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Questions tagged [finite-semigroups]

For questions about finite semigroups, finite sets equipped with an associative binary operation.

39 questions
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Is there an idempotent element in a finite semigroup?

Let $(G,\cdot)$ be a non-empty finite semigroup. Is there any $a\in G$ such that: $$a^2=a$$ It seems to be true in view of theorem 2.2.1 page 97 of this book (I'm not sure). But is there an elementary proof? Theorem 2.2.1. [R. Ellis] Let $S$ be a…
user59671
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Inclusion relations between equationally defined classes of finite semigroups

Let $S, T$ be two semigroups. In the following all semigroups are supposed to be finite. We write $S \prec T$ if there exists a surjective semigroup morphism from a subsemigroup of $T$ onto $S$. A class of finite semigroups $\mathcal V$ is defined…
StefanH
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Implicit operations in finite semigroups.

what are some examples of implicit operations in finite semigroups other than expressions involving $\omega$? Like $x^\omega y^\omega$ or $x^{\omega+1}$. By Reiterman's theorem, pseudovarieties of finite semigroups are given by a set of…
liczman
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Period of semigroup

Let $S$ be a finite semigroup of order $n$. Suppose that $S$ has index $m$ and period $r$, i.e. $S$ satisfies the identity $x^{m+r} = x^m$. Then it is quite easy to show that $m \leq n$. My question is, how are $r$ and $n$ related? More…
E W H Lee
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If $G$ is a finite semi-group and $\forall x,y,z \in G: xy=yz \implies x=z$ then $G$ is an Abelian group

If $G$ is a finite semi-group and $\forall x,y,z \in G: xy=yz \implies x=z$ then $G$ is an Abelian group. I have no idea where to start. I'm stuck! I can't prove even the existence of the identity element :|
user66733
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Show that $H_x$ is a group for all $x.$

$H_{x}$ denotes the class of $x$ for the Green relation $\mathcal{H}$. Let $S$ be a finite semigroup where all elements can be written as a product of idempotents, that is, $x=e_1 e_2\dots e_n,$ for idempotents $e_1,e_2,\dots, e_n \in S$ for any…
King Ghidorah
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A finite semigroup has only trivial subgroups iff $\mathcal{H}$ is the identity relation.

I am struggling with the rightward implication. Here is some of my working: $(\Leftarrow)$ Every subgroup is contained within a maximal subgroup that is the $\mathcal{H}$ class of that subgroup's idempotent. As each $\mathcal{H}$ class is trivial by…
Levent Michael Dasar
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Is any finite semigroup of this type a left monoid?

Let $(S, \cdot, e)$ be a semigroup $(S, \cdot)$ with binary operation $e$ in which the identities $e(x, y)\cdot x\approx x$ and $e(x, y)\approx e(y, x)$ hold. In this question I asked if any such semigroup is necessarily a left monoid. Example given…
Jakobian
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How many non-isomorphic semigroups are there of orders $2$ and $3$?

For order $2$, I have found 5. There are 16 maps from $\{a, b\} \times \{a, b\} \to\{a, b\}$. They form $10$ equivalence classes of non-isomorphic binary operations, $5$ of which are associative. Of these $2$ are monoids and $1$ is a group.They are…
Stephen Meskin
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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 decomposed, then $k = l$ and there exists a…
StefanH
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Subsemigroup generated by an element contains unique idempotent

Possible Duplicate: A cyclic subsemigroup of a semigroup S that is a group My homework: An element $s^{i+k}$ on the cycle is idempotent iff $$ s^{i+k} = s^{2i+2k} ,$$ or equivalently $$ i+k = 2i+2k \pmod p .$$ I'm stuck here (this is my…
Tegiri Nenashi
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When do $2\times2$ matrices generate a finite semigroup?

Let $A_i$, $i=1,\ldots, k$, be $2\times2$ real-valued matrices with determinant 1 or -1. Under what circumstances is the semigroup generated by these matrices finite? I can see that this will be the case if there exists $P \in GL(2,\mathbb{R})$ such…
Tom Rush
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Let $(S,*)$ be a finite semigroup with identity. Prove that $S$ is a group iff $S$ has only one element $x$ such that $x^2=x$.

Let $(S,*)$ be a finite semigroup with identity. Prove that $S$ is a group iff $S$ has only one element $x$ such that $x^2=x$. Attempt: Does this approach true? $(\Rightarrow)$ Let $S$ be a group. Denote $e_S$ be an identity element in $S$. Clearly,…
lap lapan
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Is the inverse image of a group also a group for semigroup homomorphisms

If $\varphi : S \to T$ is a surjective semigroup homomorphism between semigroups and $G \subseteq T$ is a group, then is $\varphi^{-1}(G)$ also a group? I know that this result holds if $S$ and $T$ are finite, as then I can find an idempotent…
StefanH
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A question on Finite Semigroup

Why is it so, that a finite semigroup, say $(S, \circ)$ has $a^m=a^n$ for positive integers $m$ and $n$ with $m>n$ for $a\in S$? Does it imply some sort of periodicity in the binary composition?
Subhasis Biswas
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