A note on the relationship between Huttonʹʹs quasi-uniformities and Shiʹʹs quasi-uniformities

This note studies the relationship between Huttonʹs quasi-unifor-mities and Shiʹs quasi-uniformities. It is shown that when $L$ satisfies``multiple choice principle" for co-prime elements, the category of Huttonʹs quasi-uniform spaces is a bireflective full subcategory of the category of Shiʹs quasi-uniform spaces. Especially, if the remote-neighborhood mapping defined by Shi preserves arbitrary joins, then the two categories are isomorphic to each other.