are finite sets are considered. In this case, one talks of a finite chain, often shortened as a chain. In this case, the length of a chain is the number of inequalities (or set inclusions) between consecutive elements of the chain; that is, the number minus one of elements in the chain. Thus a singleton set is a chain of length zero, and an ordered pair is a chain of length one. The dimension of a space is often defined or characterized as the maximal length of chains of subspaces. For example, the dimension of a vector space is the maximal length of chains of linear subspaces, and the Krull dimension of a commutative ring is the maximal length of chains of prime ideals. "Chain" may also be used for some totally ordered subsets of structures that are not partially ordered sets. An example is given by regular chains of polynomials. Another example is the use of "chain" as a synonym for a walk in a graph. Further concepts Lattice theory One may define a totally ordered set as a particular kind of lattice, namely one in which we have for all a, b. We then write a ≤ b if and only if . Hence a totally ordered set is a distributive lattice. Finite total orders A simple counting argument will verify that any non-empty finite totally ordered set (and hence any non-empty subset thereof) has a least element. Thus every finite total order is in fact a well order. Either by direct proof or by observing that every well order is order isomorphic to an ordinal one may show that every finite total order is order isomorphic to an initial segment of the