# Tagged Questions

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### Why are frames called “frames”?

Definition: A frame $F$ is a suplattice such that for any $x_{i}, y\in F$ (for $i\in I$, $I$ a set), we have $$y\wedge\left(\bigvee_{i\in I}x_i\right)=\bigvee_{i\in I}(y\wedge x_i).$$ Why are ...
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### What's so well in having a least element in a set?

What's so well about the principle of well-ordering? Why is this pattern of order named like this? I am unable to trace a connection between well-ordering and a set that has a least element so I ...
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### Sub-(complete lattice)? - what's the correct terminology?

Let $X$ denote a set, let $\mathcal{O}$ denote the open sets of a topological space with carrier $X$, and let $\mathcal{P}$ denote the powerset of $X$. Furthermore, let $\leq_\mathcal{O}$ denote the ...
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### Name of the order with irreflexivity, antisymmetry and transitivity?

I have an order otherwise poset aka partial order but it is irreflexive so relationships such as 1R1 and 2R2 are impossible. What is the name of this order?
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### Infinite set ordered like a “infinite tree”

Obviously I don't need to say that I'm still a newbie, so I'll try to explain my question with some pics too. I have a infinite set $T$ I have a visual idea of how this set is ordered, but I don't ...
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### Kinda well-orderedness.

Its not the case that $\mathbb{Z}$ is well-ordered (under the usual order). However, observe that Every non-empty subset of $\mathbb{Z}$ that is bounded below has a least element, and Every ...
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### What are ideals in a complete lattice?

If I have a complete lattice $L$, what conditions do I need for $I\subseteq L$ to be an ideal? In a general lattice the conditions are: $I$ is a lower set; $I$ is closed under (finite) joins. For ...
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### Extending a partial order to antichains

Let $(S, \leq)$ be a partial order. Let $T$ be the set of antichains of $S$ (i.e., subsets of $S$ whose elements are pairwise incomparable). Define a relation $\leq'$ on $T$ as follows: for all $A$, ...
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### Is there a name for this kind of “betweenness structure”?

A homeomorphism $\mathbb R\to\mathbb R$ is almost the same thing as an order isomorphism, except that a homeophorphism can also be an order anti-isomorphism. I'm wondering whether there is a natural ...
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### Problem with “tree” definitions

In my studies of choice principles, I've encountered the concept of a tree several times. Frustratingly, no two of the sources I've been working with define it in quite the same way! Notation: Given ...
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### If $\sim$ is an equivalence relation on $X$, and there is a strict total order on $X/\sim$, what kind of ordering does $X$ have?

I would like to know if there's a special name for this kind of ordering. When I say there is a strict total order on $X/\sim$, what I mean is that two distinct elements in the same equivalence ...
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### Why is 'Antisymmetry' named so?

So when we talk about order relations for the familiar number systems, we are always introduced to the antisymmetry property which is $x \le y, x \ge y \implies x=y$. When I think of the word ...
I am interested if the following concept has a name, or what do you think that a good name would be? Let $\{X_{i}: i=1,2,\ldots,n,n+1\}$ be a family of posets and $F:X_{1}\times ... 0answers 35 views ### Names of certain morphisms in Pos Pos is the category of small posets and monotone maps. I call a morphism$f:\mathfrak{A}\rightarrow\mathfrak{B}$of Pos monovalued iff it maps every atom of$\mathfrak{A}$either into an atom of ... 1answer 101 views ### A poset that's the union of the lower sets Let$(P,\leq)$be a poset, and let$\downarrow\! p = \{ x\leq p\}\subseteq P$. Let$M\subseteq P$be the subset of all maximal elements of$P$. Question: is there a specific term for a poset ... 3answers 263 views ### Is there a name encompassing both limit inferior and limit superior Is there a mathematical term which would include both liminf and limsup? (In a similar way we talk about extrema to describe both maxima and minima?) The only thing I was able to find was that some ... 4answers 125 views ### “Down-Closed”, “Down Ideal”, Something Else? Let$X$be an a set and let$\mathcal{C}$be a collection of subsets of$X$satisfying the following property: If$A$and$A^\prime$are subsets of$X$with$A \in \mathcal{C}$and$A^\prime ...
While playing around with least-fixed-point constructions on a powerset lattice, I've found this property to be useful. Let's say that $F : \mathcal{P(U)} \to \mathcal{P(U)}$, and that \$A \subseteq ...