# What is the "purposefulness"/meaning of lattices?

on wikipedia I read that a lattice is describable both as order-relation and as algebraic structure.

First I must say I have more to do with informatics but I am interested in the mathematical basics of informatics or even in mathematical logic in general.

I find it hard formulating my question, please be a little forgiving with me.

About lattices, I don't understand what it "means" or why it is of importance and interest that there is "to two elements a and b each a supremum and an infimum", that there are, defined as algebraic structure, two inner binary, commutative and associative operations. Then there obviously is an importance of the absorbtion and idempotency laws to the idea of lattices and at last the special meaning of the fact that a "algebraic structure" statement about a lattice is translatable into an "order-relation" one to begin with.

I don't understand why a lattice is the foundation for the proof of the existence of a smallest fixpoint.

What I would like to understand is what is "the special thing" about lattices. On an intuitive level.

• Very few abstract mathematical constructions are useful without modelling useful concrete constructions. They are useful precisely because they apply to many different things one night expect to encounter. So I think a few common examples of lattices could be a good idea to look for. Commented Jan 17, 2021 at 22:11
• Examples of lattices: the power set of a set, the set of subgroups of a group, the set of normal subgroups of a group, the set of ideals of a ring, the set of open subsets of a topological space, the set of closed subsets of a topological space, the set of intermediate extensions of a field extensions $K/L$, the set of positive integers under divisibility, the set of subspaces of a vector space, the set of convex subsets of a vector space, the set of partitions of a set, the set of sublattices of a lattice, the set of filters on a lattice, ... Commented Jan 17, 2021 at 22:32
• Can you give more context for the statement "a lattice is the foundation for the proof of the existence of a smallest fixpoint"? Where did you encounter this claim? Commented Jan 17, 2021 at 22:40
• @EricWofsey en.wikipedia.org/wiki/Knaster%E2%80%93Tarski_theorem here :) Commented Jan 17, 2021 at 23:38
• @vonspotz: a complete lattice is a partially ordered set where every subset has a supremum and infimum, it's a special kind of lattice and you don't need to understand the algebraic axiomatization of lattices for this fixpoint theorem. Commented Jan 18, 2021 at 1:13