Articles of semigroups

Distinguishing powers of finite functions

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 $f : F_n \to F_n$ […]

Do the idempotents in an inverse semigroup commute?

I have been looking at this for hours now. Why is it true that idempotents of an inverse semigroup commute? It seems like this should be straightforward but I just can’t get it. Any help is greatly appreciated.

Name for Cayley graph of a semigroups?

I did Google search and can’t find a good answer. I thought I should ask experts here. Cayley graph is defined for groups. My question is: Is there a special name for the Cayley graph of semigroups?

Show that $S$ is a group if and only if $aS=S=Sa$.

Let $S$ be a semigroup. Show that $S$ is a group if and only if $aS=S=Sa$ for all $a\in S$. Since it is if and only statement, we have to show that if $S$ is a group then $ aS=S=Sa$, which I already know how to do. The other part: if you have $aS=S=Sa$, then […]

The compactness of the unit sphere in finite dimensional normed vector space

We define $ (\mathbb{R}^m, \|.\|)$ to be a finite dimensional normed vector space with $ \|.\|$ is defined to be any norm in $ \mathbb{R^m}$. Let $S = \lbrace x \in \mathbb{R}^m: \| x\| = 1 \rbrace.$ Prove that $S$ is compact in $ (\mathbb{R}^m, \|.\|).$

Evans' theorem of embeddings into 2-generator semigroup.

I recently found this theorem when I am studying Semigroup presentations. Theorem: If S is a semigroup generated by denumerably many elements, then S can be embedded into a semigroup generated by two elements. As a hint for constructing the proof a theorem was given: Theorem: Let $\alpha_{1},\alpha_{2}…:\mathbb N_{+}\to \mathbb N_{+}$ be any transformations. There […]

Can a smooth function on the reals form a non-commutative semigroup?

Let $f\colon \mathbb{R}^2 \to\mathbb{R} $ be a smooth function. Can there exist an algebraic structure $(\mathbb{R}, \cdot)$ such that for $x,y \in \mathbb{R}$, $x \cdot y = f(x,y)$ that is a non-commutative semigroup that is strictly not a monoid or a group? I can’t think of an example, but it seems so unlikely that you […]

Does every continuous time minimal Markov chain have the Feller property?

Consider a Q-matrix on a countable state space. (A Q-matrix is a matrix whose rows sum up to $0$, with nonpositive finite diagonal entries and nonnegative off-diagonal entries.) As explained for example in the book of Norris on Markov chains, every Q-matrix (without any further assumptions) defines a transition function (given by the minimal nonnegative […]

The number of esquares of idempotents in the rank 2 $\mathcal{D}$-class of $M_n(\mathbb{Z}_2)$.

Here’s a variation of a question I was given during a research internship. Some Definitions: Definition 1: Let $S$ be a semigroup. For any $a, b\in S$, define Green’s $\mathcal{L}$-relation by $a\mathcal{L}b$ if and only if $S^1a=S^1b$ and define Green’s $\mathcal{R}$-relation by $a\mathcal{R}b$ if and only if $aS^1=bS^1$, where $S^1$ is $S$ with a one […]

Every right principal ideal non-emptily intersects the center — what is that?

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 […]