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.