In Gmail setting is an option that tells Gmail what to do when messages are accessed with POP.  Here is the URL to a help page that explains where that option is:

https://support.google.com/mail/answer/7104828?hl=en

Also this discusson on the Gmail help community:

https://support.google.com/mail/thread/28922658?hl=en

It doesn't appear that Gmail responds to the requests of the POP3 client but I don't know that for certain. 

I have a set up in 3 computers can access a gmail account (one ipad, two pmail on windows). Two are set to leave mail on server, although one problem is that gmail begins to block second download of the same mail, even though the download all mail button is chosen--it seems to be a transient setting. The third pmail computer is set to delete on download. But it doesn't. The mail stays in the Gmail inbox once downloaded. Is there a setting I'm missing somewhere

This is a Gmail feature. By default Gmail archives retrieved/deleted messages via POP3. If you really want do delete them you must set this option via your Gmail account Web interface.