Editing Incidence algebra (section)

===Related concepts===
An incidence algebra is analogous to a [[group ring|group algebra]]; indeed, both the group algebra and the incidence algebra are special cases of a [[category algebra]], defined analogously; [[group (mathematics)|groups]] and [[partially ordered set|posets]] being special kinds of [[category (mathematics)|categories]].

==== Upper-triangular matrices ====
Consider the case of a partial order ≤ over any {{mvar|n}}-element set {{mvar|S}}. We enumerate {{mvar|S}} as {{math|''s''<sub>1</sub>, …, ''s<sub>n</sub>''}}, and in such a way that the enumeration is compatible with the order ≤ on {{mvar|S}}, that is, {{math|''s<sub>i</sub>'' ≤ ''s<sub>j</sub>''}} implies {{math|''i'' ≤ ''j''}}, which is always possible.

Then, functions {{mvar|f}} as above, from intervals to scalars, can be thought of as [[matrix (mathematics)|matrices]] {{math|''A<sub>ij</sub>''}}, where {{math|1=''A<sub>ij</sub>'' = ''f''(''s<sub>i</sub>'', ''s<sub>j</sub>'')}} whenever {{math|''i'' ≤ ''j''}}, and {{math|1=''A<sub>ij</sub>'' = 0}} otherwise''.'' Since we arranged {{mvar|S}} in a way consistent with the usual order on the indices of the matrices, they will appear as [[upper-triangular matrix|upper-triangular matrices]] with a prescribed zero-pattern determined by the incomparable elements in {{mvar|S}} under ≤.

The incidence algebra of ≤ is then [[isomorphic]] to the algebra of upper-triangular matrices with this prescribed zero-pattern and arbitrary (including possibly zero) scalar entries everywhere else, with the operations being ordinary [[matrix addition]], scaling and [[matrix multiplication|multiplication]].<ref>{{Cite journal|last1=Kolegov|first1=N. A.|last2=Markova|first2=O. V.|date=August 2019|title=Systems of Generators of Matrix Incidence Algebras over Finite Fields|url=http://link.springer.com/10.1007/s10958-019-04396-6|journal=Journal of Mathematical Sciences|language=en|volume=240|issue=6|pages=783–798|doi=10.1007/s10958-019-04396-6|s2cid=198443199 |issn=1072-3374|url-access=subscription}}</ref>