All Questions
Tagged with order-theory definition
64
questions
7
votes
3
answers
478
views
Would the category of directed sets be better behaved with the empty set included, or excluded?
In a topology book of mine, a directed set is defined as a nonempty set $D$ equipped with a relation $R$ that is transitive, reflexive, and for all elements $x$ and $y$ of $D$, there exists a $z$ in $...
- 21.5k
2
votes
0
answers
58
views
Why is this description of the Dedekind–MacNeille completion never mentioned?
There are various ways to describe the Dedekind–MacNeille completion of a poset, the minimal complete lattice in which the poset can be embedded. I’ll first state the ones I’ve seen and then one I ...
- 239k
1
vote
1
answer
94
views
There exists a definition of order homomorphism?
Let be $(X,\preccurlyeq)$ and $(Y\curlyeqprec)$ partial ordered sets so that let be $f$ a function form $X$ to $Y$. So if $\rightthreetimes$ and $\leftthreetimes$ was an operation on $X$ and an ...
- 4,906
4
votes
1
answer
245
views
Definition of an interval in a poset
I can think of two nonequivalent ways of defining an interval in a poset:
An interval of a poset $P$ is a subset $I\subset P$ with the property that for all $x, y, z\in P$ such that $x < y < z$ ...
- 2,210
0
votes
1
answer
38
views
A formalization of the notion of Hasse diagrams
I have read many textbooks on order theory, but none of them give a formal definition of Hasse diagrams. I would like someone to give me a formalization of that notion. Just to be clear, we are ...
- 21.5k
1
vote
1
answer
74
views
Is this modification of connexity necessary, or redundant in the definition of partial ordering?
My question pertains to Fundamentals of Mathematics, Volume 1
Foundations of Mathematics: The Real Number System and Algebra, Edited by H. Behnke, F. Bachmann, K. Fladt, W. Süss and H. Kunle Vol-1 A8....
- 3,349
2
votes
2
answers
435
views
What is an interval of a lattice?
What is an interval of a lattice in order theory?
In the Wikipedia page of modular lattices, there is a theorem that uses the notion of interval (diamond isomorphism), but the term interval is not ...
- 1,243
1
vote
1
answer
134
views
Why are algebraic structures typically defined with operators/relations and laws?
Here is a definition for equivalence relations:
An equivalence relation on a set $X$ is a relation $\sim$ which is reflexive, symmetric, and transitive.
As compared to:
An equivalence partitioning ...
- 640
0
votes
2
answers
360
views
What is a z-filter?
I am trying to understand meaning of z-filters (def. below).
Why are they introduced?
Where are they used in mathematics? The only example I have found is that they are used for the construction of ...
- 2,452
4
votes
1
answer
420
views
Filters/ ultrafilters/ principal filters/ proper filters/ maximal filters - intuition and relations between them
I am studying filters and seeing a lot of definitions at once. So I would like to ask you help me.
Is my intuition correct for these guys? Suppose we have partially ordered set with some subsets:
...
- 2,452
1
vote
0
answers
31
views
Vocabulary : How to call a Strict linear order associated with a total order relation?
Let $\leq_t$ be a total order relation over a set X.
I want to define "the" strict linear order relation $<_t$ associated with $\leq_t$.
Firstly , is there a name for such a associated ...
- 369
1
vote
2
answers
78
views
What does "$A \leq B : \Longleftrightarrow A \subseteq B$ is an order relation of $\mathcal{P}(N)$" mean?
I have read the following in some exercise for discrete mathematics.
Let $N$ be a set and $\mathcal{P}(N)$ be its power set. Then $A \leq B : \Longleftrightarrow A \subseteq B$ is an order relation of ...
- 547
3
votes
0
answers
41
views
Is there anyone defining asymptotic dominance so that $f \precsim g \text{ and } f \succsim g \implies f \sim g$?
I came to understand that asymptotic negligibility, dominance and equivalence form together a partial order of real-valued functions, analogous to $=, \leqslant, <$ for the real numbers. In the ...
- 171
1
vote
1
answer
52
views
What is this algebraic structure called like? [duplicate]
Is there a name for an object $L$ that has an additive structure and a (not necessarily total) order such that for all $a,b,c \in $ L we have
$a+b = b+a$
$a+(b+c)= (a+b)+c $
$a+b \leq a+c \...
- 635
1
vote
0
answers
37
views
Notions and terms for extrema in preorders
I'm working on a computer program for making decisions that represents the user's preferences as a preorder, as is common in theories of preference. I'd like to formalize the notions of "the best ...
- 357