Let A be a poset ordered by . A subset B of A is called a chain in A if any two elements of B are comparablePlanetmathPlanetmath. In other words, if a,bB, then either ab or ba, that is, is a total orderMathworldPlanetmath on B, or that B is a linearly ordered subset of A. When ab, we also write ba. When ab and ab, we write a<b. When ab and ab, then we write a>b. A poset is a chain if it is a chain as a subset of itself. The cardinality of a chain is the cardinality of the underlying set.

Below are some common examples of chains:

  1. 1.

    𝐧:={1,2,,n} is a chain under the usual order (ab iff b-a is non-negative). This is an example of a finite chain: a chain whose cardinality is finite. Any finite setMathworldPlanetmath A can be made into a chain, since there is a bijection from 𝐧 onto A, and the total order on A is induced by the order on 𝐧. A chain that is not finite is called an infinite chain.

  2. 2.

    , the set of natural numbers, is a chain under the usual order. Here we have an example of a well-ordered set: a chain such that every non-empty subset has a minimal element (as in a poset). Other well-ordered sets are the set + of positive rationals and + of positive reals. The well-ordering principle states that every set can be well-ordered. It can be shown that the axiom of choiceMathworldPlanetmath is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath to the well-ordering principle.

  3. 3.

    , the set of integers, is a chain under the usual order.

  4. 4.

    , the set of rationals, is a chain under the usual order. This is an example of a dense chain: a chain such that for every pair of distinct elements a<b, there is an element c such that a<c<b.

  5. 5.

    , the set of reals, is a chain under the usual order. This is an example of a Dedekind complete chain: a chain such that every non-empty boundedPlanetmathPlanetmathPlanetmathPlanetmath subset has a supremumMathworldPlanetmathPlanetmath and an infimumMathworldPlanetmath.

Constructing chains

One easy way to produce a new chain from an existing one is to form the dual of the existing chain: if A is a chain, form A so that ab in A iff ba in A.

Another way to produce new chains from existing ones is to form a join of chains. Given two chains A,B, we can form a new chain AB. The basic idea is to form the disjoint unionMathworldPlanetmathPlanetmath of A and B, and order this newly constructed set so that the order among elements of A is preserved, and similarly for B. Furthermore, any element of A is always less than any element of B (See here ( for detail).

With these two methods, one can construct many more examples of chains:

  1. 1.

    Take , and form A={a}. Then A is a chain with a top element. If we take B={b}A, we get a chain with both a top and a bottom element. In fact, B is an example of a complete chain: a chain such that every subset has a supremum and an infimum. Observe that any finite chain is completePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

  2. 2.

    We can form which is a set with 1 as the top element. We can also form , which has neither top nor bottom, or , which has both a top and a bottom element, but is not complete, as , considered as a subset, has no top. Likewise, is bottomless.

The idea of joining two chains can be generalized. Let {AiiI} be a family of chains indexed by I, itself a chain. We form iIAi as follows: take the disjoint union of Ai, which we also write as iIAi. Then (a,i)(b,j) iff either i=j and ab, or i<j.

For example, let I= and Ai=, with iI. Then iIAi is a chain, whose total order is the lexicographic orderMathworldPlanetmath on 2. If we well-order I=, then iIAi is another chain called the long line.

Chain homomorphisms

Let A,B be chains. A function f from A to B is said to be a chain homomorphism if it is a poset homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath (it preserves order). f(A) is the homomorphic image of A in B. Two chains are homomorphic if there is a chain homomorphism from one to another. A chain homomorphism is an embeddingMathworldPlanetmath if it is one-to-one. If A embeds in B, we write AB. A strict embedding is an embedding that is not onto. If A strictly embeds in B, we write AB. An onto embedding is also called an isomorphismMathworldPlanetmathPlanetmathPlanetmath. If A is isomorphic to B, we write AB.

Some properties:

  • Two finite chains are isomorphic iff they have the same cardinality.

  • Top and bottom elements are preserved by chain isomorphisms. In other words, if f:AB is a chain isomorphism and if aA is the top (bottom) element, then f(a) is the top (bottom) element in B.

  • In addition, the properties of being well-ordered, dense, Dedekind complete, and complete are all preserved under a chain isomorphism.

  • AAB. More generally AiiIAi.

  • (AB)CA(BC).

  • If k is the bottom element of I, and Ak has a top element x, then there is a chain homomorphism f:iIAiAk given by f(a,i)=a if i=k and f(a,i)=x if i>k.

  • Dually, if k is the top of I and Ak has a bottom x, then there is a chain homomorphism f:iIAiAk given by f(a,i)=a if i=k and f(a,i)=x if i<k.

  • If Ak has both a bottom x and a top y, then we may define f:iIAiAk by f(a,i)=a if i=k, f(a,i)=x if i<k and f(a,i)=y if k<i.

Some examples:

  • 𝐧.

  • 𝐧𝐦𝐩 iff p=m+n for any non-negative integers m,n and p.

  • 𝐧 for any non-negative integer n.

  • .

  • Let I be the chain over under the usual order, and J the chain over under a well-ordering. Then iIjJ.

Title chain
Canonical name Chain
Date of creation 2013-03-22 12:09:12
Last modified on 2013-03-22 12:09:12
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Definition
Classification msc 03-00
Defines finite chain
Defines infinite chain
Defines dense chain
Defines complete chain
Defines join of chains
Defines chain homomorphism