Narrow Results

Exponent matrices appear in the theory of tiled orders over a discrete valuation ring. Many properties of such an order and its quiver are fully determined by its exponent matrix. We prove that an arbitrary strongly connected simply laced quiver with a loop in every vertex is realized as the quiver of a reduced exponent matrix. The relations between exponent matrices and finite posets, Markov chains, and doubly stochastic matrices are discussed.

For a given associative ring B, a two-sided ideal J ⊂ B and a finite partially ordered set P, we study the ring A = I(P, B, J) of incidence modulo J matrices determined by P. The properties of A involving its radical and quiver are investigated, and the interaction of A with serial rings is explored. The category of A-modules is studied if P is linearly ordered. Applications to the general linear group over some local rings are given.

We describe two methods to determine all generators of the additive semigroup of the non-negative exponent $n\times n$-matrices, and illustrate them finding all generating $3\times 3$-exponent matrices. The generating $4\times 4$-exponent matrices are found using a computer. We consider the Hasse diagram $H$ of the partially ordered set of non-negative matrices and prove that for an arbitrary non-negative exponent matrix $A$ there exists an oriented path in $H,$ starting in some matrix unit and ending in $A,$ which does not pass through any other exponent matrix. We also show that for any non-negative exponent matrix $A$ there exists a chain of non-negative exponent matrices $A_0\le A_1\le \cdots \le A_t=A$ such that $A_0$ is a $(0,1)$-matrix, and each $A_i$ is obtained from $A_{i-1}$ by adding a $(0,1)$-matrix.