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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.

MSC:
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
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References:
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[2] Jackobson N., Structure of Rings 37 (1956)
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[4] Voss E. R., Illinois J. Math. 24 pp 624– (1980)
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