Actually, once you close the browser, you should be logged out. The feature to remember a user just logs them in automatically when they return. Browsers handle that differently. There are also various cookie types at play. Some such that will remember items in your cart have longer expirations. Those such as "session" cookies are removed when you close the browser. You'll be looking at cookie and history preferences as well as autologin features.
I have two computers, one is a Dell desktop and the other is a HP Laptop. Both are running Windows 7 and Avast antivirus. When I log in to websites with ID and password they keep me logged in so I do not have to do it every time. The Dell desktop is fine but the laptop logs me out after closing the website down. I have checked internet options on both machines and compared all the settings and they are the same. Am I missing something? I have checked for a virus and malware but everything is good. Thanks for any help you may have.

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