# Questions tagged [lattice-orders]

Lattices are partially ordered sets such that a least upper bound and a greatest lower bound can be found for any subset consisting two elements. Lattice theory is an important subfield of order theory.

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### The open sets form a complete lattice [duplicate]

Let $X$ be a topological space and denote $O(X)$ the set of open sets of $X$. Then I read that "$O(X)$ is a complete lattice since the union of any family of open sets is again open". I don'...
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### Example of an infinite compact measurable space

Let $X$ be a nonempty set with a $\sigma$-algebra $\mathcal{A}$. The notion of $\sigma$-algebra strictly lies between Boolean algebras and complete Boolean algebras. Clearly, $\mathcal{A}$ is a ...
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### Alternative characterization of distributive lattice

Let $(X,{\leq},{\wedge},{\vee})$ be a lattice. The lattice is called distributive if for all $x,y,z\in X$ both distributive laws hold:  x \wedge (y \vee z) = (x \wedge z) \vee (y \wedge z) \quad\...
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### Möbius function of distributive lattice only takes values $\pm 1$ and $0$.

In this Wikipedia article, I found the statement [...] shares some properties with distributive lattices: for example, its Möbius function takes on only values 0, 1, −1. My question is: How it can ...
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### A seeming example of a group whose subgroup lattice is lower semimodular but not consistent: where's my error?

Corollary 5.3.12 in Schmidt's "subgroup lattices of groups" states that if groups $A,B$ have lower semimodular subgroup lattices, then so does their direct product $A \times B$. This paper ...
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### What can be said about a Bi-Heyting algebra when the complement operations are useless?

Imagine, for whatever reason, a bounded lattice[1] $(L, 0, 1, ∧, ∨)$ which is a bi-heyting algebra, i.e. there is an operation $→$ such that $x∧y ≤ z$ iff $x ≤ y→z$ (the heyting algebra structure) and ...
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### Stone-Cech compactification via lattice ideals of $Coz(X)$

While studying P. T. Johnstone's book Stone Spaces I have come across the following Proposition (Section: 3.3) Where $max C(X)$ is the set of all maximal ideals of $C(X)$ (ring of real-valued ...
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### The join of two set partitions in the refinement order

Let $X$ be a set. The set $\Pi_X$ of all partitions of $X$ is partially ordered via the refinement order, which is defined by $\alpha \leq \beta$ if and only if for each block $A\in \alpha$ there is a ...
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