Ordered set ,filed, definition of =

[plsmail2mark]

ordered set ,filed, definition of "="

Hi everyone,
an ordered set is a set S that for any x,y in S, there is a order definition "<" so that one and only one of the followings will be true:
x<y, y<x, x=y.
and also if x<y, y<z and x,y,z in S, then x<z.
but there is no such a definition that if x=y,y=z and x,y,z in S, then x=y=z and also a definition like x=x if x is in S.

An filed(F) should fullfill the axioms of addition,mulplification and distribution law.
from the addition axiom we can get the proposition that if x+y=x+z then y=z by following proof:
y=0+y=(-x+x)+y=-x+(x+y)
given above condition x+y=x+z then
y=-x+x+z=0+z=z.
but there is no such a definition or axioms that if x=y,a=b and x,y,z,b are in F, then x+a=y+b.
it really confuse me about "if x=y,y=z then x=z".
could you help me?
thanks.

 

[D H]

Where did you get the idea that "there is no such a definition that if x=y,y=z and x,y,z in S, then x=y=z and also a definition like x=x if x is in S"? Equality is reflexive, symmetric, and transitive.

 

[plsmail2mark]

Where did you get the idea that "there is no such a definition that if x=y,y=z and x,y,z in S, then x=y=z and also a definition like x=x if x is in S"? Equality is reflexive, symmetric, and transitive.

Hi DH,
thank you for your reply.
I get the definition of order, ordered set, filed, from rudin's book principles of mathematical analysis. In this books there is no definition or axiom like if x=y,y=z. then x=z. I did not see any definition of equality with reflexive,symmetric and transitive.

 

[HallsofIvy]

You will not see it in a definition of field (not "filled" or "filed") because it is part of the definition of "=" itself which is primitive to field.