0
votes
1answer
28 views

What is the name of this property of relation?

What is the name of property of a binary relation $R$ that states that $\lnot(a\mathrel{R} b) \iff \lnot(b \mathrel{R} a)$ for all $a, b$?
0
votes
0answers
34 views

Is a set of some $m \times n$ matrices a relation?

A relation between sets $A_i, i = 1, \dots, n$ is defined as a subset of $\prod_i A_i$. Given $m, n \in \mathbb N$, is a set of (some or all) $m \times n$ matrices over $\mathbb R$ considered a ...
2
votes
1answer
40 views

Terminology on pullbacks

I'm quite confused with the use of pullbacks, and in particular I wonder which terminology I shall use in the following examples. Let $X$ and $Y$ be arbitrary sets. Suppose that $f,g:X\to Y$ and I ...
1
vote
1answer
25 views

Name for a generalized relation to be a multiset?

A relation between two sets $A$ and $B$ is a subset of $A \times B$. If taking a multiset subset of $A \times B$, e.g. allowing $(a,b)$ appears twice in the subset, is there a name for such a ...
0
votes
1answer
18 views

What's the difference between a partial function and a relation?

My understanding of a partial function is that it is one which only maps a subset of some set $A$ to another set $B$ (where $B$ could be $A$). On the Wikipedia page, the below image is given as an ...
0
votes
0answers
22 views

Is there a particular name for the set of all relations?

I know that a relation on a set $S$ is a subset $R \subseteq S \times S$ such that for all $(s,s') \in S \times S$, $(s,s') \in R$ iff $sRs'$, therefore the set $T$ of all relations on $S$ is the set ...
3
votes
1answer
51 views

Rel: the category of relations

$\text{Rel}$ is the standard name for the category of sets and relations. Confusingly in "Abstract and concrete categories" (ACC), page 22, $\text{Rel}$ is defined as a category whose objects are ...
0
votes
1answer
207 views

Is an anti-symmetric and asymmetric relation the same? Are irreflexive and anti reflexive the same?

I don't understand the difference between an anti symmetric and asymmetric relation. From my understanding, it is asymmetric if there is not any element where: if (x,y) (y,x). But what if you have ...
1
vote
1answer
51 views

symmetric/antisymmetric

according to both the text and my professor, these properties are not mutually exclusive. i.e. a relation can be both symmetric and antisymmetric. I understand the properties themselves, but I don't ...
0
votes
0answers
11 views

Name for symmetric irreflexive binary relation

I have an irreflexive relation $\prec$ called unpreference: if $x\prec y$ then I say $x$ is unpreferred (or not preferred) to $y$. I wish to give a name to the symmetric part of the relationship, ...
0
votes
1answer
29 views

Partial order up to equivalence

In certain contexts one runs into something like a partial order, but the antisymmetry property is weakened as follows: if $x \preceq y$ and $y \preceq x$ then $x \simeq y$, where $\simeq$ is a given ...
2
votes
1answer
133 views

Is there a name for this property of a binary relation? $\forall x\forall y(x\mathsf{R}y\to\exists z(x\mathsf{R}z\land y\mathsf{R}z)))$

Consider a binary relation $\mathsf{R}$ such that $x\mathsf{R}y$ is the case only if there is some $z$ such that both $x\mathsf{R}z$ and $y\mathsf{R}z$ are the case. Is there a well-known name for the ...
0
votes
0answers
41 views

Embedding vs restriction

Embedding is the morphism $( A ; B ; \operatorname{id}_A)$ of the category $\mathbf{Rel}$ for sets $A \subseteq B$. I call restriction the morphism $( A ; B ; \operatorname{id}_B)$ for sets $A ...
1
vote
5answers
234 views

Reflexivity: How can something be related to itself?

Background: I'm a philosophy student. I'm comfortable with math, but don't have much of a background in it. One of the topics I'm writing about (I-relation in theories of identity) closely mirrors ...
2
votes
1answer
37 views

What is the term for a relation whose inverse relation is serial?

