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Questions tagged [greens-relations]

In abstract algebra, Green's relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate. Use it along with (semi-group) and (abstract-algebra). DO NOT confuse it with (greens-theorem) or (greens-identities).

In abstract algebra, Green's relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate. Use it along with (semi-group) and (abstract-algebra). DO NOT confuse it with (greens-theorem) or (greens-identities). See https://en.wikipedia.org/wiki/Green%27s_relations for more details.

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How does $(eMe)m_1(eMe) = (eMe)m_2(eMe) \Rightarrow Mm_1M = Mm_2M$?

I am studying the book "Representation Theory of Finite Monoids" by Benjamin Steinberg, I've tried searching online for the solution but I don't know how to search for it so google shows no results. In Lemma 1.7 the author says that is clear…
Diogo Santos
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Expressing Green's relations in regular semigroups

Let $S$ be a semigroup and $a \in S$. An element $a' \in S$ is called an inverse of $a$ if $$ aa'a = a \qquad a'aa' = a'. $$ Denote the set of all inverses by $V(a)$. A semigroup where every element has an inverse is called a regular semigroup.…
StefanH
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Example of an infinite semigroup where $\mathcal{D} \neq \mathcal{J}$

Let $S$ be a semigroup and take elements $a, b$ in $S$. Consider the following two Green's relations $a\mathcal{J}b \iff SaS = SbS$ $a\mathcal{D}b \iff \exists c \text{ such that } aS = cS \text{ and } Sb = Sc$ In any finite semigroup we have…
user86420
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semigroup in which Green's relations are different

I am beginning to study the Green relations and I would like to know if there is a semigroup in which the 5 Green relations are different. I would like to find an example where this occurs, I have tried some easy semigroups and have had no luck. …
Userxdxd
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Question on proof related to inclusion relations of principal ideals generated by elements.

Let $S$ be a semigroup, then an ideal $W$ is a subset $W \subseteq S$ such that $sWt \in W$ for all $s,t \in S^{\bullet}$, where $S^{\bullet}$ is $S$ with an additional unit element added if $S$ does not has one (hence forms a monoid). Further we…
StefanH
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Show that Green's relations coincide on a group, in particular, that $\mathcal{L}=\mathcal{R}$.

I am trying to prove that Green's relations on a group coincide, that is, $\mathcal{L}=\mathcal{R}=\mathcal{H}=\mathcal{D}=\mathcal{J}$. It can be easily deduced that $\mathcal{H},\mathcal{D},\mathcal{J}$ conicide with $\mathcal{L}$ and…
tashakinns
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$\mathcal{J}-$ trivial elements

What does it mean by saying that … idempotents are $\mathcal{J}-$ trivial. Indeed, by searching old docs and this one, we see that: A semigroup $S$ is $\mathcal{J}-$ trivial if two elements of $S$ which are $\mathcal{J}-$ equivalent are…
Mikasa
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About Green's relations

Let $(S,\cdot)$ be a semigroup. Then $x \in S$ is regular if there exists $y \in S$ such that $xyx=x$. A semigroup $S$ is regular if for all $x\in S,$ $x$ \is regular. If $S$ is a semigroup and $x,y \in S$ we have $$(x,y) \in \mathcal{L} \iff…
Natali Delgado M
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What can we say about Green's relations on a semilattice?

Note: This is a soft-question in the flavour of, say, "what does $X$ look like?" and "Is there a description of $Y$?" - so, hopefully, it is not too broad. Let's focus on the basics if necessary. The Details: Definition: A (meet/join) semilattice…
Shaun
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Green's second identity with complex functions

Let $\phi$ and $\psi$ be two smooth scalar fields: $$\phi:\mathbb{R}^2\rightarrow\mathbb{R}$$ $$\psi:\mathbb{R}^2\rightarrow\mathbb{R}$$ Consider a closed surface $S$ in $\mathbb{R}^2$ bounded by a contour $C$. Let the following be the Green's…
Marcos Henriques
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Doubt in Location Lemma in Greens Relation Abstract Algebra!

I am unable to prove one part of rectangular lemma in green's relations. Let $S^1$ be a monoid. Then I need to prove that $m.m' \in D(m) \iff m.m'\in R(m) \cap L(m')$. How should I go about proving this? $R(m)$ is Right equivalence classes of m.…
A J
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$\mathcal{D}-$class in a semigroup is a union of $\mathcal{L}$-classes

I am reviewing section 2.2 of fundamentals of semigroup theory at the beginning they say that "Each $\mathcal{D}$-class in a semigroup is a union of $\mathcal{L}$-classes and also a union of $\mathcal{R}$-class." This statement is telling us that if…
Userxdxd
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Green relations in semigroups : how to interpret $J(x) \setminus J_x$

I read in semigroup theory that given a semigroup $S^1$ (which has an identity), the $\mathcal{J}$ Green relation has an associated function $J(x)$: $$ J(x) = S^1xS^1 $$ which is the principal ideal generated by $x$ given the Green relation…
Link L
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Equal nr of $\mathscr D$-classes, different nr of idempotents

Are there examples of (finite) semigroups $S$ and $T$ such that they have the same 'number' of $\mathscr D$-classes, $S$ has idempotents and $T$ doesn't? Alternatively, they both may have idempotents, but one of them has multiple idempotents in one…
AlvinL
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What is the definition of generating the same ideals?

Taken from An Introduction to Semigroup Theory by J.M. Howie, aLb if and only if a and b are generating the same principal ideals. What does it really means to generate the same ideal? If someone has asked the same matter and has been answered,…
Beatri Gita Ananda
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