Why does Garling's definition of partial order omit reflexivity?

[Math Amateur]

I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume I: Foundations and Elementary Real Analysis ... ...

I am focused on Chapter 1: The Axioms of Set Theory ... ...

I need some help to clarify an aspect of Garling's definition of a partial order ...Garling's definition of a partial order reads as follows:View attachment 9028
View attachment 9029Garling thus defines a partial order relation as possessing transitivity and anti-symmetry ... but a number of other authors define a partial order as having the property of reflexivity as well ...

... ... an example is Daniel W. Cunningham: Set Theory: A First Course ... who defines a partial order as follows:View attachment 9030Can someone explain why Garling does not include reflexivity in his definition of a partial order ...?

Is it perhaps that it is possible to derive reflexivity from transitivity and anti-symmetry ... but how do we do that ...?Help will be appreciated ...

Peter

 

[Evgeny.Makarov]

Note that the definitions of antisymmetry in Garling and Cunningham differ. Garling does not even call it antisymmetry.

 

[Math Amateur]

Note that the definitions of antisymmetry in Garling and Cunningham differ. Garling does not even call it antisymmetry.

Oh! Indeed, Evgeny ... think I see what you mean ...

After transitivity, Garling's second condition for a partial order is as follows:

(ii) $$a \leq b$$ and $$b \leq a \ \ $$ if and only if $$ \ \ a = b$$ ... so if $$a = b$$ we have $$a \leq a$$ ... that is, we have reflexivity ... I should have read Garling's condition more carefully ... Thanks for the help, Evgeny ...

Peter