Timeline for Monoids (or semigroups) with a "finite decomposition" property
Current License: CC BY-SA 3.0
20 events
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| May 2, 2017 at 12:03 | comment | added | Salvo Tringali | (...) idempotent. Not to mention that the OP is seeking information about semigroups $H$ with the property that, for every $x \in H$, there exists $n_x \in \mathbf N^+$ s.t., if $m \ge n_x$ and $x=x_1\cdots x_m$, then $x_i^2=x_i$ for some $i \in [\![1,m]\!]$: He writes, "I would like to know whether such a condition (or a condition close to it) has been encountered before, and if so, when." So, the only question should be: Do reduced BF-monoids satisfy, among many others, the condition stated in the OP? | |
| May 2, 2017 at 9:53 | comment | added | Salvo Tringali | @MarkSapir From my previous comment, "Every monoid with the property of the OP must be reduced, so units are not at all an issue." To wit, it's clear to me that there can't be non-trivial units, so "units are not at all an issue", in that we may assume since the outset that, for the purposes of this thread, monoids are reduced. As for the rest, I'll give up: You continue to repeat the refrain that the OP is (only) interested in monoids with many idempotents, and to ignore that (i) 2 out of 3 of the OP's examples are reduced BF-monoids and (ii) some of his non-examples have no non-trivial (...) | |
| May 1, 2017 at 9:07 | comment | added | Aleš Bizjak | @MarkSapir The context in which these monoids would be useful is models of separation-like logics. There monoid elements represent "resources" and idempotent elements (called duplicable in the separation logic literature) are those resources which can be shared. Thus a lot of monoids which are used have very many idempotents, but typically have no units. In general the monoids are not cancellative either (e.g., an often used one is natural numbers under maximum). | |
| May 1, 2017 at 8:26 | comment | added | user6976 | If $a^{-1}$ exists, $a\ne 1$, then $a=aa^{-1}aa^{-1}...a$, and none of the factors is an idempotent. Non-trivial units cannot exist. If there are no idempotents, then the monoid simply is a BF monoid without non-trivial units. But that is not what the OP wants apparently. | |
| May 1, 2017 at 3:48 | comment | added | Salvo Tringali | @MarkSapir Every monoid with the property of the OP must be reduced, so units are not at all an issue. On the other hand, I'm not so sure about your statement that the OP "likes many idempotents": 2 out of 3 of the examples he mentions (namely, free monoids and the natural numbers under multiplication) are reduced BF-monoids. So, I sympathize with you that the OP would be better to clarify his requests, but this doesn't change that reduced BF-monoids are a legitimate answer to the question (in fact, they supply a non-trivial and wide class of interesting examples), in the form it stands. | |
| Apr 30, 2017 at 23:52 | comment | added | user6976 | The OP likes many idempotents and does not like units at all. I do not know the reason (it would be interesting to know) but it seems clear from the question. A typical example: a Rees quotient of the direct product of a reduced BF-monoid with trivial group of units and a semilattice. | |
| Apr 30, 2017 at 20:23 | comment | added | Salvo Tringali | @MarkSapir I maintain it does. But I could have given more details: I did it now, you may want to give it a second try. In short, assume $H$ is a reduced BF-monoid: The basic point of this answer is that, for every fixed element $x \in H$, there exists $n_x \in \mathbf N^+$ s.t., if $m \ge n_x$ and $x=x_1\cdots x_m$ for some $x_1,\ldots,x_m \in H$, then at least $m-n_x$ of the factors $x_i$ must be equal to the identity, and the identity is an idempotent, so... | |
| Apr 30, 2017 at 19:06 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
fixed a mistake (I had overlooked one of the examples in the OP)
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| Apr 30, 2017 at 17:56 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
fixed some typos
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| Apr 30, 2017 at 17:48 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
added some further explainations
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| Apr 30, 2017 at 17:19 | history | undeleted | Salvo Tringali | ||
| Apr 30, 2017 at 17:10 | history | deleted | Salvo Tringali | via Vote | |
| Apr 30, 2017 at 16:39 | comment | added | user6976 | This does not seem to answer the question. The OP likes idempotents and does not like units. | |
| Apr 30, 2017 at 15:38 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
added 2 characters in body
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| Apr 30, 2017 at 15:32 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
added another example
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| Apr 30, 2017 at 15:25 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
added some examples of BF-monoids
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| Apr 30, 2017 at 15:04 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
added 2 characters in body
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| Apr 30, 2017 at 14:59 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
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| Apr 30, 2017 at 14:52 | history | edited | Salvo Tringali | CC BY-SA 3.0 |
added 2 characters in body
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| Apr 30, 2017 at 14:47 | history | answered | Salvo Tringali | CC BY-SA 3.0 |