# 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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### Let $L$ be a lattice. Is there a union closed family of subsets of $[n]$ ordered by inclusion that correspond to $L$.

Let $\mathcal{C}$ be a collection of subsets of $[n]$ that is closed under taking unions (including taking the union of the empty set which is the empty set). Then $(\mathcal{C},\subseteq)$ is a ...
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### Understanding Tarski's fixed-point theorem.

I changed my question slightly. (Tarski Fixed Point Theorem). Let $X=\prod^{N}_{i=1} X_{i}$ where each $X_{i}$ is a compact interval of $\mathbb{R}$. Suppose $\phi : X \rightarrow X$ is an increasing ...
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### What is an example of a nonempty subset of $\mathbb{R}$ that is bounded above that does not contain its least upper bound?

What is an example of a nonempty subset of $\mathbb{R}$ that is bounded above that does not contain its least upper bound? This is an on-a-review sheet for my final. I thought the completeness axiom ...
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### In a compact metric space, every pair of homeomorphic open subsets has isomorphic basis?

Let X be a compact metric space, V and W be open subsets. Suppose there is an homemorphism from V to W. Let B be a countable basis for X and B(V), B(W) the relativized basis for V and W, respectively. ...
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### Are de Morgan algebras or lattices always bounded?

In some textbooks, de Morgan lattices are defined to be bounded distributive lattice satisfying the involution law and the de Morgan's laws. But in some textbooks, there is no requirement for ...
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### Join in Lattice of Subobjects

In an elementary topos the join $A \vee B$ of two subjects $A \to X$ and $B \to X$ is defined to be the image of the induced morphism $A \sqcup B \to X$. For sets it holds, that this is the same as ...
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### Example of non distributive module [closed]

I'm looking for a module that has three submodules $P,Q,R$ such that $P\cap (Q+R) \neq P\cap Q +P\cap R$ I'm struggling to find this example because I'm not that familiarized in module theory Can ...
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### How is it the case that: Any complete lattice is a Boolean algebra.

In the book “A Functorial Model Theory” by Nourani (pg152), it is stated that However, I didn’t understand what does he mean? Because a complete lattice is not even necessarily distributive whereas ...
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### Duality in Lattices

I am studying Lattices using the algebraic definition i.e. A set with 2 binary operations $\wedge , \vee$ that satisfies: Commutative of $\wedge$ and $\vee$ Associativity of $\wedge$ and $\vee$ ...
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### Special lattices

In a lattice one has a meet and a join operation. I would like to know more about lattices that have the following additional property: $b\leq (a \vee a')$ is equivalent to $(b\leq a \ or\ b\leq a')$...
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### A general setting to state the isomorphism theorems for (complete) (semi)lattices

Lattices, complete lattices and (complete) lower and upper semilattices are all very similar algebraic structures. The homomorphisms between lattices of the same type and also the Isomorphism Theorems ...
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### Is a finite lattice where each element has exactly one complement distributive? Why or why not?

While reading the paper LATTICES WITH UNIQUE COMPLEMENTS by R. P. DILWORTH, I get to know that any number of weak additional restrictions are sufficient for a lattice with unique complement to be a ...
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### Defining arbitrary join on the set of complete ideals of a Heyting Algebra

Given a Heyting Algebra $H$ we define a complete ideal (or c-ideal) $I$ to be a subset of $H$ satisfying. $\bot \in I$ $b \in I$ and $a \leq b$ implies $a \in I$ $X \subseteq I$ and $\bigvee X$ ...
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### Need a clarification of the proof that the prime ideal space of a distributive bounded lattice is compact

11.19 Theorem, from B. A. Davey, H. A. Priestley, Introduction to lattices and order, Let $L$ be a bounded distributive lattice, then the prime ideal space $\langle \mathcal{I}_p(L); \tau \rangle$ ...
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### Symmetric relations form a CABA

Fix a set X and consider the collection of all symetric relations on it. I also assume that the empty relation is by definiyion symmetric. Well, it is true that the above collection forms a complete ...
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### Which are the latticial properties of the quartered aztec diamond?

The aztec diamond is an area of a 2-dimensional square lattice. The quarter aztec diamond it's a part of this area, it can be seen in the following picture: Triangular arrangement of a 2-dimensional ...
1 vote
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### Lattice under a product oder

I have been told that under the product order, {(0,0),(1,0),(0,1),(2,1),(1,2),(2,2)} is not a lattice. I know that a lattice is when joins and meets exist for any pair of elements, and I suspect the ...
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### $L$ finite and distributive lattice, then $\mathcal{J}(L)$ (join-irreducible's) is isomorphic, as poset, to $\mathcal{M}(L)$ (meet-irreducible's)
Show that for any finite distributive lattice $L$, $\mathcal{J}(L)$, that is the associated poset of join-irreducible elements of $L$, is isomorphic to $\mathcal{M}(L)$, the associated poset of meet-...