chain
Introduction
Let be a poset ordered by . A subset of is called a chain in if any two elements of are comparable. In other words, if , then either or , that is, is a total order on , or that is a linearly ordered subset of . When , we also write . When and , we write . When and , then we write . 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:
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is a chain under the usual order ( iff is non-negative). This is an example of a finite chain: a chain whose cardinality is finite. Any finite set can be made into a chain, since there is a bijection from onto , and the total order on is induced by the order on . A chain that is not finite is called an infinite chain.
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, 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 choice is equivalent to the well-ordering principle.
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, the set of integers, is a chain under the usual order.
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, 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 , there is an element such that .
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, 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 bounded subset has a supremum and an infimum.
Constructing chains
One easy way to produce a new chain from an existing one is to form the dual of the existing chain: if is a chain, form so that in iff in .
Another way to produce new chains from existing ones is to form a join of chains. Given two chains , we can form a new chain . The basic idea is to form the disjoint union of and , and order this newly constructed set so that the order among elements of is preserved, and similarly for . Furthermore, any element of is always less than any element of (See here (http://planetmath.org/AlternativeTreatmentOfConcatenation) for detail).
With these two methods, one can construct many more examples of chains:
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Take , and form . Then is a chain with a top element. If we take , we get a chain with both a top and a bottom element. In fact, 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 complete.
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We can form which is a set with 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 be a family of chains indexed by , itself a chain. We form as follows: take the disjoint union of , which we also write as . Then iff either and , or .
For example, let and , with . Then is a chain, whose total order is the lexicographic order on . If we well-order , then is another chain called the long line.
Chain homomorphisms
Let be chains. A function from to is said to be a chain homomorphism if it is a poset homomorphism (it preserves order). is the homomorphic image of in . Two chains are homomorphic if there is a chain homomorphism from one to another. A chain homomorphism is an embedding if it is one-to-one. If embeds in , we write . A strict embedding is an embedding that is not onto. If strictly embeds in , we write . An onto embedding is also called an isomorphism. If is isomorphic to , we write .
Some properties:
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Two finite chains are isomorphic iff they have the same cardinality.
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Top and bottom elements are preserved by chain isomorphisms. In other words, if is a chain isomorphism and if is the top (bottom) element, then is the top (bottom) element in .
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In addition, the properties of being well-ordered, dense, Dedekind complete, and complete are all preserved under a chain isomorphism.
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. More generally .
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.
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If is the bottom element of , and has a top element , then there is a chain homomorphism given by if and if .
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Dually, if is the top of and has a bottom , then there is a chain homomorphism given by if and if .
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If has both a bottom and a top , then we may define by if , if and if .
Some examples:
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iff for any non-negative integers and .
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for any non-negative integer .
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Let be the chain over under the usual order, and the chain over under a well-ordering. Then .
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 |