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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Lattices and Boolean algebra

I have read in a text book that the set of natural numbers form a lattice under divisibility. How can it possibe, since there is no upper bound and therefore a Sup of the set?
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Prove every finite lattice has a greatest element - without induction

I have to prove that every finite lattice (L, ≤) has a greatest element. I have seen a lot of proofs proving this by using induction, however, I have to prove it without induction since our ...
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Do DeMorgan's laws hold for pseudo-complement in Bi-Heyting Algebra?

A textbook says in Heyting Algebra, The pseudo-complement of an element $a$ is denoted as $a^{\ast}$. One of the DeMorgan's law $\left(\vee a_{i}\right)^{\ast}=\wedge a_{i}^{\ast}$ holds ...
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How can I show that a lattice $(S, \leq)$ must have a greatest element?

How can I show that a lattice $(S, \leq)$, where $S$ is a finite set, must have a greatest element? What I mean is an element $x$ such that $a \leq x$ for all $a \in S$.
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23 views

Website or book with Hasse diagrams of subgroups

I need to look at Hasse diagrams of very many groups, especially high powers of small symmetric groups. Is there any place where I could look them up? Calculating them myself would be a huge amount of ...
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How far a lattice is from distributive/Boolean

Is there a standard quantification of how far a lattice is from being distributive? And Boolean? Or anyway how non-exact the representation is? In other words, is there an object (or number) which is ...
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When is a join semilattice not a meet semilattice?

When would a join semilattice not also be a meet semilattice (and vice versa)? I can't think of any way. An example or two would probably be very helpful.
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Basis of irreducibles of non algebraic lattices

Let $L$ be a complete lattice. An element of lattice $x$ is called $\wedge$-irreducible if $x=y\wedge z$ implies $x=y$ or $x=z$. Similarly, it is completely $\wedge$-irreducible if $x=\bigwedge x_i$ ...
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Flip/flop of finite joins and finite meets of lattices

Let $\mathfrak{A}$, $\mathfrak{B}$, $\mathfrak{C}$ be lattices (in fact in the example I have in mind, they are distributive and even co-Heyting lattices). Let maps ...
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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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Understanding the Basic Theorem on Concept Lattices

In Ganter and Wille's Applied Lattice Theory: Formal Concept Analysis, one can find the following definition: Basic Theorem on Concept Lattices. Let $K := (G, M, I)$ be a formal context. Then ...
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If $P$ has all binary joins, and all chain-shaped joins, is $P$ necessarily a complete lattice?

Question. Let $P$ denote a poset with all binary joins, and all chain-shaped joins (of arbitrary cardinality). Is $P$ necessarily a complete lattice? Motivation. Let $f$ and $g$ denote ...
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How do I prove that every chain has an upper bound?

Let $A$ be a non-empty set. Let $X$ be the collection of bijections $f:U→V$ where $U,V$ are disjoint subsets of $A$. Define the relation $≥$ as follows: $$(f:U→V) ≥ (f′:U′→V′) \text{ iff } U′⊆ U ...
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Lattice of Flats of Graphic Matroid and Intersection Lattice of Graphic Arrangement

Let $G$ be a simple graph. We will be looking at hyperplane arrangements in $\mathbb{R}^d$. Suppose $\mathcal{H}$ is the graphic hyperplane arrangement arising from $G$. Let $L(\mathcal{H})$ be the ...
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necessity of $f(0)=0$ and $f(1)=1$ in homomorphisms of boolean algebras

Let $A,B$ be boolean algebras and let $f \colon A \rightarrow B$. $f$ is a homomorphism of boolean algebras if $f$ is a homomorphism of the corresponding lattices and $f(0)=0$ and $f(1)=1$. Why is it ...
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Explicit formulas for meets and joins of uniform spaces

I want explicit formulas for meets and joins (and finite meets and joins) for sets of uniform spaces (where uniformities are ordered by inclusion). And also for proximity spaces. I am also ...
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Is a finite lattice uniquely atomistic iff it is boolean?

