Realcompactification of frames

for each a ∈ A. We denote the top and bottom elements of a σ-frame A respectively by 1A and 0A. σ-Frame homomorphisms preserve countable joins and finite meets. The resulting category is denoted σFrm. Extending the above notions by allowing arbitrary subsets and arbitrary joins in the definitions leads to the notions of a frame and a frame homomorphism, and the corresponding category Frm of frames. For further details on frames and σ-frames we refer to Johnstone [4] and Banaschewski-Gilmour [1]. Given a bounded distributive lattice A, and supposing a, b ∈ A, then a is said to be rather below b (written a ≺ b) if there exists s ∈ A such that a∧s = 0A and b ∨ s = 1A. We say a is completely below b (written a ≺≺ b) if there is a family {xi | i ∈ Q ∩ [0, 1]} of elements in A satisfying x0 = a, x1 = b and i < j implies xi ≺ xj . The lattice A is called normal if for each pair a, b of elements of L with a ∨ b = 1A, there exists u, v ∈ A such that a∨ u = 1A = b ∨ v and u∧ v = 0A. In a normal lattice, the two relations ≺ and ≺≺ coincide.