A Compendium of Continuous Lattices

Abstract

Abstract. Binary trees are very useful tools in computer science for estimating the running time of so-called comparison based algorithms, algorithms in which every action is ultimately based on a prior comparison between two elements. For two given algorithms A and B where the decision tree of A is more balanced than that of B, it is known that the average and worst case times of A will be better than those of B, i.e., T A(n) ≤ T B (n) and T W A (n) ≤ T W B (n). Thus the most balanced and the most imbalanced binary trees play a main role. Here we consider them as semilattices and characterize the most balanced and the most imbalanced binary trees by topological and categorical properties. Also we define the composition of binary trees as a commutative binary operation, *, such that for binary trees A and B, A * B is the binary tree obtained by attaching a copy of B to any leaf of A. We show that (T, * ) is a commutative po-monoid and investigate its properties.2000 AMS Classification: 06A12, 06F05, 16B50.

M-Scott Convergence and M-Scott Topology on Posets

“…If (M 1) and (AP P ) hold, then the five conditions are equivalent to each other. (3) by Proposition 4.3 (5) and (3) ⇐⇒ (4) see [11,12]. (1) =⇒ (2): We only need to prove that σ M (L) is a continuous lattice.…”

P-Compactness in L -Topological Spaces

“…An element a in L is called prime element if a ≥ b ∧ c implies a ≥ b or a ≥ c. a in L is called co-prime element if a is a prime element [3]. The set of non-unit prime elements in L is denoted by P (L).…”

Some properties of complete congruence lattices

“…Given a complete lattice C, let us denote by Con(C) the complete lattice of complete congruences on C, and by JCon(C) and MCon(C) the complete lattices of, respectively, join-and meet-complete congruences on C. Recall that an equivalence relation [¤ C × C is a join-complete congruence if (Öi. The main result of this paper gives a sufficient condition on the complete lattice C in order that Con(C) is a complete sublattice of JCon(C): It turns out that this is true whenever C is continuous (in the standard sense of the book by Gierz et al [8]). From this result, by duality, we also get that if C is co-continuous then Con(C) is a complete sublattice of MCon(C).…”