Generalized matrix rings and generalization of incidence algebras

D. T. Tapkin

Čebyševskij sbornik, Tome 16 (2015) no. 3, pp. 422-449.

Résumé

The paper presents generalization of incidence algebras, which includes the case of generalized matrix rings. Constructions similar to partial ordering and quasi-ordering were introduced — $\eta$-poset and $\eta$-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 case of generalized incidence algebra over a field was examined explicitly. In this case we can introduce equivalence relation on the underlying set, under which generalized incidence algebra would have block structure. As with incidence algebras, there is close connection between algebras over $\eta$-posets and $\eta$-qosets. For example, if we know sizes of equivalence classes, then we can reconstruct algebra over $\eta$-qoset by corresponding algebra over $\eta$-poset. It was shown that generalized incidence algebras can be viewed as subalgebras of some formal matrix rings of the same size as the underlying set. The problem of isomorphism was studied and it was shown that it can be reduced to the problem of isomorphism of generalized incidence algebras over $\eta$-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 to be 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. Bibliography: 16 titles.

Détail

Keywords: incidence algebras, generalized matrix rings, formal matrix rings, Mobius function.

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