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Finitary incidence algebras. (English) Zbl 1180.16021
Let \(K\) be a field, \(P\) a poset and for \(x\leq y\in P\), \([x,y]=\{z\in P\mid x\leq z\leq y\}\). The authors call \(I(P)=\{\sum_{x\leq y}\alpha(x,y)[x,y]\mid\alpha(x,y)\in K\}\) the incidence space of \(P\) over \(K\). A series in \(I(P)\) is a finitary series if for any \(x\leq y\), there are only a finite number of intervals \([u,v]\subseteq[x,y]\) with \(u\neq v\) and \(\alpha(u,v)\neq 0\).
The authors show that the collection of finitary series, \(FI(P)\), under componentwise addition and convolution, is an algebra (which is a generalization of the usual incidence algebra of a locally finite partially ordered set), and then show several properties of this algebra. In particular, the isomorphism question holds for these finitary incidence algebras.

16S60 Associative rings of functions, subdirect products, sheaves of rings
06A07 Combinatorics of partially ordered sets
16G20 Representations of quivers and partially ordered sets
16S50 Endomorphism rings; matrix rings
16S34 Group rings
Full Text: DOI arXiv
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[2] Jackobson N., Structure of Rings 37 (1956)
[3] Stanley R. P., Enumerative Combinatorics 1 (1986) · Zbl 0608.05001
[4] Voss E. R., Illinois J. Math. 24 pp 624– (1980)
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