Lim-Inf Convergence 1
Bartomiej Skorulski · 2001
[9] provide the notation and terminology for this paper. One can prove the following propositions: (1) For every complete lattice L and for every net N in L holds infN ≤ liminfN. (2) Let L be a complete lattice, N be a net in L, and x be an element of L. Suppose that for every subnet M of N holds x = liminfM. Then x = liminfN and for every subnet M of N holds x ≥ infM. (3) Let L be a complete lattice, N be a net in L, and x be an element of L. Suppose N ∈ NetUniv(L). Suppose that for every subnet M of N such that M ∈ NetUniv(L) holds x = liminfM. Then x = liminfN and for every subnet M of N such that M ∈ NetUniv(L) holds x ≥ infM. Let N be a non empty relational structure and let f be a map from N into N. We say that f is greater or equal to id if and only if: (Def. 1) For every element j of N holds j ≤ f(j). The following three propositions are true: (4) For every reflexive non empty relational structure N holds idN is greater or equal to id. (5) Let N be a directed non empty relational structure and x, y be elements of N. Then there exists an element z of N such that x ≤ z and y ≤ z. (6) For every directed non empty relational structure N holds there exists a map from N into N which is greater or equal to id. Let N be a directed non empty relational structure. One can verify that there exists a map from N into N which is greater or equal to id. Let N be a reflexive non empty relational structure. Observe that there exists a map from N into N which is greater or equal to id. Let L be a non empty 1-sorted structure, let N be a non empty net structure over L, and let f be a map from N into N. The functor N · f yields a strict non empty net structure over L and is defined by the conditions (Def. 2).