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The Completely Distributive Lattices And The Complete Rings of Sets

Yu Zhang

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Abstract

Let L be a complete lattice.Then SL is called a basis of L iff x∈L,SxS satisfying ∨Sx=x,and L is called a basisquasiatomic lattice,if x∈S and x≠1,y∈L satisfying xy and so xy.This paper by using the wedge below relation  proves: A completely distributive lattice L is a complete ring of sets iff L has a basis SL so that L is a basisquasiatomic lattice.Again with the topological methods,we prove the topological characterization: A completely distributive lattice is a complete ring of sets iff the interval topology θ(L)(or the Lawson topology λ(L),or the double Scott topology σω(L)) of L is totally disconnected.

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Let L be a complete lattice.Then SL is called a basis of L iff x∈L,SxS satisfying ∨Sx=x,and L is called a basisquasiatomic lattice,if x∈S and x≠1,y∈L satisfying xy and so xy.This paper by using the wedge below relation  proves: A completely distributive lattice L is a complete ring of sets iff L has a basis SL so that L is a basisquasiatomic lattice.Again with the topological methods,we prove the topological characterization: A completely distributive lattice is a complete ring of sets iff the interval topology θ(L)(or the Lawson topology λ(L),or the double Scott topology σω(L)) of L is totally disconnected.

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Available abstract

Let L be a complete lattice.Then SL is called a basis of L iff x∈L,SxS satisfying ∨Sx=x,and L is called a basisquasiatomic lattice,if x∈S and x≠1,y∈L satisfying xy and so xy.This paper by using the wedge below relation  proves: A completely distributive lattice L is a complete ring of sets iff L has a basis SL so that L is a basisquasiatomic lattice.Again with the topological methods,we prove the topological characterization: A completely distributive lattice is a complete ring of sets iff the interval topology θ(L)(or the Lawson topology λ(L),or the double Scott topology σω(L)) of L is totally disconnected.

Key concepts: Lattice (music), Distributive property, Distributive lattice, Mathematics, Topology (electrical circuits), Congruence lattice problem, Combinatorics, Basis (linear algebra)

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