For each $n \in \mathbb{N}$, let $F_n$ be a finite set with $n$ elements. For any function $f : F_n \to F_n$ and $k \in \mathbb{N}$, $f^k$ is the result of composing $f$ with itself $k$ times. Say that $n$ distinguishes powers $i$ and $j$ iff there is some function $f : F_n \to F_n$ […]

Is there an example of cancellative Abelian monoid $M$ in which we may find two elements $x$ and $y$ such that they have a least common multiple but not a greatest common divisor?

This is a follow-up to Do Lipschitz/Hurwitz quaternions satisfy the Ore condition? Jyrki Lahtonen answered the question in the positive by noticing that every right principal ideal in either ring has a non-empty intersection with the center. Does this property have a name? Has it been studied? Are there Ore domains without it? Are there […]

If $\mathcal{C}$ is a category with finite coproducts (including an initial object $0$), then the set of isomorphism-classes of $\mathcal{C}$ becomes a commutative monoid with $0 := [0]$ and $[x] + [y] := [x \oplus y]$. Is there a category $\mathcal{C}$ with finite coproducts such that the associated commutative monoid of isomorphism-classes is isomorphic to […]

I’m only in high school, so excuse my lack of familiarity with most of these terms! A monoid is defined as “an algebraic structure with a single associative binary operation and identity element.” A binary operation, to my understanding, is something like addition, subtraction, multiplication, division i.e. it involves 2 members of a set, a […]

This may be a trivial question. Every group is isomorphic to its opposite using the isomorphism that sends $x$ to $x^{-1}$. Now does this hold even if the condition of existence of inverses is dropped. More precisely: Is every monoid isomorphic to its opposite ? I am expecting the existence of a counterexample, but I […]

(The set of natural numbers $\mathbb{N}$ starts at $0$ for me.) Let $X$ denote a set, and define $X_\bot = X \uplus \{\bot\}.$ Let $\mathbf{else}$ denote the binary operation on $X_\bot$ defined as follows. If $x \in X$, then $x \mathop{\mathbf{else}} y = x$. If $x = \bot$, then $x \mathop{\mathbf{else}} y = y$. It […]

I’ve been trying to figure out what having a monad in a monoid (i.e. a category with one object) would mean. As far as I can tell it would be a homomorphism (functor) $T : M → M$, with two elements (natural transformation components) $\eta, \mu : M$, such that $\forall x. \eta x = […]

Let $A$ be the following matrix:$$A=\dfrac{1}{2}\ \left( \begin{array}{cccccccccc} -1 & -1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & -1 & 0 & 0 […]

As I am not a native English speaker, I sometimes am bothered a little with the word “monoid“, which is by definition a semigroup with identity. But why this terminology? I searched some dictionaries (Longman for English, Larousse for Francais, Langenscheidts for Dentsch) but didn’t find any result, and it seems to me that it […]

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