All Questions
Tagged with order-theory terminology
146
questions
2
votes
0
answers
50
views
Converting "improper" partial order to total order
I suspect that if I knew what to search for, this would be easy to find an answer to, but I don't know what the proper name is for the input portion of the problem statement.
I have a set and a ...
- 663
1
vote
0
answers
34
views
Does this poset property have a name?
I have a poset with the following property:
For any infinite descending chain $x_1 > x_2 > \dots$ and any $y$ that is a lower bound for the chain ($y < x_i$ for all indices $i$), there ...
- 4,216
2
votes
0
answers
74
views
Terminology for "transposition" of monomorphism to epimorphism in simplex category?
Recall that the simplex category $\Delta$ is dual to the category of intervals $\mathbb{I}$. By $\Delta$ I mean the category of finite ordinals $\mathbf{n} \in \omega$ with monotone functions between ...
- 804
0
votes
0
answers
30
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Terminology for non-empty suprema preserving function
Is there an established name for a map of complete lattices $f : L \to L'$ that preserves nonempty suprema? I.e. for all $U \subseteq L$ with $U \neq \emptyset$,
$$ f( \bigvee U) = \bigvee_{u \in U} f(...
- 469
5
votes
1
answer
123
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Is there a name for this refinement of the subset ordering?
Lately I have been considering a certain partial ordering on the subsets of a totally-ordered set. My question is:
Does this ordering have a name?
The ordering is defined as follows:
If $\langle S,...
- 65.8k
4
votes
1
answer
366
views
Sharp vs. tight
What is the difference between a "sharp bound" and a "tight bound"? Are the two adjectives synonyms in mathematical prose? Otherwise, when would you use one and when the other?
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- 3,557
0
votes
1
answer
44
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Is there some name in your field for an element in a bounded poset that is just “one step” above/below the minimum/maximum?
I say that a poset is lower-bounded (resp., upper-bounded) if it has a minimum (resp., maximum) element, and I say that the poset is bounded if it is both lower and upper-bounded. In algebra, ...
5
votes
1
answer
74
views
Two poset properties: are they related?
A bunch of infinite posets $P$ with $\hat 0$ have the following property
For every $x\in P$, the principal filter $\{ y\in P : y\ge x\}$ is isomorphic as a poset to $P$ itself.
Examples include ${\...
- 1,794
0
votes
0
answers
32
views
Is there a name for this ordering on integer vectors?
Let $\mathbf{k} \in \mathbb{[n]}^{u}$ be $u$-dimensional arrays, where $[n] = \{0,1,\dots,n\}$. Now let us assume that $\mathbf{k}$ are generated under $u$ nested for loops running from 0 to n. For ...
- 1,799
1
vote
0
answers
34
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What is the name of this family of subsets (obtained from partition refinement)?
Let $[q]=\{1,\dots,q\}$.
We obtain a set hierarchy $H$ (aka rooted tree) on $[q]$ using partition refinements.
This means that the vertices $v\in V(H)$ of $H$ are subsets $v\subseteq[q]$ of $[q]$, and ...
- 3,633
1
vote
1
answer
51
views
Is there a formal term for a "subset connected by comparability" in a poset?
Suppose I have a poset $P$. A subset $Q$ of the elements of $P$ has the property that, for any two elements $a,b \in Q$, $a$ is "connected to" $b$ through a chain of comparisons with the ...
- 472
0
votes
0
answers
97
views
Name of this notion in a total order?
Let me start with a prototypical situation, with $S \subset \mathbb{R}$. I'd like to divide $S$ into the "smaller" and "bigger" parts, by specifying the boundary $t \in \mathbb{R}$,...
- 225
3
votes
1
answer
65
views
Semilattice whose Subsets are All Closed -- does it have a special name?
Context: self-education.
I am currently getting my head round semilattices.
My understanding is that a semilattice $(S, \odot)$ is a semigroup whose operation $\odot$ is both commutative and ...
- 5,057
0
votes
0
answers
25
views
Are the odd integers cofinal in the integers with the usual order?
I think the definition I read of cofinal says there must always be an element of the subset which is greater than any given element. The odd numbers have this property, so they're cofinal, right?
...
- 9,299
0
votes
0
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38
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Name of property: $\phi (x)\geq x$ [duplicate]
Let $X$ be a preordered set and $\varphi : X\to X$ a function (can assume monotone if useful for the answer). Does the property of $\forall x\in X: \varphi (x) \geq x$ have a standard name?
