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In the book "Combinatorial theory" by Martin Aigner (from 1979), the standard algebra of a poset is introduced as the subalgebra of the incidence algebra of a poset consisting of the functions that are constant on isomorphic intervals.

(It was answered in the comments by Darij Grinberg that standard algebras are called reduced incidence algebras today (which answered one original question), so I removed a part of the questions.)

Questions: Are the quiver and relations of a reduced incidence algebra of a poset known? Are there some examples in some textbooks for the algebraic structure of those algebras?

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    $\begingroup$ Reduced incidence algebra, if I correctly understand your definition. $\endgroup$ – darij grinberg Jan 4 at 22:46
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    $\begingroup$ Not necessary -- the old question wasn't mathematical, while the new one is. I don't know the answer, though. $\endgroup$ – darij grinberg Jan 4 at 23:15
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    $\begingroup$ At first sight it would seem that the identity is the only idempotent here and that the ring is local. So the Quiver would be a bunch of loops. The non units are the strictly triangular matrices. I am not sure how many loops there are. The element which is 1 on all two element intervals should give a loop. I am not sure there are others. $\endgroup$ – Benjamin Steinberg Jan 4 at 23:29
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    $\begingroup$ An non zero idempotent would have to have a 1 on the diagonal. But then by your reduced condition all diagonal entries are 1. $\endgroup$ – Benjamin Steinberg Jan 4 at 23:32
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    $\begingroup$ Any basic algebra embeds in upper triangular matrices so the Quiver doesn't behave well under going to subalgebras. $\endgroup$ – Benjamin Steinberg Jan 4 at 23:33

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