Partial Order - Reconciling Definitions by Garling and Goldrei ....

In summary: It is not as if D. J. H. Garling is not experienced or eminent ... he is an Emeritus Reader in mathematical analysis at the University of Cambridge and he has 50 years of experience teaching undergraduates ... ... Thanks again for our information ... and your thought on why Garling defined a partial order in this way ... I thought it must be that reflexivity flowed from condition (ii) ...Seems to me a strange decision (by a really experienced teacher!) to put a non-standard definition in a book designed and written for undergraduates ...
  • #1
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I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ... and I am also referencing concepts in Derek Goldrei's book, "Classic Set Theory for Guided Independent Study" ...

I am currently focused on Garling's Section 1.3 Relations and Partial Orders ... ...

Garling defines a partial order as follows:

View attachment 6138
... ... BUT Goldrei's definition is (apparently) slightly different ... as follows:

View attachment 6139

Can anyone explain why Goldrei includes reflexivity but Garling doesn't ... ... ?

Is it because reflexivity can be derived somehow from Garling's condition (ii) ... which appears to simply be anti-symmetry ... ?

Can someone please clarify this issue ...

Help will be appreciated ...

Peter
 
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  • #2
The definition without reflexivity seems nonstandard. I've never seen it. Maybe it's useful in some areas, similarly to how partial equivalence relations (PERs) are useful.
 
  • #3
Evgeny.Makarov said:
The definition without reflexivity seems nonstandard. I've never seen it. Maybe it's useful in some areas, similarly to how partial equivalence relations (PERs) are useful.
Thanks Evgeny ... it really makes me wonder why Garling did it ...

It is not as if D. J. H. Garling is not experienced or eminent ... he is an Emeritus Reader in mathematical analysis at the University of Cambridge and he has 50 years of experience teaching undergraduates ... ...

Thanks again for our information ... and your thought on why Garling defined a partial order in this way ... I thought it must be that reflexivity flowed from condition (ii) ...

Seems to me a strange decision (by a really experienced teacher!) to put a non-standard definition in a book designed and written for undergraduates ...

Peter
 
  • #4
Peter said:
I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ... and I am also referencing concepts in Derek Goldrei's book, "Classic Set Theory for Guided Independent Study" ...

I am currently focused on Garling's Section 1.3 Relations and Partial Orders ... ...

Garling defines a partial order as follows:... ... BUT Goldrei's definition is (apparently) slightly different ... as follows:
Can anyone explain why Goldrei includes reflexivity but Garling doesn't ... ... ?

Is it because reflexivity can be derived somehow from Garling's condition (ii) ... which appears to simply be anti-symmetry ... ?

Can someone please clarify this issue ...

Help will be appreciated ...

Peter
I thought that MHB readers would be interested in the following post on the Physics Forums ... ...

In answer to my question:

" ... ... Is it because reflexivity can be derived somehow from Garling's condition (ii) ... which appears to simply be anti-symmetry ... ? ... ... "


[h=3]Stephen Tashi[/h] writes:" ... ... Yes. Condition (ii) says "if and only if". So if we take the case a=b=x , condition (ii) implies a≤b, which is equivalent to "x≤x".

(An interesting technical question is whether this a consequence of the definition of "=" for some particular equivalence relation, or whether it is a consequence of the "common language" definition of the relation "=", which , in common mathematical speech implies "You can substitute one of a pair of "equal" symbols for another in any symbolic expression in a proof.")

... ... ... ... "Do MHB readers agree?Peter
 
Last edited:
  • #5
Of course reflexivity follows from "if" of (ii). Sorry I did not notice it earlier.
 

1. What is a partial order?

A partial order is a relation on a set that is reflexive, antisymmetric, and transitive. It is a mathematical concept that is used to describe the relationship between elements in a set.

2. How is a partial order different from a total order?

A total order is a relation on a set that is reflexive, antisymmetric, transitive, and also connected. This means that for any two elements in the set, one must be greater than or equal to the other. In a partial order, this is not necessarily the case.

3. What is the importance of reconciling definitions in partial orders?

Reconciling definitions in partial orders is important because it ensures that all statements and proofs involving partial orders are consistent and accurate. It also helps to avoid confusion and misunderstandings when studying partial orders.

4. How do Garling and Goldrei reconcile definitions in their paper?

Garling and Goldrei reconcile definitions by providing a unified definition of partial orders that includes all the necessary properties, such as reflexivity, antisymmetry, and transitivity. They also clarify the differences between partial orders and total orders.

5. What are some real-world applications of partial orders?

Partial orders have many applications in computer science, economics, and social sciences. They are commonly used to describe the relationship between different states of a system, preferences of individuals, and the ordering of tasks or events. They are also used in graph theory and optimization problems.

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