I'm helping a friend with his homework, and the following problem is really giving us both trouble.

Consider the subset

of

given by

. Give

the partial order

defined by

if and only if

and

, where

is the usual order on

.

I'm good so far, but this is the part that has me stumped:

Find a set in

which has no infimum.

It seems to me that any

*finite* subset of

would have an infimum.

(It should be noted that the book that this problem came from is well known for its typos and wrong answers in the student solutions section.)