GENERALIZED MATRIX RINGS AND GENERALIZATION OF INCIDENCE ALGEBRAS
The paper presents generalization of incidence algebras, whichincludes the case of generalized matrix rings. Constructions similar to partial ordering and quasi-ordering were introduced -γ-poset and γ-qoset respectively. The question of invertibility of elements of generalized incidence algebras was studied. The algorithm of ﬁnding inverse element and clear formula were found. This formula holds for incidence algebras, in particular. The caseof generalized incidence algebraovera ﬁeldwas examined explicitly. In this case we can introduce equivalence relation on the underlying set, under whichgeneralized incidence algebra would have blockstructure. As with incidence algebras, there is close connection between algebras over γ-posets and γ-qosets.Forexample,ifweknowsizesofequivalence classes,thenwecan reconstruct algebra over γ-qosetbycorresponding algebraover γ-poset. Itwas shown that generalized incidence algebras canbe viewed as subalgebras of some formal matrix rings of the same size as the underlying set. The problemof isomorphismwas studied anditwas shown thatit canbe reduced to the problem of isomorphism of generalized incidence algebras over γ-posets. Partial solution to this problem was found. The paper introduces Mobius function of generalized incidence algebra. Analogue of Mobius inversion formula was found and it was shown that basic properties of classical Mobius function are remain tobe true. Generalized incidence algebras with so-called {0, 1}-multiplicative system are of peculiar interest. There is good reason to believe that all generalized incidence algebras over a ﬁeld are isomorphic to algebras with {0, 1}-multiplicative system.