Partial Order - Reconciling Definitions by Garling and Goldrei

In summary, the conversation discusses the definitions of partial orders in D. J. H. Garling's "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis" and Derek Goldrei's "Classic Set Theory for Guided Independent Study." While Garling defines a partial order as transitive and anti-symmetric, Goldrei's definition includes reflexivity as well. The conversation then delves into the implications of Goldrei's inclusion of reflexivity and the confusion it may cause. One possible solution is to redefine a weak partial order as reflexive, anti-symmetric, and transitive, and a strict partial order as anti-symmetric and transitive without reflexivity. However, this may lead to further confusion as there are other types of orders
  • #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:
?temp_hash=bde0420746728c796c40c6980af783e4.png

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

?temp_hash=bde0420746728c796c40c6980af783e4.png


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
Math Amateur said:
Is it because reflexivity can be derived somehow from Garling's condition (ii)

Yes. Condition (ii) says "if and only if". So if we take the case ##a = b = x ## , condition (ii) implies ##a \le b##, which is equivalent to "##x \le 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.")
 
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  • #3
Math Amateur said:
Can anyone explain why Goldrei includes reflexivity but Garling doesn't ... ... ?
I read it as:

Garling (1) = Goldrei (3) = transitivity
Garling (2) = Goldrei (2) = anti-symmetry
Garling automatically satisfies Goldrei (1).

Garling doesn't distinguish between a strict inclusion (##\subsetneq##) and one that allows equality as well (##\subseteq##).
So Garling calls Goldrei's weak partial order simply a partial order whereas Goldrei also speaks of a strict partial order.

Now starts my confusion alike. My reading is, that a strict partial order (## \subsetneq##) cannot be reflexive (Goldrei (1).)
So Goldrei's formulation of a strict partial order which involves a weak partial order isn't a good one. In my opinion he should have defined a weak partial order to be transitive, anti-symmetric and reflexive, and a strict partial order, one that is not reflexive without to refer to weak partial oders. The usage of the word "strict" indicates this intention, I think. Therefore the definition of a strict partial order to be a weak partial order and ... is very misleading for he contradicts himself on reflexivity. He should have taken it simply as:
  1. weak ##\Longleftrightarrow## reflexive, anti-symmetric and transitive ##\Longleftrightarrow## Garling's partial order
  2. strict ##\Longleftrightarrow## anti-symmetric and transitive and reflexivity excluded, i.e not allowed
I guess he thought a similar confusion may arise here: How does it come, that "weak" satisfies three conditions and "strict" only two of them? Shouldn't this be the other way around? But in this case, strict with conditions (1,2,3) and weak with conditions (2,3), how can we call a strict inclusion (##\subsetneq##) then a weak partial order, but an inclusion that allows equality (##\subseteq##) a strict order in contradiction to our usage on the word strict on inclusions.

Either you follow logical principles and contradict the language used for your main and most important example, or you adapt the definitions according to this example and get into logical trouble for strict satisfies less conditions than weak or to be more exact: explicitly exclude one property of weak. And if you compromise as Goldrei tried to, then a clever student from the end of the world shows up and complaints, as well. :wink: Whatever you do ...

I start to understand why Garling abstained from the distinction. There are already total orders waiting around the corner, Archimedian orders and this ominously well-ordering ...

P.S.: Sorry for the end of the world. :oops:
 
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  • #4
fresh_42 said:
I read it as:

Garling (1) = Goldrei (3) = transitivity
Garling (2) = Goldrei (2) = anti-symmetry
Garling automatically satisfies Goldrei (1).

Garling doesn't distinguish between a strict inclusion (##\subsetneq##) and one that allows equality as well (##\subseteq##).
So Garling calls Goldrei's weak partial order simply a partial order whereas Goldrei also speaks of a strict partial order.

Now starts my confusion alike. My reading is, that a strict partial order (## \subsetneq##) cannot be reflexive (Goldrei (1).)
So Goldrei's formulation of a strict partial order which involves a weak partial order isn't a good one. In my opinion he should have defined a weak partial order to be transitive, anti-symmetric and reflexive, and a strict partial order, one that is not reflexive without to refer to weak partial oders. The usage of the word "strict" indicates this intention, I think. Therefore the definition of a strict partial order to be a weak partial order and ... is very misleading for he contradicts himself on reflexivity. He should have taken it simply as:
  1. weak ##\Longleftrightarrow## reflexive, anti-symmetric and transitive ##\Longleftrightarrow## Garling's partial order
  2. strict ##\Longleftrightarrow## anti-symmetric and transitive and reflexivity excluded, i.e not allowed
I guess he thought a similar confusion may arise here: How does it come, that "weak" satisfies three conditions and "strict" only two of them? Shouldn't this be the other way around? But in this case, strict with conditions (1,2,3) and weak with conditions (2,3), how can we call a strict inclusion (##\subsetneq##) then a weak partial order, but an inclusion that allows equality (##\subseteq##) a strict order in contradiction to our usage on the word strict on inclusions.

