On incidence algebras and directed graphs
The incidence algebra I(X,ℝ) of a locally finite poset (X,≤) has been defined and studied by Spiegel and O'Donnell (1997). A poset (V,≤) has a directed graph (Gv,≤) representing it. Conversely, any directed graph G without any cycle, multiple edges, and loops is represented by a partially order...
Saved in:
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2002-01-01
|
| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171202007925 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1832564162587262976 |
|---|---|
| author | Ancykutty Joseph |
| author_facet | Ancykutty Joseph |
| author_sort | Ancykutty Joseph |
| collection | DOAJ |
| description | The incidence algebra I(X,ℝ) of a locally finite poset (X,≤) has been defined and studied by Spiegel and O'Donnell (1997). A poset (V,≤) has a directed graph (Gv,≤) representing it. Conversely, any directed graph G without any cycle, multiple edges, and loops is represented by a partially ordered set VG. So in this paper, we define an incidence
algebra I(G,ℤ) for (G,≤) over ℤ, the ring of integers, by I(G,ℤ)={fi,fi*:V×V→ℤ} where fi(u,v) denotes the number of directed paths of length i from u to v and fi*(u,v)=−fi(u,v). When G is finite of order n, I(G,ℤ) is isomorphic to a subring of Mn(ℤ). Principal ideals Iv of (V,≤) induce the subdigraphs 〈Iv〉 which are the principal ideals ℐv of (Gv,≤). They generate the ideals I(ℐv,ℤ) of I(G,ℤ). These results are extended to the incidence algebra of the digraph representing a locally finite weak poset both bounded and unbounded. |
| format | Article |
| id | doaj-art-f463f475b9d84cf289083a76d75d2f06 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2002-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-f463f475b9d84cf289083a76d75d2f062025-02-03T01:11:40ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252002-01-0131530130510.1155/S0161171202007925On incidence algebras and directed graphsAncykutty Joseph0Department of Mathematics, St. Dominic's College, Kanjirapally 686512, Kerala, IndiaThe incidence algebra I(X,ℝ) of a locally finite poset (X,≤) has been defined and studied by Spiegel and O'Donnell (1997). A poset (V,≤) has a directed graph (Gv,≤) representing it. Conversely, any directed graph G without any cycle, multiple edges, and loops is represented by a partially ordered set VG. So in this paper, we define an incidence algebra I(G,ℤ) for (G,≤) over ℤ, the ring of integers, by I(G,ℤ)={fi,fi*:V×V→ℤ} where fi(u,v) denotes the number of directed paths of length i from u to v and fi*(u,v)=−fi(u,v). When G is finite of order n, I(G,ℤ) is isomorphic to a subring of Mn(ℤ). Principal ideals Iv of (V,≤) induce the subdigraphs 〈Iv〉 which are the principal ideals ℐv of (Gv,≤). They generate the ideals I(ℐv,ℤ) of I(G,ℤ). These results are extended to the incidence algebra of the digraph representing a locally finite weak poset both bounded and unbounded.http://dx.doi.org/10.1155/S0161171202007925 |
| spellingShingle | Ancykutty Joseph On incidence algebras and directed graphs International Journal of Mathematics and Mathematical Sciences |
| title | On incidence algebras and directed graphs |
| title_full | On incidence algebras and directed graphs |
| title_fullStr | On incidence algebras and directed graphs |
| title_full_unstemmed | On incidence algebras and directed graphs |
| title_short | On incidence algebras and directed graphs |
| title_sort | on incidence algebras and directed graphs |
| url | http://dx.doi.org/10.1155/S0161171202007925 |
| work_keys_str_mv | AT ancykuttyjoseph onincidencealgebrasanddirectedgraphs |