A semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup with an identity element is called monoid. This tag is most frequently used for questions related to the concept of semigroups in the context of abstract algebra. Please use the more ...

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282 views

How much do idempotent ultrafilters generate in terms of semigroups?

It is known that the set of ultrafilters on, say, the natural numbers $\mathbb{N}$, can naturally be endowed with the structure of a compact topological left semigroup (which fails to be anything ...
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86 views

What can we learn about a magma by studying these monoids?

Given a magma $(X,*)$, we get three monoids in the following way. First, define a pair of functions $L,R : X \rightarrow (X \rightarrow X).$ $$(Lx)(y) = x*y,\quad (Rx)(y) = y*x$$ Then each of the ...
6
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141 views

Is there an upper bound to the number of rings that can be obtained from a semigroup with zero by defining an additive operation?

Let $\mathscr S$ be the class of all semigroups with zero. For $(S,\times,0)\in\mathscr S,$ I want to count additive operations $+$ on $S$ such that $(S,+,\times,0)$ is a ring (possibly without unity)....
5
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292 views

Inverse of the Grothendieck group construction?

Is there an "inverse" of the Grothendieck group construction that would generate a (somehow "simplest") Abelian semigroup given some (suitably qualified, if necessary) Abelian group? (I realize that ...
4
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60 views

Seeking more information regarding the function $\varphi(n) = \sum_{i=1}^n \left[\binom{n}{i} \prod_{j=1}^i(i-j+1)^{2^j}\right].$

Define a function $\varphi : \mathbb{N} \rightarrow \mathbb{N}$ as follows. $$\varphi(n) = \sum_{i=1}^n \left[\binom{n}{i} \prod_{j=1}^i(i-j+1)^{2^j}\right]$$ The motivation is that according to ...
4
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163 views

semidirect product of semigroups

Let $S,T$ be two semigroups. A function $f:S\to T$ is called multiplicative if for any $x,y\in S$ we have $f(xy)=f(x)f(y)$. If $T=S$ then $f:S\to S$ is called automorphism on $S$. A function $f:S\to\...
4
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67 views

Can we say anything about the structure of the semigroup of non-coprime pairs after this?

Let $S = \{(a,b) : \ a, b \in \Bbb{Z} \wedge \gcd(a,b) \neq 1 \}$. Then it forms a semigroup under componentwise multiplication and if we add an exception, that even though $\gcd(1,1) = 1$, we ...
4
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89 views

Is the universal inverse semigroup of a commutative semigroup an embedding?

The question of existence of a universal inverse semigroup of an arbitrary semigroup has been answered before (this is a construction similar to the Grothendieck group). Let's refer to the universal ...
3
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110 views

Monoids, Semigroups, and a Reflective Subcategory.

The following is reflective in the category $\mathbf{Sem}$ of semigroups and their homomorphisms: we add a new neutral element to each semigroup, even monoids; then the reflections are identity ...
3
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33 views

Proof of commutativity for topological I-semigroups

A topological semigroup on a closed interval I and order topology is called I-semigroup if 1 acts as an identity and 0 as an annihilator. I've seen in several articles that such I-semigroups should ...
3
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166 views

Unitary operators - convergence problem

Let $\mathcal{U}:=\left\{ U(t) \colon t \geq 0\right\}$ be a family of unitary operators on a Hilbert space $\mathcal{H}$ where $U(0)=I$. Assume that $\left| \left<\left( \frac{U(t)-I}{t} - A \...
3
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155 views

Topology of maximal ideal space

We know that strictly positive integers with multiplication form a semigroup $S$. Let $\mathscr A=\ell ^1(S)$ with convolution. What is $\mathscr A$'s maximal ideal space? It seems enough to find ...
3
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70 views

A set of trajectories as a linear subspace of Hilbert space

Let $\left\{S(t)\right\}_{0 \leqslant t \leqslant \theta}$ be a strongly continuous semigroup of linear continuous operators in Hilbert space $H$, $S(0) = I$. Let $x$ be some element of $H$. Then its ...
3
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194 views

Constructing a semigroup from a small category

The following was given as an example for a semigroup without an identity: Finite sets of matrices of varying dimensions, where the product $A*B=\{PQ \mid P \in A, Q \in B \text{ and } dim(Q)=...
2
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92 views

Have semigroups with actions on themselves that have a dual to the compatibility axiom ever been studied?

[Update: I've now asked the same question on mathoverflow.] For a semigroup $G$ with a left action on itself, the axiom for compatibility becomes: $$ \forall f,g,h\in G:hg(f)=h(g(f)) $$ Now suppose ...
2
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18 views

Is the unit ball of $H^\infty(\mathbb{D})$ a metrizable topological semigroup under multiplication?

The space $H^\infty(\mathbb{D})$ of all bounded holomorphic functions on the open unit disc carries many different topologies. One such topology is given by uniform convergence on compact subsets; ...
2
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30 views

Doubt about semigroups in this article (anyone can help).