A finite lattice $(L,\wedge,\vee)$ is atomistic if every nonzero element is a join of atoms. Let $A = \{a_1, \dots , a_n \} \subset L$ be the subset of atoms, then $L$ is called uniquely ...
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Is there a term in lattice theory for this?

Let $\mathfrak{A}$ be a lattice with join denoted $\cup$, meet denoted $\cap$, and least element $\bot$. Consider a set $S\in\mathscr{P}\mathfrak{A}$ such that the following property holds (for every ...
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39 views

Distributive subgroups lattice

Let $G = \langle a \rangle \times \langle b \rangle $ ($a,b \in G$), where $ |\langle a \rangle| = n, |\langle b \rangle| = m$ and $gcd(n,m) = d > 1$. I need to show that subgroup lattice of $G$ is ...
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Are there results for relations between upward and downward closed partitions of some powerset?

I stumbled upon this, given some set $X$ and its powerset $\mathcal{P}(X)$ and some incomparable set $\mathbb{S}\subseteq\mathcal{P}(X)$, i.e. for any $S,S'\in\mathbb{S}$ we have $S\setminus ...
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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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21 views

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 ...
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Most general form of Cayley's theorem?

For many classes of algebraic structures, there exists a family of structures such that any member of the class can be embedded in some member of the family (groups and symmetric groups, unital rings ...
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Complete lattices: equivalence between join and meet existence

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 ...
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How many different functions we have by only use of $\min$ and $\max$?

We can making many functions of three variable by only use and combining of $\min$ and $\max$ functions. But many of them are not different , like : ...
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How to find an $R$-order $\Lambda$, such that $A=k\Lambda$?

Let $R$ be a Noetherian integral domain with quotient field $k$. An $R$-order $\Lambda$ is a finitely generated torsionless non-zero $R$-module, which is at the same time an $R$-algebra. Why is ...
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Is there standard terminology for any or all of these concepts designed to help with thinking about Riemann integrability?

When trying to determine whether a function $f$ is Riemann integrable or not, we're typically in the following situation: Write $U_f(P)$ and $L_f(P)$ for the corresponding "upper Riemann sum" and ...
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Linear algebra set-theoretic intersection

Today,I took a linear algebra test and I had the following question in the test: $W,U,V$ are all vector spaces. Prove the statement is true : $W \cap [(W+V) \cap U + (U+V) \cap W]=(U+V)\cap W$ ...
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Applications of graph theory to algebra?

Seeing as graphs model relations and algebra is essentially entirely based on relations, one would think that the two fields would inform each other. I know that algebra has many applications to graph ...
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Compactly generated complete lattice is Heyting Algebra

Let $L$ be a lattice. $a\in L$ is said to be compact if whenever $a\leq \bigvee A$ for some $A\subseteq L$, then there is some finite subset $B\subseteq A$ such that $a\leq \bigvee B$. A lattice is ...
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An intuition connected with Heyting implication

Suppose $L$ is a bounded lattice and let $\Rightarrow$ be its Heyting implication, i.e. the operation defined in the standard way: $x\Rightarrow y$ is the largest object of the set $\{u\in L\mid ...
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Subrings of the product of rings of algebraic integers

Let $K,L$ be a number fields and $\mathcal{O}_K,\mathcal{O}_L$ their rings of algebraic integers, and $n_K=[K:\mathbb Q]$, $n_L=[L:\mathbb Q]$. Suppose that $R$ is a subring of ...
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What is a “lattice” in set theory??? [closed]

NOTE: There is another question asking "What is a Lattice?" but when reading the question, it has to do with programming, and that is not what my question has to do with. The answer provided to that ...
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Finding a poset with an antichain and failing pairwise join

In an introductory exercise on posets and lattices, I am asked to give an example of a poset $(P,\leq)$ in which there are three elements $x,y,z$ s.t. $\{x,y,z\}$ is an antichain (EDIT: $A\subseteq ...
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Finding the congruences of a lattice

Part of the excercise I am currently doing is finding the congruences of the following lattice: The problem I struggle with the most is what happends when $1 \sim d$ - how to find what is the ...