Meet – Continuous Lattices 1

The aim of this work is the formalization of Chapter 0 Section 4 of [11]. In this paper the definition of meet-continuous lattices is introduced. Theorem 4.2 and Remark 4.3 are proved. Let X, Y be non empty sets, let f be a function from X into Y , and let Z be a non empty subset of X. One can verify that f • Z is non empty. Let us note that every non empty relational structure which is reflexive and connected has also g.l.b.'s and l.u.b.'s. Let C be a chain. Note that Ω C is directed. Next we state a number of propositions: (1) Let L be an up-complete semilattice, D be a non empty directed subset of L, and x be an element of L. Then sup {x} D exists in L. (2) Let L be an up-complete sup-semilattice, D be a non empty directed subset of L, and x be an element of L. Then sup {x} D exists in L. (3) For every up-complete sup-semilattice L and for all non empty directed subsets A, B of L holds A ≤ sup(A B). (4) For every up-complete sup-semilattice L and for all non empty directed subsets A, B of L holds sup(A B) = sup A sup B. (5) Let L be an up-complete semilattice and D be a non empty directed subset of [: L, L :]. Then {sup({x} π 2 (D)); x ranges over elements of L: x ∈ π 1 (D)} = {sup X; X ranges over non empty directed subsets of L: x : element of L (X = {x} π 2 (D) ∧ x ∈ π 1 (D))}. (6) Let L be a semilattice and D be a non empty directed subset of [: L, L :]. Then {X; X ranges over non empty directed subsets of L: x : element of L (X = {x} π 2 (D) ∧ x ∈ π 1 (D))} = π 1 (D) π 2 (D).