0
votes
0answers
63 views

Relational algebraic structures

Recently I came across the notion of relational $\beta$-algebra, defined as a set $S$ and a binary relation $\xi:\beta S-S$, where $\beta S$ denotes the set of ultrafilters on $S$ (and $\beta$ is the ...
2
votes
1answer
102 views

The closures of a binary relation

I have the following dilemma concerning the equivalence closure of a binary relation. Let $X$ be a nonempty set and $R\subseteq X\times X$ (i.e., a binary relation over $X$). Consider the following ...
1
vote
2answers
31 views

How do you call this feature and property of relation?

If an operator can be defined for $k$ operands, $k \in \mathbb N$, how do you call this feature of the operator? For example, "+" is such an operator. Similarly, for a relation $R$ on a set $X$, $R$ ...
3
votes
1answer
48 views

Finite poset maximum and minimum element

Let $(P,\le)$ be a finite poset. An element $z \in P$ is an upper bound for $x,y \in P$ if $x \le z$ and $y \le z$. How do I prove that if every two elements in $P$ have an upper bound then ...
2
votes
1answer
42 views

Let $S=\{a,b\}$. Of all the relations on $S$ which are symmetric? Reflexive? Transitive?

The relations are as follows: 1.) $\{(a,a)\}$ 2.) $\{(a,b)\}$ 3.) $\{(b,a)\}$ 4.) $\{(b,b)\}$ 5.) $\{(a,a),(a,b)\}$ 6.) $\{(a,a),(b,a)\}$ 7.) $\{(a,a),(b,b)\}$ 8.) $\{(a,b),(b,a)\}$ 9.) ...
-1
votes
1answer
60 views

Show that ≡ is an equivalence relation, Show that ⊕ is well-defined, and Show that ⊕ is a commutative and associative operation.

Let $(a,b),(x,y) \in\Bbb R\times\Bbb R$ and define $(a,b) \equiv (x,y)$ iff $a+b = x+y$. a. Show that $\equiv$ is an equivalence relation. Define the operation $\oplus$ on the equivalence classes as ...
2
votes
1answer
136 views

Smallest Congruence Relation generated by a set

$\newcommand{\cl}{\operatorname{cl}}$ Let $R \subset S \times S$ be a binary relation, the smallest i) reflexive relation containing it is $$ \cl_\mathrm{ref} = R \cup \{ (x,x) : x \in S \} $$ ii) ...
3
votes
3answers
310 views

New kind of identities?

I found a new kind of identities which are half logic and half algebraic while working on a proof of NP-completeness. They are like this: $$ \frac{a+mb}{n+m} < \frac{a}{n} \iff b < ...
2
votes
0answers
81 views

How to denote the set of binary relations of which a particular ordered pair is a member?

Given a universe $U$ and two subsets $S$ and $T$ (also, both members of $U$), what is the name given to denote the set of all binary relations in $U$ where the ordered pair $(S,T)$ is a member? The ...
0
votes
1answer
61 views

an infinite queue preserving equality.

Is there any well-ordered set $(A,\leq)$ such that: $(A,\leq^{-1})$ is well-ordered. $A$ is infinite. there's exactly one function $\theta:A\rightarrow \{0,1\}$ such that 1) for each $a < M$, ...
0
votes
1answer
127 views

Quotient set univocally defined

I know that what I am going to ask is pretty basic, borderline stupid, nevertheless it is bugging me. By definition I know that given a set $A$ and a equivalence relation $\rho$, then the items ...
2
votes
2answers
94 views

Verifying prime factorization equivalence class

I define a relation on $\Bbb N$ as follows: $x \sim y \Longleftrightarrow \ \exists \ j,k \in \Bbb Z$ s.t. $x \mid y^j \ \wedge \ y \mid x^k$ I have shown that $\sim$ is an equivalence relation ...
1
vote
1answer
407 views

Guidelines for finding maximum, minimum, maximal and minimal elements of a poset

What I am struggling the most these days is determining maximum, minimum, maximal and minimal elements of a poset. I realize I'm often misled by the definition of total order given by the well known ...
2
votes
1answer
254 views

Spotting maximum, minimum, maximal and minimal elements in a poset

Let $(\mathbb N^* \times \mathbb N^*, \varphi)$ be a poset defined as follows: $$\begin{aligned} (a,b)\varphi(c,d)\Leftrightarrow ab<cd \text{ or } (a,b) = (c,d)\end{aligned}$$ Check if $\varphi$ ...
2
votes
5answers
594 views

General questions about equivalence classes and partitions

1) If a set is partitioned into non-overlapping, non-empty subsets, then those subsets are equivalence classes. If each element in a set is unique, how can a set be partitioned into subsets with ...
2
votes
1answer
296 views

Check if a relation is a partial or total order and find minimum, maximum, minimal and maximal elements

Given $\mathbb Z^-=\{x\in \mathbb Z:x<0\}$ and $T = \mathbb Z^-\times \mathbb N$, let the binary relation $\odot$ be defined as follows: $$\begin{aligned} (a,b) \odot (c,d) \Longleftrightarrow a ...
4
votes
2answers
193 views

An analogy between subgroups and equivalence relations.

I have noticed a certain analogy between subgroups of a group $G$ and equivalence relations on a set $X$. I would like to know if there's an explanation for this analogy or a common generalization of ...
7
votes
1answer
213 views

Can we extend the definition of a homomorphism to binary relations?

This is going to be quite a long post. The actual questions will be at the end of it in section "Questions." INTRODUCTION After receiving an answer to this question about extending the definition of ...
4
votes
2answers
244 views

The fibres of a map form a partition of the domain.

This is a question from the free Harvard online abstract algebra lectures. I'm posting my solutions here to get some feedback on them. For a fuller explanation, see this post. This problem is from ...
0
votes
3answers
189 views

Two questions about equivalence relations

Question 1: Let $x,y \in S$ such that $x\sim y$ if $x^2 =y^2\pmod6 $. Show that $\sim$ is an equivalence relation. This is what I tried: Reflexive: $x^2\pmod6 = x^2$ implying $x\sim x$ ...
7
votes
5answers
655 views

Dependence of Axioms of Equivalence Relation?

This question is problem 11(a) in chapter 1 in 'Topics in Algebra' by I.N. Herstein. These are the properties of equivalence relation given in this book. Prop 1 $a \sim a$ Prop 2 $a \sim b$ ...
4
votes
3answers
178 views

Books and Papers that have treatment of properties like Idempotence and related operations

Please recommend resources to study Idempotence and other similar properties of processes and operations in depth. I want to know what other properties like Idempotence are there for an operation. I ...