Formula omitted]-uniform spaces versus [formula omitted]-uniform spaces

Abstract

This paper studies relationships between the categories of I ( L ) -uniform spaces, L-uniform spaces and uniform spaces. We construct two adjunctions: Φ L ⊣ Ψ L between the category of I ( L ) -uniform spaces and the category of L-uniform spaces and φ L ⊣ ψ L between the category of L-uniform spaces and the category of uniform spaces (with L a complete lattice with an order-reversing involution in both cases), which with L = { 0 , 1 } and L = I = [ 0 , 1 ] , respectively, reduce to the adjunction from the category of I -uniform spaces to the category of uniform spaces investigated by Katsaras. If L is any complete lattice (not necessarily with an order-reversing involution), then so is the L-unit interval I ( L ) , and we have another adjunction, viz. φ I ( L ) ⊣ ψ I ( L ) from the category of I ( L ) -uniform spaces to the category of uniform spaces. We show that the following two factorizations hold: φ I ( L ) = Φ L ∘ φ L and ψ I ( L ) = ψ L ∘ Ψ L . When L is a meet-continuous lattice with an order-reversing involution, there is also a natural link between Φ L ⊣ Ψ L and the existing adjunction Ω L ⊣ I L from the category of I ( L ) -topological spaces to the one of L-topological spaces, via the forgetful functors. An essential tool in this paper is the theory of Galois connections. It is emphasized that no kind of distributivity is assumed on the lattice L in this paper.