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Questions tagged [semigroups]

A semigroup is an algebraic structure consisting of a set together with a single 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/universal algebra. Please use the more specific tag (semigroup-of-operators) whenever appropriate.

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, and semigroups with the notion of inverses are known as regular semigroups and inverse semigroups.

Semigroups are used in various areas of mathematics. $C_0$-semigroups are important in partial differential equations. Semigroups have also connections to automata theory.

Topological (and left/right topological) semigroups are also studied. Perhaps the best know result in this area is Ellis-Numakura lemma. Using Ellis-Numakura lemma, existence of idempotent ultrafilters can be shown.

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Does there exist a semigroup such that every element factorizes in this way, which nonetheless lacks a left identity?

If a semigroup $S$ has a left identity-element, then for any $y \in S$ we can write $y = xy$ for some $x \in S$. Just take $x$ to be any of the left identities, of which there is at least one, by hypothesis. Does there exist a semigroup $S$ such…
goblin GONE
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Are there published results providing characterizations of commutative non-cancellative Archimedean semigroups with no idempotents?

The first post below provides examples of commutative non-cancellative Archimedean semigroups with no idempotents. Can anyone provide a reference to a characterization theorem for commutative non-cancellative Archimedean semigroups with no…
Bruce Ebanks
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The generator of a Cauchy problem

In a Banach space $E$, we consider the Cauchy problem $$u'(t)=Au(t),\quad u(0)=u_0 \quad (t\geq0)$$ where $A:D(A)\subseteq E \to E$ denote a linear operator with domain dense. If the problem is uniformly well posed we define the solution operator…
Hidda Walid
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Numerical semigroups generated by two elements

I have a question about numerical semigroups. It is known that if $a, b\in \mathbb{N}$ and $\gcd(a, b)=1$, then the numerical semigroup $\langle a, b \rangle$ has genus $\frac{(a-1)(b-1)}{2}$. My question is: If $a, b, c \in \mathbb{N} $ are such…
E. R. A. M
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Index of an element in direct product of finite semigroups

Let $S$ and $T$ be finite semigroups and let $(x,y)\in S\times T$. What is the index of $(x,y)$ ? Is it equals $$ \max\{\mathrm{index}(x),\mathrm{index}(y)\}\ ? $$ My proof: If $\mathrm{index}(x)=i$, $\mathrm{period}(x)=p$, $\mathrm{index}(y)=j$…
boaz
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Left and Right Regular Representations are Permutations therefore Identity and Inverses Exist

Let $(S, \circ)$ be a semigroup. Let $a \in S$. It's a straightforward exercise to show that the left and right regular representations $\lambda_a$ and $\rho_a$ with respect to $a$ are permutations if $a$ is invertible. However, it is not so obvious…
Prime Mover
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Size of the image (rank) in $T_n$

For $\alpha, \beta \in T_n$ (full transformation semigroup/monoid - set of all maps from $\{1,2,\ldots, n\}$ to itself), show that $|\text{Im}(\alpha\beta)| \leqslant |\text{Im}(\alpha)|$ and $|\text{Im}(\alpha\beta)| \leqslant…
Sam
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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 $i$. I want to apply the Bergman's Diamond lemma to…
user73789
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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 is the following: there exists constant $M_0$…
Dina
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Finite semi-group generating set

Let $S$ be a finite semi-group which basically means that it is an associative binary operation. $A$ will be called as a generating set of $S$ if every element can be written as a multiplication of elements of $A$. If $S$ is a group then there…
user639830
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Semigroup isomorphism between $(\{1,2,\dots \},\times)$ and $(\{0,1,2,\dots \},+)$.

I know that the two semigroups $(\{0,1,2,\dots \},\times)$ and $(\{0,1,2,\dots \},+)$ are not isomorphic because if we want to map identity elements together then it can be see that we can't have injective function between them,but what can we say…
a.p
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Adjoining an identity to a cancellative semigroup

Does adjoining an identity to a cancellative semigroup that is not already a monoid always give a cancellative monoid? The only way that this could fail is if a product in a cancellative semigroup that is not a monoid is equal to one of the…
Geoffrey Trang
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A question about semigroup union

I ask the question about semigroup in here The semigroup of all order-preserving and decreasing transformations in full transformations semigroup $T_n$ is denoted $C_n$. I consider the idempotent set $A=\{\begin{bmatrix}2\\1 …
1Spectre1
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Understanding the definition of an ideal of a semigroup

Liapin wrote: Two-sided ideals are also all the possible unions of principal two-sided ideals. I get this as: If $I$ is an ideal of $S$ then for $a_1...a_k\in S$ we have $I= \cup(a_i)$. Also, I think $a_i\in S$ can be chosen from $D$-classes'…
Gozal
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Pseudo-Frobenius numbers acting as the maximal elements with respect to $\leq_S$ of $\mathbb{Z} \setminus S$

According to Rosales and Garcia, an integer is a pseudo-frobenius number if $x \not \in S$ and $x + s \in S$ for all $s \in S \setminus \{0 \}$ without the 0 element. Furthermore, let $a \leq_S b$ if $b - a \in S$. The book claims that the set of…
Tarang Saluja
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