0
votes
1answer
26 views

Lattice inside a finite dimensional vector space

I have an integral domain $R$ and its field of fractions $K$. Let $V$ be a finite dimensional $K$ vector space. Let $M$ be a finitely generated $R$-module contained in $V$. Why is $K\cdot M=V$ ...
3
votes
2answers
30 views

All tree orders are lattice orders?

Say that a set is tree ordered if the downset $\downarrow a =\{b:b\leq a\}$ is linearly ordered for each $a$. In a comment, Keinstein says that such sets are also semi-lattices, provided they are ...
4
votes
4answers
68 views

Number of join-irreducible elements of a lattice: is it monotonic?

Let $\mathcal L$ be a sub-lattice of $\mathcal P(X)$, where $X$ is a finite set. Denote by $\mathcal I(\mathcal L)$ the set of union-irriducible elements of $\mathcal L$ (i.e. $A\in \mathcal ...
0
votes
1answer
109 views

Is every sub-lattice of $\mathcal P(X)$ isomorphic to a sub-lattice of $\mathcal P(X')$ containing singleton sets?

Let $X$ be a finite set. $\mathcal{P}(X)$ denotes the set of all subsets of $X$. Let $\Gamma$ be a sub-lattice of $\mathcal{P}(X)$, i.e. $\Gamma$ is a collection of subsets of $X$ closed under union ...
1
vote
1answer
12 views

extension of an increasing function over a lattice

Let $X$ be a finite set. $\mathcal{P}(X)$ denotes the set of all subsets of $X$. Let $\Gamma$ be a sub-lattice of $\mathcal{P}(X)$, i.e. $\Gamma$ is a collection of subsets of $X$ closed under union ...
0
votes
0answers
45 views

Exercise 1.17 from Bell & Slomson's Models and Ultraproducts

I'm attempting to prove the following theorem left as an exercise from Bell & Slomson's Models & Ultraproducts (1969). I'd like to know whether my attempted proof is correct, and if not, I'd ...
3
votes
2answers
83 views

Examples of Stone algebras which are not Boolean algebras

Grätzer, in his Lattice Theory: Foundation, describes a Stone algebra as a distributive lattice with pseudocomplementation $L$ which satisfies the Stone identity: for every $a \in L$, $\neg a \vee ...
0
votes
0answers
21 views

Operation table of Hasse diagram

Consider the following Hasse diagram: My book gives the following join and meet operation tables for this diagram: $$\begin{array}{|c || c | c|} \hline Subset & x \wedge y & x \vee y \\ ...
0
votes
1answer
29 views

How to show that complement of prime filter is ideal? [closed]

How to show that in any lattice L, F is a prime filter if an only if its complement L\F is an ideal?
1
vote
0answers
31 views

Characterization of subgroup dual to Frattini Subgroup

Let $G$ be a group and let $\mathcal{L}(G)$ denote the complete lattice of subgroups of $G$. We have that every automorphism of $G$ induces a lattice-automorphism on $\mathcal{L}(G)$. From here we see ...
0
votes
0answers
60 views

Ascending chain condition holds in a lattice implies every ideal is principal

Proof: Suppose for contradiction that ACC holds for a lattice L, but there exists an ideal which is not principal. Thus $\exists$ I $\subset$ L s.t. ~$\exists$x: x $\le$a for a$\in$L. Thus $\exists$I ...
0
votes
1answer
52 views

Definition of finite direct decomposition of elements and indecomposable elements at arbitrary lattice

How can i define finite direct decomposition of elements and indecomposable elements at arbitrary lattice . I think i can say an element of lattice is finite direct decomposition of elements if it be ...
1
vote
2answers
62 views

Name for this axiom $(X \bullet R) \sqcup (Y \bullet S) \equiv (X \sqcup Y) \bullet (R \sqcup S)$

