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
61 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
60 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
90 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
43 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
66 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
56 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
52 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.
1
vote
3answers
56 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 ...
4
votes
0answers
76 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
87 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
62 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
59 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
100 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
54 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
163 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
225 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
111 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
96 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
206 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
47 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
33 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
96 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
261 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
122 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
119 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 ...