Either you follow logical principles and contradict the language used for your main and most important example, or you adapt the definitions according to this example and get into logical trouble for strict satisfies less conditions than weak or to be more exact: explicitly exclude one property of weak. And if you compromise as Goldrei tried to, then a clever student from the end of the world shows up and complaints, as well. :wink: Whatever you do ...

I start to understand why Garling abstained from the distinction. There are already total orders waiting around the corner, Archimedian orders and this ominously well-ordering ...

P.S.: Sorry for the end of the world. :oops:
Thanks for the help and the thoughts fresh_42 ...

My thought ... maybe the definition for a strict partial order could be thought to have 3 conditions ... anti-symmetry, transitivity and anti-reflexivity ... then weak and strict partial orders have 3 conditions each ... not quite your preferred ideal of a strict partial order having more conditions however ...

Peter
 
  • #5
Stephen Tashi said:
Yes. Condition (ii) says "if and only if". So if we take the case ##a = b = x ## , condition (ii) implies ##a \le b##, which is equivalent to "##x \le 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.")
Thanks so much for that help Stephen ... that clarifies the issue .. what a relief ...!

I thought that D. J. H. Garling would be too well informed and experienced to put a non-standard definition in a book for undergraduates ... he is an Emeritus Reader in mathematical analysis at the University of Cambridge and he has 50 years of experience teaching undergraduates ... ...

Thanks again ... it all makes sense now ...

Peter
 
  • #6
Math Amateur said:
Thanks for the help and the thoughts fresh_42 ...

My thought ... maybe the definition for a strict partial order could be thought to have 3 conditions ... anti-symmetry, transitivity and anti-reflexivity ... then weak and strict partial orders have 3 conditions each ... not quite your preferred ideal of a strict partial order having more conditions however ...

Peter
Hi Peter,
I agree, although I would not call it anti-reflexive, because it is a forbidden reflexivity, while the "anti" in the anti-symmetry has actually to do with symmetry. Let's leave it to linguists ... To be honest I got confused myself each time I read my own statement for linguistic corrections.
For some time I even thought Goldrei wanted to distinguish between sets like ##\{1,2,3, \dots\}## and ##\{\frac{2}{2}\frac{4}{4}\frac{6}{6}, \dots \}## but that's too absurd for obvious reasons. I prefer Garling's definition and reserve "strict" as a property of possible inclusions as examples of a partial order, or for progression, comparisons in size and so on. It's fine enough to have partial orders in Garling's sense and distinguish between ##\subseteq## and ##\subsetneq## as examples, if needed.
 

1. What is Partial Order?

Partial Order is a mathematical concept used to describe a relationship between elements of a set. It is a binary relation that satisfies three properties: reflexivity, antisymmetry, and transitivity.

2. What is the difference between Partial Order and Total Order?

The main difference between Partial Order and Total Order is that in Partial Order, not all elements of a set are necessarily related, while in Total Order, all elements are related. This means that in Partial Order, some elements may be incomparable, while in Total Order, every element can be compared to one another.

3. How can Partial Order be represented graphically?

Partial Order can be represented graphically using a Hasse diagram. In this diagram, each element of the set is represented by a node, and the relationship between elements is represented by arrows connecting the nodes. Elements that are incomparable are placed on the same level, while elements that are related are placed on different levels.

4. What is the importance of Partial Order in mathematics?

Partial Order is an important concept in mathematics as it allows for the comparison and ordering of elements in a set. It is used in various fields such as computer science, economics, and statistics, to name a few. It also serves as the basis for other mathematical concepts such as lattices and order-preserving functions.

5. What are some real-life examples of Partial Order?

There are many real-life examples of Partial Order, such as the relationship between height and weight, where taller people are generally heavier, but there can be exceptions. Another example is the relationship between age and intelligence, where older people are typically more intelligent, but there can be exceptions. In both cases, not all elements are comparable, making it a Partial Order.

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