Articles of congruence relations

What can we actually do with congruence relations, specifically?

Let $T$ denote a Lawvere theory and $X$ denote a $T$-algebra. Under my preferred definitions: A subalgebra of $X$ consists of a $T$-algebra $Y$ together with an injective homomorphism $Y \rightarrow X$. A quotient of $X$ consists of a $T$-algebra $Y$ together with a surjective homomorphism $X \rightarrow Y$. Also: A congruence on $X$ is […]

Solving a congruence without Fermat's little theorem

Given $n\in\Bbb N$, what is the least $a>1$ with $a^{2^n}\equiv1\bmod2015$? Is there a solution not using Fermat’s little theorem or the Chinese remainder theorem, any ideia?

What is the interpretation of $a \equiv b$ mod $H$ in group theory?

I.N. Herstein has defined: Let $G$ be a group, $H$ a subgroup of $G$; for $a,b \in G$ we say $a$ is congruent to $b \mod H$, written as $a \equiv b \mod H$ if $ab^{-1} \in H$. Let $G$ be a group, $H$ a subgroup of $G$; for $a,b \in G$ we say $a$ […]

Two congruent segments does have the same length?

The answer to the question in the title seems an obvious ”Yes by definition !”. And this really is the definition from Wikipedia: Two line segments are congruent if they have the same length. But in Hilbert’s Foundations of Geometry congruence is defined without use of metric notions, by the axioms of Group IV (chapter […]

$M_3$ is a simple lattice

I’d like to prove (exercise 9.5 in Roman’s Lattices and Ordered Sets, p.203) that the lattice $M_3$ is simple, meaning that the only congruences on $M_3$ are the trivial ones (the ‘equality’ congruence, i.e. $\{(x,x); x\!\in\!M_3\}$, and the ‘everything’ congruence, i.e. $\{(x,y); x,y\!\in\!M_3\}$). Attempt of proof: By the symmetry of $M_3$, it suffices to prove […]

Lattices are congruence-distributive

$\newcommand{\r}[1]{\mathrel{#1}}$ First, a few definitions. Given a lattice $L$, a congruence on $L$ is an equivalence relation $\theta$, compatible with the lattice operations, i.e. if $x_1\r{\theta}x_2$ and $y_1\r{\theta}y_2$, then $x_1\wedge x_2\r{\theta}y_1\wedge y_2$ and $x_1\vee x_2\r{\theta}y_1\vee y_2$. The congruences on $L$ form a lattice, with meet being intersection and join being the equivalence envelope. I’m trying […]