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

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Discussion Overview

The discussion centers on the definitions of partial orders as presented by D. J. H. Garling and Derek Goldrei. Participants explore the implications of Garling's omission of reflexivity in his definition compared to Goldrei's inclusion of it, examining the reasoning behind these differences.

Discussion Character

  • Debate/contested
  • Technical explanation

Main Points Raised

  • Peter questions why Goldrei includes reflexivity in his definition of partial orders while Garling does not, suggesting that reflexivity might be derivable from Garling's condition (ii), which emphasizes anti-symmetry.
  • Some participants note that Garling's definition without reflexivity seems nonstandard and may have specific applications, similar to partial equivalence relations (PERs).
  • Stephen Tashi argues that condition (ii) implies reflexivity by stating that if a = b = x, then it follows that a ≤ b, which is equivalent to x ≤ x.
  • Another participant acknowledges that reflexivity does indeed follow from the "if" part of condition (ii), indicating a realization of this connection later in the discussion.

Areas of Agreement / Disagreement

Participants express differing views on the standardization of Garling's definition and whether reflexivity can be derived from his conditions. While some agree that reflexivity follows from condition (ii), the overall discussion remains unresolved regarding the implications of these definitions.

Contextual Notes

The discussion highlights the potential for varying definitions of partial orders in mathematical literature and the implications of these differences on understanding foundational concepts in analysis.

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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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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.
 
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
 
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:
Of course reflexivity follows from "if" of (ii). Sorry I did not notice it earlier.
 

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