Tagged Questions

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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How to draw a lattice for the divisors of big numbers?

An exercise ask to find atoms and join-irreducible elements for the set of divisors of 360. I know how to find them by drawing the lattice but it seems difficult in this case. Is there another way to ...
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Category whose objects are subsets of A with (some) morphisms as subset or superset proofs [on hold]

In the process of trying to solve some other problem I found myself constructing the following category, which seems a little baroque but quite interesting. I'm wondering if this example is known and ...
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Order Theory and Lattice Theory Synonymous?

Is Order Theory the same as Lattice Theory? Can anyone recommend good beginners text book on either?
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What operator has these algebraic properties?

I am working in a space $V$ of objects that behaves like a vector space with a partial ordering $\preceq$. I have discovered an operator $f:V\times V \rightarrow V$ with the following properties: For ...
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How to find the sublattice?

Let say I have this equation, Yt+1=Yt +sign(Xt) Xt+1=Xt+a(Yt+1)+b where a,b are integers and 0 < b < a. If gcd(a,b)=d, then how can I find the sublattice by looking at the coordinates? This ...
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How to determine the Supremum and Infimum in a Hasse Doagram?

In order to find the supremum or infimum of a Hasse diagram we follow the outgoing lines from the elements up for supremum or down for infimum until the lines meet each other. My question is, do we ...
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Do all geometric lattices admit an order-theoretic lattice structure?

Wikipedia defines the geometric notion of a lattice as a discrete subgroup of $\mathbb{R}^n$ (i.e. a subgroup isomorphic to $\mathbb{Z}^n$. This can be viewed as the span of a basis for $\mathbb{R}^n$ ...
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Are there any infinite (virtually) polycyclic groups with lattice orders that are not linear orders?

I am interested in noetherian group algebras, so I am learning about polycyclic groups. Specifically, I want to generalize some ideas that work well with $k[\mathbb{Z}^n]$ utilizing the lattice ...
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Is polynomial ring a lattice?

My prof says it's not. But I can't find a polynomial pair of $f,g$ such that $max(f,g)$ or $min(f,g)$ is not in $R[x]$. Define uniform order: $f\leq g$,if for all $x, f\leq g$.
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Order of Galois connections between two boolean lattices

Is the poset of Galois connections between two boolean lattices itself a boolean lattice? If not, does it hold for: complete boolean lattices? atomic boolean lattices? atomistic boolean lattices?
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Galois connections between boolean lattices - an alternative representation

Let $\mathfrak{A}$ and $\mathfrak{B}$ be (fixed) boolean lattices (with lattice operations denoted $\sqcup$ and $\sqcap$, bottom element $\bot$ and top element $\top$). I call a boolean funcoid a ...
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Two alleged counterexamples (about boolean algebras)

Trying to solve this question, I propose two possible counter-examples. Please help me to understand whether these cases are really counter-examples. Let $\mathfrak{A}$ and $\mathfrak{B}$ be (fixed) ...
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More on a construction on two boolean lattices

Let $\mathfrak{A}$ and $\mathfrak{B}$ be (fixed) boolean lattices (with lattice operations denoted $\sqcup$ and $\sqcap$, bottom element $\bot$ and top element $\top$). I call a boolean funcoid a ...
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Boolean lattices vs boolean rings

Which kinds of theorems about boolean algebras are easier to prove with boolean rings (than with actual boolean lattices)? Give me at least one example, as an answer.
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Semicontinuous ordered topological space which is not continuous ordered

I'm reading article Mobs, trees, and fixed points by J. E. Ward. A partially ordered topological space (POTS) is defined to be a space $X$ together with a partial order $\le$ defined on $X$ such that ...
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Proving a characterization for the existence of a supremum in a sublattice of the powerset lattice.

Suppose you have a class $\mathcal{C}$ over some set $A$, closed under intersection and union. This class forms a lattice with inclusion as the order relation, and union and intersection as join and ...
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Structure of vector topologies and locally convex topologies on a fixed vector space

Let $X$ be a real or complex vector space and consider the partially ordered sets $lc(X) \subseteq v(X) \subseteq t(X)$ of respectively locally convex topologies, vector topologies and all topologies ...
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Supremum of family of semilinear sets

Consider the class of semilinear sets. Because semilinear sets are closed under intersection and union, this class forms a lattice with inclusion as the order relation. I am interested in (infinite) ...
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Sublattice of complete lattice

Suppose you have a complete lattice $(A, \preceq)$ and a sublattice $(B, \sqsubseteq)$. By definition finite joins and meets are the same in $A$ and $B$. I wounder how infinite joins and meets relate ...
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Lattice without 0 , 1 [closed]

I am looking for an example of Lattice that has no $0$ , $1$ elements.
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Why can we act like functions are totally ordered by their orders?

For simplicity, consider only functions from $\Bbb N$ to $\Bbb R^{>0}$. Let $f\preceq g$ if there is an $A>0$ such that for all sufficiently large $n$, $f(n)\le A g(n)$. We normally would write ...
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Can a Formal Concept Lattice Have two Empty Sets?

Let $G=\{cat,apple\}$ and $M=\{Meows,Fruit\}$ then $C(G,M,I) = \{(\emptyset,\{Meows,Fruit\}), (\{Cat\},\{Meows\}),(\{Apple\},\{Fruit\}),(\{Cat,Apple\},\emptyset)\}$ And the lattice looks like this ...
Let $(L, \sqsubseteq)$ be a poset. In every textbook on lattice theory, you find a property of complete lattices stating that the following three are equivalent: $\forall X \subseteq L$, there ...