I am trying to give a name to this axiom in a definition: $(X \bullet R) \sqcup (Y \bullet S) \equiv (X \sqcup Y) \bullet (R \sqcup S)$ (for all $X, Y, R, S$) where $\sqcup$ is the join of a ...
3
votes
0answers
37 views

Proof of lattice distributivity by a grading function

Looking for an algorithm for recognizing finite distributive lattices, I came across Linear Time Recognition Algorithm for Distributive Lattices by Michel Habib and Lhouari Nourine. Just before the ...
2
votes
1answer
40 views

Looking for an algebraic structure

I'm looking for the name of algebraic structures (in which the elements are partially ordered) with the following properties: Top element defined, bottom optional; Join defined for all elements, ...
0
votes
1answer
77 views

Condition for the boolean algebra of clopen sets to be extremely disconnected.

Let $X$ be a topological space and let $\Gamma \mathcal O(X)$ be it's boolean algebra of clopen subsets. For compact totally disconnected space, show that $\Gamma \mathcal O(X)$ is complete (as a ...
0
votes
1answer
29 views

lowering of a semilattice

In these Lecture Notes the notion of lowering a semilattice is introduced, there it is stated: Sometimes "broken" elements need to be looked at and computed with. Now any semilattice can have an ...
0
votes
1answer
29 views

Are ideals of an sup semilattice always non-empty?

I am trying to do an exercise from the book A Compendium of Continuous Lattices. Exercise: Let $L$ be a set with a transitive relation, and let $A,B$ be ideals of $L$. (i) $A\cap B$ is an ideal of ...
0
votes
1answer
129 views

Showing that a group is lattice-ordered

Say I have a set $S$ with a group operation $\cdot$ and lattice ordering $\leq$. Suppose further that: $x\leq 1\implies xy\leq y$ $x\geq 1\implies xy\geq y$ For all $x,y$. Does it follow that ...
1
vote
2answers
54 views

Notation for “incommensurate” elements?

Say that $x\wedge y\not=x,y$, that is neither $x\leq y$ nor $y\leq x$. Is there a way to denote this? I've been saying $x<>y$ but that's completely made up.
0
votes
1answer
214 views

What is a subdirect product?

I'm having trouble understanding what a subdirect product is. Say $G$ is a subdirect product of $H=\prod H_i$ - this means that the homomorphisms $f_i:G\to H_i$ are surjective, which can be ...
1
vote
2answers
105 views

The kernel of the kernel.

From Wikipedia-Entry on Equivalence Relatin:Lattices The possible equivalence relations on any set X, when ordered by set inclusion, form a complete lattice, called Con X by convention. The ...
2
votes
2answers
168 views

Smallest Congruence Relation generated by a set

$\newcommand{\cl}{\operatorname{cl}}$ Let $R \subset S \times S$ be a binary relation, the smallest i) reflexive relation containing it is $$ \cl_\mathrm{ref} = R \cup \{ (x,x) : x \in S \} $$ ii) ...
1
vote
1answer
62 views

Proof that the lattice of fully invariant congruences is a sublattice of the lattice of all congruences

Let $\mathfrak U$ be an algebra (i.e. a set, called universe, together with several $n$-ary operations) in the sense of universal algebra. Denote by $\operatorname{Con} \mathfrak U$ the set of all ...
5
votes
2answers
96 views

Are filters in lattices exactly the homomorphic preimages $\varphi^{-1}(1)$ of top elements?

Say I got a lattice L, a bounded lattice K with top-element $1$ and a homomorphism $\varphi : L \to K$, then $\varphi^{-1}(1)$ is a filter in L. I wondered whether I can represent every filter $F ...
3
votes
0answers
50 views

How can we define “trivially orthogonal” groups?