I need help in this article. My doubt is very arithmetical and I think follows directly from the definitions. So I think anyone could help me. The author defines what is a semigroup, gaps and non-...
2
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63 views

Shift operators and c0 semigroups

I am asked where the bounded continuous functions on $\mathbb{R}$ (with sup norm) with the right shift operators ( $T_t(f)(x)=f(x-t)$ ) form a c0 semigroup. I believe the answer is no, but I am ...
2
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40 views

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 ...
2
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31 views

How to calculate when a specific semigroup is closed under conjugation by this group.

Let $\mathcal{T}_n$ be the set of $n \times n$ matrices such that $X\in \mathcal{T}_n$ can be written as: $$X = \bigl( \begin{array}{cc} 1 & \overline{0} \\ T & \underline{0} \end{array} \...
2
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306 views

Partial differential equations and semigroups: explanation of an example.

The semigroup theory (as presented in Pazy's book) give us theorems that ensures existence of solutions for the abstract cauchy problem $$\left\{\begin{align*} \frac{d}{dt}u(t)=A[u(t)];&&t\in[...
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52 views

Generator of a transport semigroup on the torus

Consider on the Banach space $C( \mathbb T;\mathbb R)$ of continuous functions on the 1-dimensional torus $\mathbb T:=\mathbb R/\mathbb Z$ (equipped with uniform norm) the operator $L$ given by $$ L ...
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24 views

Limit relevant to parametrised semi-group 2

Let $s\geq 1, \epsilon >0, T>0$ and $f \in \mathcal{C}([0,T], H^s(\mathbb{T}))$. Define the function $$g(t,x):= \int_0^t(Id-\exp\left(-i\tau\epsilon \Delta)\right)f(\tau,x)d\tau.$$ I want to ...
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15 views

The Krull-Schmidt-Remak Theorem for Semigroups and Monoids

For finite groups, the Krull-Schmidt-Remak-Theorem holds, i.e. if $$ H_1 \times H_2 \times \ldots \times H_k \cong G_1 \times G_2 \times \ldots \times G_l $$ where the $H_i, G_i$ could not be further ...
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0answers
11 views

Ring like structure with non-associative “addition”

Is there any formally defined algebraic structure which consists of a set and two binary operations, where one is a commutative magma ("addition"), and the other is a semigroup ("multiplication"), (...
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20 views

Looking for concrete examples of semigroups, with explicit computations done to help make the theory more understandable.

I would appreciate if someone can help me find a resource which has a bunch of worked out examples showing how to find, for example, the smallest congruence containing a relation on the semi groups, ...
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16 views

Realizing $\mathbf{C}$ as 1-dimensional normal toric variety

Take $N = \mathbf{Z}$, $N_\mathbf{R} = \mathbf{R}$. Then I know that the only cones are the intervals $\sigma_+ = [0,\infty)$, $\sigma_- =(-\infty,0]$, and $\tau = \{0\}$. Consider the fans $\{\...
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51 views

Semigroup of matrices and expander Cayley graphs

I am interested in proving or disproving that certain Cayley graphs are expander. Let $S$ be the multiplicative semigroup of matrices generated by $A = \left( \begin{array}{cc} a & b \\ 0 & ...
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34 views

How do the axioms of naturally ordered semigroup imply commutativity?

Semigroup $(S,*)$ is naturally ordered (natural ordering is supposed to behave like natural numbers under addition) iff: $1$. ${\forall}I\,{\subseteq}\,S:{\exists}m\,{\in}\,I:{\forall}n\,{\in}\,I:m\,{...
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41 views

Does subsets of a semigroup with strictly smaller cardinality insure infinitely many disjoint translation copies of the set?

Let $S$ be a semigroup with infinite cardinality, $A\subset S$ with $|A|<|S|$. Under what condition we may find a infinite net $\{s_\alpha; \alpha\in \Gamma\}$, such that $s_\alpha A \cap s_\beta A ...
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45 views

Membership in Semigroup Generated by a Set of Matrices

I have a mathematical problem which can be expressed in the following form. I am given a small number (less than 10) of typically 50x50 matrices. Each column of each of these matrices has one 1 and ...
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60 views

What is the right (a good) definition for a dual monoid?

Suppose we have the free abelian monoid $S = \{a^m : m \in \mathbb{N}_0\}$ on the set of one element $X = \{a\}$. The binary operation on the monoid is denoted by $\cdot$. If $(T,\ast)$ is another ...
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26 views

Showing that an operator is bijective

Assume that $ A $ generates a contraction semigroup on a Hilbert space $ X $, and B is a bounded linear operator on $ X $. I want to show that $ A + B - 2|| B ||I $ with the domain equal to the domain ...
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20 views

“Universal solution” to factorization of arithmetic mean through a semigroup homomorphism.

