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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:
 [1] Doubilet P., Proc. of the Sixth Berkely Symp. on Math. Stat. and Probab pp 267– (1972) [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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