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Exponent matrices and their quivers

Exponent matrices appeared in the study of tiled orders over discrete valuation rings. Many properties of such orders are formulated using this notion. We think that such matrices are of interest in them own right, in particular, it is convenient to write finite partially ordered sets (posets) and finite metric spaces as special exponent matrices. Note that when we defined a quiver Q(E) of a reduced exponent matrix E , E corresponds to a reduced tiled order Λ, a matrix E(1) corresponds to a Jacobson radical R of Λ, and E(2) corresponds to R2. Then the adjacency matrix [Q] = E(2) − E(1) defines a structure of the Λ-bimodule V = R/R2. Note that investigations on tiled orders over discrete valuation rings and finite posets are discussed in [10]. The bibliography about tiled orders see in [2] and [3].