Headnote A: my apologies for the long winded exposition; I couldn't find a more compact way to convey what I'm looking for, why I'm looking for it, and what have I tried myself. Suggestions for ...
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59 views

Schrödinger Semigroup is compact if potential goes to infinity

Several papers (e.g. this one: arXiv:0810.3275v1 [math.SP] 17 Oct 2008 ) claim that if $H=-\Delta+V$ and $V(x)\to\infty$ if $|x|\to\infty$, then the semigroup $e^{-tH}$ is eventually compact. Does ...
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76 views

Krull dimension and Hilbert polynomial of graded rings

Let $P \subseteq \mathbb{R}^n$ be a polytope with integral vertices, and $Q := \mathbb{R}_+ (P \times \{1\}) \cap \mathbb{Z}^{n+1}$ be the monoid of all lattice points of the cone generated by $P$ in $...
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0answers
53 views

Reference request on numerical semigroups

I watched some talks about numeric semigroups, and thei relation whti algebraic geometry (such as Weierstrass semigroup of a curve), and I'm interested in take a deeper look in this topic, can anyone ...
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105 views

How many associative ternary operations there are on a finite set?

We know that algebraic operation is a function $f:\underbrace{\left ( X\times X\times \cdots \times X\right )}_{t\ \text{times}}\rightarrow{X}$ If $X$ is a set and and cardinality is $|X|=n$ then ...
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0answers
96 views

Is a core for the generator of a Feller semi-group invariant under the resolvent?

Let $\{T_t:t\geq 0\}$ be a Feller semi-group acting on $C_0(\mathbb{R})$ with generator $(A,\mathcal{D}_A)$. We know a subspace $D\subset \mathcal{D}_A$ is a core for $A$ if $(\lambda-A)D$ is dense in ...
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46 views

Rotation semigroup and identity element.

Let $\Gamma =\{z\in\mathbb{C}:|z|=1\}$, and $X=C(\Gamma)$. The rotation semigroup $\{T(t)\}_{t\geq 0}$, is defined as $$T(t)f(z)=f(\mathrm{e}^{it}z),\quad f\in X.$$ $Z\in X$, s.t. for all $z\in\Gamma$,...
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71 views

looking for books on topological semigroup:

I'm looking for several books on topological semigroup: Topological semigroups: history, theory, applications. Karl Heinrich Hofmann Mathematics Research Library, Tulane University. The ...
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109 views

fraction power of operators in $C_0$ semigroup.

Let $X$ be Banach space, and $\{Z(t)\}_{t\geq 0}\subseteq B(X)$ be the $C_0$-Semigroup of operators defined on $X$. Moreover, let $A$ be the infinitesimal generator of $\{Z(t)\}_{t\geq 0}$. A ...
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20 views

Ergodicity and the generator.

For a markov process $X_t$ adapted to ${\cal F}_t$, let $A$ satisfy $$\lim_{h\longrightarrow 0}[E\{f(X_{t+h})|{\cal F}_t\}-f(X_t)]/h = Af(X_t).$$ What are (references to?) conditions that allow one ...
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51 views

What operations is this set closed under? The set of all $\{(a,b) : (a,b) \in \Bbb{Z}^2, \gcd(a,b) \neq 1\}$.

Let $S = \{(a,b): a,b \in \Bbb{Z}, \gcd(a,b) \neq 1 \}$. Under what binary operations is $S$ closed (that we can come up with)? I came up with the following: $$ \begin{align*} (a,b)(c,d) = \\ &...
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40 views

Proper term for a monotonic magma

Let $M$ be the multiplication table for a finite magma. Entries are labeled $0,\ldots,n-1$. M has the property that $M(i,j)\ge i, M(i,j) \ge j$. What is the proper term for this kind of magma and ...
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54 views

A combinatoric theorem about semigroups

I am struggling with the following theorem about semigroups, so I was hoping someone could give me a hand. The theorem states: "Let $S$ be an arbitrary semigroup such that for every $a\in{}S$ it ...
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251 views

Strong continuity in time vs uniform continuity in time

I have a problem with understanding the definition of strong continuity and uniform continuity for the families of operators, e.g. semigroups. Let $(X_t)_{t \geq 0}$ be a family of bounded linear ...
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0answers
68 views

irreducible words in a semigroup

Let $S$ be a monoid generated by $\{x_1,x_2,x_3,x_4,x_5\}$ with the following relations: $x_i^2=0$ for all $i$, $x_ix_j=x_jx_i$ for all $|i-j| \geq 2$ and $x_ix_{i+1}x_i=x_{i+1}x_ix_{i+1}$ for all $i$...
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0answers
64 views

Trivial question about semigroups

I have a question about analytic semigroups. I know that if $A: D(A)\subset X\to X$ is a sectorial operator on $X$, it generates an analytic semigroup $e^{tA}$. One of the properties of this semigroup ...
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0answers
52 views

Weak limits and structure of a generated semigroup

I am getting acquainted with the beautiful theorem known as Jacobs–de Leeuw–Glicksberg decomposition. A special case of this theorem is the following: Theorem. (Jacobs–Glicksberg–de Leeuw ...