All Questions
Tagged with lattice-orders discrete-mathematics
106
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If sub-universe $S$ of lattice has congruence $\theta$, does the lattice have a congruence $\lambda = \theta \cap S^2$? [duplicate]
Let $(L, \lor , \land )$ be a lattice and $S$ a sub-universe of the lattice. A sub-universe of a lattice will be any subset of the lattice set that is non-empty and closed under $\land$ and $\lor$. ...
- 3,468
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40
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Understanding the definition of congruences over a lattice
Let $(L, \land, \lor)$ a lattice and $\theta$ a binary relation over $L$. We say $\theta$ is a congruence iff
$$
x_0\theta x_1, y_0 \theta y_1 \Rightarrow (x_0 \lor y_0)\theta(x_1 \lor y_1)
$$
(and ...
- 3,468
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45
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Number of lattices over a finite set
I'm interested in finding a general formula for the number of lattices possible over a finite set $S$ as a function of the set's cardinality.
For instance, how many lattices over $\{1, 2, 3\}$ are ...
- 3,468
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49
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Help undsersanding matroid closure, loops, contraction and duality.
These ideas are being used a lot, but I cannot justify why they are correct:
If M is a matroid and $T$ a subset of $E(M).$ Then $$(a)\ cl(T) = T \cup \{e \in E(M) - T: e \text{ is a loop of M/T}\}.$$ ...
- 3,127
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36
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Infinite lattice with every totally ordered set finite
Construct a lattice L such that L is infinite but every totally ordered subset of L is finite? I really don't know how to proceed because i don't see how every totally ordered set would be finite ...
- 79
3
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3
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145
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Are all complete lattices a pointed complete partial order, and vice versa?
A friend of mine asked for my help in drawing a venn diagram that includes the notions of partial orders (PO) in general, complete partial orders (CPO), pointed complete partial orders (CPPO), total ...
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159
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Rigorously proving a lattice is sublattice of another
Consider the following two lattices, $L_1$ (top) and $L_2$ (bottom):
I apologize for the bad image arrangement.
We are asked whether $L_1$ is a sublattice of $L_2$. This can be visually observed ...
- 3,468
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194
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For a distributive lattice, show $x ~\hat{\land}~ a = y ~\hat{\land}~ a$ and $x ~\hat{\lor}~ a = y ~\hat{\lor}~ a$ imply $x = y$.
Let $\hat{\land}, \hat{\lor}$ be binary operators denoting the infimum and supremum of two elements in a poset. I was given the following problem.
For a distributive lattice $\langle ~L, ~\hat{\lor}~,...
- 3,468
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47
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Is the number of atoms in a finite lattice always less than or equal to the length of the lattice?
My attempt: Let $\{v_1, v_2,...,v_m\}$ is the set of atoms in a finite lattice L. Now the chain
$0<v_1 \lt (v_1 \lor v_2) \lt (v_1 \lor v_2 \lor v_3) \lt...\lt (v_1\lor v_2 \lor ...\lor v_m) \leq I$...
- 33
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155
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Sublattice of an Atomic Lattice [closed]
Is the sublattice of an atomic lattice atomic? If not, then what additional condition is necessary for a sublattice to be atomic?
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3
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165
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Dual definition for distributivity lattice conditions
I am starting discrete mathematics and it came to me that distributive lattices has 2 equicalent representations for their definition:
$a\wedge (b\vee c) = (a \wedge b)\vee (a\wedge c)$
$a\vee (b\...
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30
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If $(f, g)$ is a Galois connection between two bounded lattices, then if $T$ is an ideal we have $f^{-1}(T)$ is an ideal
Let $\mathcal{I}(L)\:$ and $\mathcal{I}(N)\:$ be the ideal lattices of the bounded lattices $L$ and $N$ and let $(f, g)$ be a Galois connection between $L$ and $N$, then show that $\:\forall \:\: T \...
- 173
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330
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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 ...
- 21
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77
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let $L$ be a bounded distributive lattice with dual space $(X:=\mathcal{I}_p(L), \subseteq, \tau)$, then the clopen downsets of $X$ are $X_a, a \in L$
11.22 Lemma, from B. A. Davey, H. A. Priestley, Introduction to lattices and order,
let $L$ be a bounded distributive lattice with dual space $(X:=\mathcal{I}_p(L), \subseteq, \tau)$ and $X_a = \{I \...
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37
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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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