# Questions tagged [congruence-relations]

For questions about general congruence relations, i.e. equivalence relations on an algebraic structure that are compatible with the structure. Please DO NOT use this for questions about integer modular arithmetic.

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### Congruence lattice of a semiring

A famous result of Funayama and Nakayama states that the congruence lattice of any lattice is a distributive lattice [1]. Also, it can be proved that the lattice is a frame/ complete Heyting algebra 2....
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### Do quotient inverse semigroups exist?

A congruence on a semigroup $S$ is an equivalence relation $\sigma\subseteq S\times S$ that respect to the multiplication. In other words $$(a,b), (c,d)\in\sigma \implies (ac, bd)\in\sigma.$$ Given a ...
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### Congruence relation of an Algebra

Let $2=\{0,1\}$, $+$ denote addition modulo $2$, $g\left(x\right)=x+1$. Consider the following Algebra $A=\langle 2,+,g\rangle$. I want to calculate congruence relation on $A\times A$. Can someone ...
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### What differs a ring ideal from a congruence class in the integers?

I know the following definitions: Congruence: $a \equiv b \pmod{n}$ if $n \mid a-b$ A subring $I$ of a ring $R$ is an ideal if whenever $r \in R$ and $a \in I$, then $ar \in I$ and $ra \in I$. ...
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### Varieties in which congruences are characterized by subalgebras

For some varieties $V$, for any algebra $\mathbb{A}\in V$, there exists a lattice isomorphism $f:\text{Con}(\mathbb{A})\to \mathbb{B}$ where $\mathbb{B}\leq \text{Sub}(\mathbb{A})$ between the lattice ...
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### Universal Algebra: an application of a Maltsev's algorithm to characterize congruences

In "Universal Algebra: Foundamentals and Selected Topics (pg. 107)" of Clifford Bergman the following Maltsev's theorem is intoduced for constructing congruences. Let $\textbf{A}$ be an ...
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### The only congruence is the identity congruence [From Algebraic Methods in Philosophical Logic, Dunn and Hardegree]

In the book "Algebraic Methods in Philosophical Logic" by Dunn and Hardegree I was very much confused by the remark 2.6.7 on page 22. In this book a relational structure $\mathbf{A}$ is ...
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### Let R and S be binary relations on a set A. Solve for each case: If R and S are i) reflexive, ii) symmetric, iii) transitive, what would R∪S be? [duplicate]

I had a similar question just now and it peaked my interest on binary relations. Solving i) "Assuming R and S are reflexive so: Let x ∈ A. Then (x, x) ∈ R and (x, x) ∈ S (reflexive property of R ...
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### Qutoient monoid of kernel pair of monoid homomorphism between multiplicative naturals to additive naturals (sum of prime numbers).

Let $M = \Bbb{N}^{\times}$ and $N = (\Bbb{N}\setminus \{1\} \cup \{0\})^+$ be the multiplicative, and respectively, the additive naturals and let $\varphi : M \to N$ be defined by taking $1$ to $0$ ...
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### Question On Solutions for a Specific Congruence $ax \equiv 1 \pmod p$ when $p$ is prime and $p \not\mid a$

I have been working through exercises within a book on polynomials. Part (f.) of a particular exercise asks the following : Show that, if $p$ is a prime, and $a$ is not a multiple of $p$, then there ...
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I want to resolve the diophantine equation: $\sqrt{x^2+5x+12} ≡ x-2\pmod 5$ I have thought 2 ways: 1. $(\sqrt{x^2+5x+12})^2 ≡ (x-2)^2\pmod 5$ $x^2+5x+12 ≡ x^2 -4x+4\pmod 5$ $9x+8 ≡ 0\pmod 5$ \$ 4x+3 ...