In any lattice-ordered group, we say that two elements are orthogonal if their meet is 1. I've been thinking of groups who have only "trivial" orthogonal relations, i.e. $x\perp y\implies x=1$ or ...
1
vote
1answer
75 views

Show that the stabilizer is a prime subgroup

We define a subgroup $H$ as being convex if $g\in H\implies h\in H$ for all $1\leq h\leq g$. A convex subgroup $P$ is prime if for any two convex subgroups $X,Y$: $X\cap Y \subseteq P$ implies that ...
1
vote
0answers
63 views

Intuition behind prime subgroups

In any lattice ordered group, we say that a convex subgroup $P$ is prime if for any two convex subgroups $X,Y$: $X\cap Y \subseteq P$ implies that $X\subseteq P$ or $Y\subseteq P$. This is analogous ...
1
vote
1answer
22 views

Show $g^{-1} \wedge h^{-1}=(g\vee h)^{-1}$

Glass' Partially Ordered GroupsLemma 2.3.2 says: Let G be a p.o. group and $g,h\in G$. If $g\vee h$ exists, then $g^{-1} \wedge h^{-1}=(g\vee h)^{-1}$ Proof: If $f\leq g^{-1},h^{-1}$ then ...
0
votes
1answer
25 views

Show that $\langle G^+\rangle=G$ in a directed group

Lemma 2.1.8 of Glass' Partially Ordered Groups states: $G$ is a directed group if and only if $\langle G^+\rangle=G$ (where $G^+=\{x:x\geq1\}$) This doesn't make any sense to me. For example, ...
1
vote
1answer
83 views

When is “mod $n$” a congruence relation on the lattice $(\Bbb N,\gcd,\text{lcm})$?

For which $n\in \Bbb N$, $$a\equiv b,a'\equiv b'\quad \text{implies} \quad \gcd(a,a')\equiv \gcd(b,b'), \text{lcm}(a,a')\equiv \text{lcm}(b,b')$$ all mod $n$. For $n=2$ it is true.
5
votes
1answer
171 views

A Question on the Young Lattice and Young Tableaux

Let: $\lambda \vdash n$ be a partition of $n$ $f^\lambda$ - number of standard Young Tableaux of shape $\lambda$ $\succ$ - be the covering in the Young Lattice (that is, $\mu \succ \lambda$ iff ...
8
votes
2answers
219 views

Example of non-abelian partially ordered group

What is a simple example of a non-abelian partially ordered group?
0
votes
1answer
63 views

an infinite queue preserving equality.

Is there any well-ordered set $(A,\leq)$ such that: $(A,\leq^{-1})$ is well-ordered. $A$ is infinite. there's exactly one function $\theta:A\rightarrow \{0,1\}$ such that 1) for each $a < M$, ...
2
votes
1answer
71 views

Composition length of surjective inverse limits

Let $M$ be a left module over some ring $R$ and suppose that $M$ is an inverse limit of a family of modules $M_i$ with $i\in I$. We suppose also that the maps of the inverse limit $\pi_i:M\to M_i$ are ...
5
votes
2answers
2k views

Given the Hasse diagram tell if the structure is a lattice

Let's consider the following Hasse diagram: I need to tell whether this is a lattice. By lattice definition I can prove the above shown structure $M_5$ to be a lattice if and only if $\forall x,y ...
0
votes
2answers
386 views

How to apply the lattice definition and show if a poset is a lattice

Let $S = \{1,2,3\}$ and let the poset $(\wp(S)\setminus\{\emptyset\}, \sim)$ be defined as follows: $$\begin{aligned} X \sim Y \Leftrightarrow X = Y \text{ or } \max(X) < \max(Y) \end{aligned}$$ ...
3
votes
1answer
167 views

How to derive distributivity from Boolean algebra laws

Let $(L,\le,\bot,\top)$ be a bounded lattice and $\neg: L \rightarrow L$ be a map that satisfies the following laws: $a \wedge b = \bot \Leftrightarrow a \le \neg b$ $\neg\neg a =a$ I'd like to ...
1
vote
2answers
297 views

