0
votes
0answers
22 views

Is there accepted notation and/or terminology for the smallest cover of $S$ with cells from $P$?

Let $X$ denote a set. Then for $S \subseteq X$ and $P$ a partitioning of $X$, define $P \diamond S$ as the smallest cover of $S$ with cells from $P$. Explicitly: $$P \diamond S = \bigcup\{Q \in P ...
0
votes
1answer
46 views

Why we use ANY in the definition of a maximal element?

I am confused about the following definition: "a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S." I do not understand ...
2
votes
3answers
148 views

Name the property $f(x) \ge x$

It's a really one of the simplest properties you could imagine for a function. But I haven't been able to find a name for it. What do you call a function $f$ with the following property: $$f(x) \ge ...
1
vote
0answers
46 views

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 ...
6
votes
1answer
85 views

Terminology in forcing

In the context of forcing one reads the relation $p \leq q$ in a poset $P$ as "$p$ extends $q$". A typical example is the poset $P$ of finite partial functions, where one defines $p \leq q$ when $q ...
1
vote
0answers
23 views

Has an order with this property a special name?

If $a$ is an element in a preorder then you can eventually go 'a step back' (and repeat this) in the sense of finding an element with $b\leq a$ and not $a\leq b$. Is there a special name for ...
1
vote
1answer
28 views

Notation/terminology: Existence of a nonleast element which is less of any element of a set

Let $A$ is a subset of a partial order $X$. Are there any name and/or notation for the following predicate $P(A)$? $P(A)$ iff there is a non-least element $x$ of $X$ which is a subelement of each ...
3
votes
1answer
78 views

What are some examples of “homogeneous” linear orders, other than $\mathbb{R}$ and $\mathbb{Q}$?

Let $L$ denote a linear order that is unbounded. Then it may or may not satisfy: Globally homogeneous. For all $x,y \in L \cup \{-\infty,\infty\}$, if $x < y$ then the interval $(x,y)$ is ...
1
vote
1answer
63 views

A set with a supremum and an infinum inside

What is the name of a set $A$ that has its infimum $i \in A$ and a supremum $s \in A$? Examples: $[0,1]$ has a supremum $1$ and an infimum $0$ which are inside $[0,1]$. $\{0,1,2,3\}$ has a supremum ...
1
vote
2answers
101 views

Initial Segments and Initial Sections of Posets

For a set A with a partially ordering <=, define the following 1) A subset s(x) of A = {y in A such that y <=x} 2) A subset S of A with the property that for every x in S then all y in A ...
2
votes
1answer
48 views

In the transition from set theory to order theory, what is the appropriate generalization of “group”?

Given a set $X$, the collection of all bijective endofunctions on $X$ forms a group, called the symmetric group on $X.$ Furthermore, Cayley's theorems says that every group embeds into a subgroup of ...
2
votes
0answers
70 views

Do subhomomorphisms / subfunctors have a standard name, and where can I learn more?

Sorry for all the mistakes in the original! I think they're mostly fixed now. Thank you for your patience. Part 1. If $A$ and $B$ are models in $\mathrm{Pos}$ of an algebraic signature $\sigma$, ...
0
votes
1answer
57 views

What is the terminology for this type of order-preserving function?

As I roughly know, a function $f$ in $\mathbb{R}$ is called order-preserving if $f(x)>f(y)$ for $x>y$, $x,y \in \mathbb{R}$. May I ask, if anybody knows the terminology for two functions $f$ ...
1
vote
2answers
54 views

Notation for “incommensurate” elements?

Say that $x\wedge y\not=x,y$, that is neither $x\leq y$ nor $y\leq x$. Is there a way to denote this? I've been saying $x<>y$ but that's completely made up.
2
votes
3answers
100 views

When we talk of e.g. the natural numbers equipped with a non-standard order , what does “equipped” mean?

A question for "real" mathematicians who have become better acculturated to math-speak than this philosopher! If you read a phrase like ... the natural numbers equipped with the ...
5
votes
1answer
102 views

Terminology for metric space with “anti-symmetric” distance

I'm interested in spaces that have a two-place function $d$ with non-negative real values, satisfying the following three conditions (for all $x$, $y$, $z$): $d(x, x) = 0$ $d(x, y) + d(y, z) \geq ...
1
vote
3answers
89 views

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 ...
0
votes
1answer
67 views

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 ...
2
votes
2answers
63 views

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?
3
votes
0answers
105 views

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 ...
4
votes
2answers
61 views

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 ...
6
votes
1answer
58 views

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 ...
5
votes
1answer
174 views

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 $A$, ...
15
votes
4answers
240 views

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 ...
3
votes
1answer
117 views

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 ...
3
votes
2answers
99 views

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 ...
2
votes
3answers
210 views

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 ...
0
votes
0answers
48 views

Does this monotonicity-type concept have a name?

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 ...
1
vote
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 ...
2
votes
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 ...
3
votes
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 ...
2
votes
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 ...
2
votes
1answer
144 views

What's the name of this function property?

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 ...