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Are there any examples of a semigroup (which is not a group) with exactly one left(right) identity (which is not the two-sided identity)? Are there any “real-world” examples of these (semigroups of some more or less well-known mathematical objects) or they could only be “manually constructed” from abstract symbols (a, b, c…) subject to operation given by a Caley table?

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    Certainly, yes.2012-09-15
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    Can you give some examples of semigroups you would consider "real-world" examples as opposed to "manually constructed" examples?2012-09-15
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    I have already discribe “manually-constructed” examples in my question. You're free to consider all examples not match this secription as a “real-world” ones.2012-09-15
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    How can we tell whether a given manually constructed examples does not show up in any real-world context?2012-09-15
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    We can't, but untill the fact of showing up in any real-world context is found, we would consider an example as “manually constructed”.2012-09-15
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    @tomasz this surely is ad-hoc example, but thank you anyway!2012-09-15
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    @Artem: There's no need to specify in your question that the semigroup is not allowed to be a group, since a group cannot satisfy the condition you ask for.2012-09-15
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    @tomasz: If you rename $a$ to $1$ and $b$ to $0$, you'll notice that you've described the multiplication of the two-element field.2012-09-15
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    And therefore it is not an example of what Artem is asking for, since $a$ is a two-sided identity.2012-09-15
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    @celtschk: Touche. Change it to $\{a,b,c\}$ with $a\cdot x=x$, $y\cdot x=b$ if $y\neq a$. :) That's what I had thought of in the first place, but I wanted to make it minimal and I overdid it. :)2012-09-15
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    @celtschk oops, sorry, I meant the comment by tomasz2012-09-16
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    @tomasz I suggest you convert your comment to the answer. Good one. Thanks.2012-09-16

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