Draw Hasse diagram as two elements have same image

Let $S=\{1,2,3,4,5,6,7,8,9,10\}$, $P=\{y \in \mathbb N : y \text { is a prime number}\}$, consider the map $f$ defined as follows: $$\begin{aligned} f:x\in S \rightarrow f(x) \in \wp (P) ...
5
votes
0answers
130 views

Prove $(\mathbb Z \times \mathbb Z, \Sigma)$ to be a partial order and tell if its subset $T'$ is a lattice

Let $T = (\mathbb Z\times\mathbb Z, \Sigma) $ be defined as follows: $$\begin{aligned} (a,b) \text{ } \Sigma \text { } (c,d) \Leftrightarrow (a,b) = (c,d) \text{ or } a^2b^2<c^2d^2\end{aligned}$$ ...
4
votes
2answers
345 views

partially ordered group, positive cone, quotient (exercise)

Definitions: A partially ordered group or po-group is a po-set $(G,\leq)$, such that $G$ is a group and $\forall x,y,a,b\!\in\!G\!:x\!\leq\!y\Rightarrow axb\!\leq\!ayb$, i.e. a po-set that is a group ...
1
vote
1answer
306 views

MacNeille completion of a totally ordered set: Dedekind cuts

If $X$ is any partially ordered set with $A\!\subseteq\!X$ and $x\!\in\!X$, define $x\!\leq\!A :\Leftrightarrow \forall a\!\in\!A\!: x\!\leq\!a$ and $A\!\leq\!x :\Leftrightarrow \forall a\!\in\!A\!: ...
2
votes
1answer
78 views

Diamonds of ideals, part 3

I'd like to wrap up the line of questioning started first in this question and then continued in this question. The only variant left to try is: "How close can you get to the Diamond lattice ...
4
votes
2answers
198 views

An analogy between subgroups and equivalence relations.

I have noticed a certain analogy between subgroups of a group $G$ and equivalence relations on a set $X$. I would like to know if there's an explanation for this analogy or a common generalization of ...
2
votes
1answer
148 views

correspondence for universal subalgebras of $U/\vartheta$

Let $U$ be a universal algebra of type $T$, and denote $\mathrm{Con}(U)\!=\!\{\text{congruence relations on }U\}$ and $\mathrm{Sub}(U)\!=\!\{\text{subalgebras of }U\}$. Let "$\leq$" mean "subalgebra". ...
3
votes
1answer
128 views

$M_3$ is a simple lattice

I'd like to prove (exercise 9.5 in Roman's Lattices and Ordered Sets, p.203) that the lattice $M_3$ is simple, meaning that the only congruences on $M_3$ are the trivial ones (the 'equality' ...
2
votes
1answer
112 views

ideals of a ring form a modular lattice

We know that if $M$ is a left $R$-module, then $(\{\text{submodules of }M\},\subseteq)$ is a modular lattice. Taking $M\!=\!R$, we deduce that $(\{\text{ideals of }R\},\subseteq)$ is a modular ...
16
votes
2answers
633 views

Why are modular lattices important?

A lattice $(L,\leq)$ is said to be modular when $$(\forall a,b\in L) x \leq b \implies x \vee (a \wedge b) = (x \vee a) \wedge b,$$ where $\vee$ is the join operation, and $\wedge$ is the meet ...
1
vote
2answers
245 views

Followup to “Examples of rings with ideal lattice isomorphic to $M_3$, $N_5$”

In this post: Examples of rings with ideal lattice isomorphic to $M_3$, $N_5$ a nice example was given of a non-distributive ring. The lattice of ideals turned out to be the Diamond lattice $M_3$ with ...
1
vote
3answers
193 views

How to manage without specifying a particular algebraic system?

My long standing question: How to eliminate writing $\cap^L$ instead of plain $\cap$ when we deal with more than one lattice? (and likewise with other (finite and infinite) structures) It is ...