All due respect, but I think you’re mistaken. My “twin” sheets fit a mattress which is more than half of a full mattress width, and the mattress they fit was likewise labeled a “twin” mattress, and fits on a “twin” bedframe.
TheInternational Mattress Sizes chart on wikipedia indicates that a single and a twin are the same size. There’s something smaller than a single/twin in the UK called a “mini single”, but even that is wider than half a double/full (UK: “small double”). ETA: A mini single is half of a UK sized King, is that what you were thinking of, perhaps?