Thanks for the replies.
I checked the settings in IE and found that, as I had expected, cookies ARE enabled for all sites and that should not be the problem.
I have done some more testing and found that if I go to the perlmonks.org site then all of the links across the top and any hyperlink on any web page causes me to be "logged off" when I get to the next page.
If I go to perlmonks.com, then I do not have this problem. I stay logged in as expected.