A relation $R$ is serial iff $\forall x, \exists y, xRy$. What is the name of the inverse property stating that $\forall y, \exists x, xRy$? And is there a name for the property which is the ...
1
vote
1answer
49 views

$xRx'$ and $yRy'$ implies $f(x,y)Rf(x',y')$

Let $R$ be a binary relation. Is there a name for the following property? $f(x,y)Rf(x',y')\quad$ if $xRx'$ and $yRy'$ Note:$f$ is a function.
1
vote
2answers
147 views

The bijective property on relations vs. on functions

I recently encountered what seems to me like an inconsistency in the usage of the term bijective. This is pretty basic stuff, but it somehow never occurred to me before. I'd like to make sure I'm ...
6
votes
3answers
174 views

Unambiguous terminology for domains, ranges, sources and targets.

Given a correspondence $f : X \rightarrow Y$ (which may or may not be a function) I generally use the following terminology. $X$ is the source of $f$ $Y$ is the target $\{x \in X \mid \exists y \in ...
5
votes
0answers
74 views

Is there a standard name for the relation $X \times Y$?

Given sets $X$ and $Y$, is there a standard name for the relation $f : X \rightarrow Y$ whose graph equals $X \times Y$? Something like the full/maximum/top/total/complete relation?
2
votes
1answer
66 views

Is there a name for this type of relation?

Let $S$ be a set. Let $\sim$ be a binary relation on $S$. Suppose $\sim$ follows these three rules. $x\sim x$ for all $x\in S$ (reflexivity). If $x\sim y$, then $y\sim x$ for all $x, y \in S$ ...
2
votes
0answers
41 views

Terminology for some special sets

The following predicates are used in the definition of total boundness of uniform spaces: a space is bounded if the predicate holds for every entourage, specifically for the first predicate (the ...
7
votes
2answers
157 views

Is there a name for relations with this property?

Is there a name for relations $\rho : X \rightarrow Y$ such that for all $x,x' \in X$ and all $y,y' \in Y$ we have that the following conditions $$xy \in \rho$$ $$x'y \in \rho$$ $$xy' \in \rho$$ imply ...
1
vote
1answer
2k views

Definition of correspondence

A one-to-one correspondence is an alternative name for a bijection between two sets, but to what does the term 'correspondence' alone refer? As far as I can see, it seems to be another term for ...
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
1answer
112 views

Mono's and Epi's in the category Rel?

Sorry to ask such a trivial question, but I can't find the answer anywhere. Question. What are the monomorphisms/epimorphisms in Rel? Furthermore, what's the standard terminology for describing ...
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 ...
12
votes
2answers
336 views

When does “pairwise” strengthen and when does it weaken?

"Pairwise disjoint" is stronger than "disjoint"; it sometimes happens that $\displaystyle\bigcap\limits_{i\in I} A_i=\varnothing$ but for every $i,j$, or at least for some, one has $A_i \cap ...
1
vote
1answer
563 views

Empty set as a relation

The empty set is an $n$-ary relation for every $n$, right? How should we call a pair $(n;r)$ consisting of some number $n$ and an $n$-ary relation $r$? To specify $n$ is necessary only when $r$ is ...
1
vote
1answer
151 views

Name of binary relation: if $(x, y)\in R$ then there is no $z$ such that $(y, z)\in R$

Is there a term for a binary relation $R\subset A^2$ on some set $A$ such that if $(x, y)\in R$ then there is no $z$ such that $(y, z)\in R$ ? Are there any examples of it? Are there any related ...
0
votes
1answer
41 views

Two terminology question about relations

Is there a name for constructing a set from a relation (or, more generally speaking, from a set of pairs that are tuples)? For example, let $R = \{(0, 1), (1, 2), (2, 3)\}$; if you collect all the ...
3
votes
3answers
173 views

What's the name for the equivalence induced by a function on its domain?

Any function $f$ with domain $X$ induces an equivalence relation on $X$, with classes $$\{f^{-1}(\{y\})\,:\, y \in \operatorname{im}f\;\} .$$ Is there a name for this equivalence? Thanks!
1
vote
0answers
53 views

Term for generalized antisymmetry?

As I understand it, a binary relation $R$ over a set $A$ is antisymmetric if for all $a, b \in A: aRb \land bRa$ implies $a = b$. Now, suppose that I have an equivalence relation $E$ over the set ...
2
votes
1answer
165 views

Is there a name for this type of binary relation?

Suppose that $X$ is a set and $\sim$ is a binary relation on $X$ that satisfies for all $x,y \in X$; if $x \sim y$ then $x \sim x$ and $y \sim y$. Is there a name for this type of relation? I am ...