All Questions
Tagged with lattice-orders supremum-and-infimum
21
questions
2
votes
1
answer
41
views
The join of two set partitions in the refinement order
Let $X$ be a set. The set $\Pi_X$ of all partitions of $X$ is partially ordered via the refinement order, which is defined by $\alpha \leq \beta$ if and only if for each block $A\in \alpha$ there is a ...
- 23.1k
1
vote
1
answer
39
views
Properites of Lattices imply eachother.
Assume S is a finite poset.
A lattice is a poset S such that
S is bounded.
∀ x, y ∈ S there exists x ∧ y (existence of Meet/Infimum).
∀ x, y ∈ S there exists x ∨ y (existence of Join/Supremum).
...
- 13
9
votes
1
answer
508
views
Can the supremum of an uncountable family of measures be replaced by the supremum over a countable subfamily?
Consider a measurable space $(X,\mathcal{A})$. Let $\mathcal{M}$ denote the family of all countably additive measures $\mu\colon \mathcal{A}\to [0,+\infty]$. This family can be made into a partially ...
- 561
0
votes
1
answer
65
views
Clarifications needed in an exercise about semilattice and abelian monoids in Arbib and Manes' text
The following question is taken from Arrows, Structures and Functors the categorical imperative by Arbib and Manes
Exercise: A $\textbf{semilattice}$ is a poset in which every finite subset has a ...
- 3,683
3
votes
1
answer
116
views
Right unitor in the monoidal category of sup-lattices
In short: I am trying to find a right unitor for the monoidal category $\mathsf{SupLat}$ of sup-lattices.
1. Preliminaries
Let $L,M$ be sup-lattices.
Denote by $L^*$ the set $L$ with order $l \leq_{L^*...
- 3,068
1
vote
1
answer
107
views
Does $(f^{0})^{0} = f$ hold for suplattice homomorphisms?
Let $X$ and $Y$ be suplattices and $f: X \rightarrow Y$ a suplattice homomorphism (i.e. $f(\bigvee S)=\bigvee \{f(s)|s \in S\}$ for every subset $S \subseteq X$). Denote by $X^0$ and $Y^0$ the ...
- 1,769
3
votes
1
answer
153
views
Why is product of infimum and supremum of a commuting pair of elements in a lattice-ordered group equal to the product of the elements?
Context: Self-study.
Seth Warner's Modern Algebra (1965), question $15.11$ gives:
If $(G, \circ \preccurlyeq)$ is a lattice-ordered group and if $x$ and $y$ are commuting elements of $G$, then $$\sup ...
- 5,057
1
vote
1
answer
148
views
What is a relation of category of complete sublattices to category of lattices?
What is a relation of category of complete sublattices to category of lattices?
I haven´t found much about a category of lattices, but I assume objects = lattices, morphisms = lattice homomorphisms.
...
- 2,452
3
votes
2
answers
120
views
When does downward closure commute with supremum?
Let $A$ be a suplattices, and suppose we have a family $\{a_i\}_{i\in I}\subseteq A.$
Is $\bigcup_{i\in I}(\operatorname{\downarrow}a_i) = \operatorname{\downarrow} \sup_{i\in I}(a_i)$ in general?
...
- 18.4k
0
votes
2
answers
131
views
What does being smaller that to a join means in a distributive lattice?
This is a follow up question to my previous question with more restrictions. It was answered negatively for arbitrary lattices, but mentioned that the result holds "only" in distributive ...
- 18.4k
2
votes
1
answer
61
views
What does being smaller that to a join means in a lattice?
This sounds like a very naïve question, but I couldn't find a correct argument to prove/disprove it rigorously.
Suppose we have a a subset $A$ of a lattice (or any join-semilattice), and some $x\le\...
- 18.4k
1
vote
1
answer
114
views
$lub(a, b)$ and $glb(a, b)$ of non- comparable lattice elements
In an algebraic lattice $(L, \vee, \wedge, 0, 1)$, the binary operations $\vee$ and $\wedge$ are commutative and associative, satisfying absorption laws. The elements $0$ and $1$ are top and bottom ...
- 1,352
1
vote
1
answer
110
views
Ordered set has a maximum and every subset has infimum $\implies$ every subset has supremum
Let $M$ be an ordered set such that
$a)$ $M$ has a maximum element
$b)$ Every (non-empty) subset of $M$ has an infimun.
Prove that every non-empty subset of $M$ also has a supremum.
I tried fixing a ...
- 342
2
votes
1
answer
103
views
About dense subalgebras of boolean algebras.
I started studying about Boolean algebras and got stuck doing an exercise. Maybe it's trivial but really I can't do it. First, a definition
A subset $S$ of $B$, where $B$ is a Boolean algebra, is ...
- 3,935
1
vote
1
answer
93
views
Understanding infimum in a complete lattice
For any two formal concepts, $(A_1,B_1)$ and $(A_2,B_2)$ of a formal context, the standard definition for the supremum and infimum in a complete lattice are as ...
- 173