Tiled Orders Research Articles

The cone of quasi-semimetrics and exponent matrices of tiled orders

Finite quasi semimetrics on n can be thought of as nonnegative valuations on the edges of a complete directed graph on n vertices satisfying all possible triangle inequalities. They comprise a polyhedral cone whose symmetry groups were studied for small n by Deza, Dutour and Panteleeva. We show that the symmetry and combinatorial symmetry groups are as they conjectured.Integral quasi semimetrics have a special place in the theory of tiled orders, being known as exponent matrices, and can be viewed as monoids under componentwise maximum; we provide a novel derivation of the automorphism group of that monoid. Some of these results follow from more general consideration of polyhedral cones that are closed under componentwise maximum.

Computing normalizers of tiled orders in Mn(k)

Computing normalizers of tiled orders in Mn(k)

Проективнi гратки черепичних порядкiв

Tiled orders over discrete valuation ring have been studied since the 1970s by many mathematicians, in particular, by Yategaonkar V.A., Tarsy R.B., Roggenkamp K.W, Simson D., Drozd Y.A., Zavadsky A.G. and Kirichenko V.V. Yategaonkar V.A. proved that for every n > 2, there is, up to an isomorphism, a finite number of tiled orders over a discrete valuation ring O of finite global dimension which lie in $M_n(K)$ where K is a field of fractions of a commutatively discrete valuation ring O. The articles by R.B. Tarsy, V.A. Yategaonkar, H. Fujita, W. Rump and others are devoted to the study of the global dimension of tiled orders. H. Fujita described the reduced tiled orders in Mn(D) of finite global dimension for n = 4; 5. V.M. Zhuravlev and D.V. Zhuravlev described reduced tiled orders in Mn(D) of finite global dimension for n = 6: This paper examines the necessary condition for the finiteness of the global dimension of the tile order. Let A be a tiled order. The kernel of the projective resolvent of an irreducible lattice has the form M1f1 +M2f2 + ::: +Msfs, where Mi is irreducible lattice, fi is some vector. If the tile order has a finite global dimension, then there is a projective lattice that is the intersection of projective lattices. This condition is the one explored in the paper.

The max-plus algebra of exponent matrices of tiled orders

An exponent matrix is an n×n matrix A=(aij) over N0 satisfying (1) aii=0 for all i=1,…,n and (2) aij+ajk≥aik for all pairwise distinct i,j,k∈{1,…,n}. In the present paper we study the set En of all non-negative n×n exponent matrices as an algebra with the operations ⊕ of component-wise maximum and ⊙ of component-wise addition. We provide a basis of the algebra (En,⊕,⊙,0) and give a row and a column decompositions of a matrix A∈En with respect to this basis. This structure result determines all n×n-tiled orders over a fixed discrete valuation domain. We also study automorphisms of En with respect to each of the operations ⊕ and ⊙ and prove that Aut(En,⊕,⊙,0)≅Aut(En,⊕)≅Aut(En,⊙)≅Sn×C2, n>2.

Minimal tiled orders of finite global dimension

A directed graph is associated to any basic tiled order, and it turns out that the graph is connected for all known examples of tiled orders of finite global dimension. It is proved that the minimal connected tiled orders of finite global dimension in a fixed algebra are of global dimension two, and that up to isomorphism, these minimal orders are characterized by their unoriented graph which is a tree. Their irreducible representations are in one-to-one correspondence with the possible orientations of this tree.

On exponent matrices of tiled orders

We describe two methods to determine all generators of the additive semigroup of the non-negative exponent [Formula: see text]-matrices, and illustrate them finding all generating [Formula: see text]-exponent matrices. The generating [Formula: see text]-exponent matrices are found using a computer. We consider the Hasse diagram [Formula: see text] of the partially ordered set of non-negative matrices and prove that for an arbitrary non-negative exponent matrix [Formula: see text] there exists an oriented path in [Formula: see text] starting in some matrix unit and ending in [Formula: see text] which does not pass through any other exponent matrix. We also show that for any non-negative exponent matrix [Formula: see text] there exists a chain of non-negative exponent matrices [Formula: see text] such that [Formula: see text] is a [Formula: see text]-matrix, and each [Formula: see text] is obtained from [Formula: see text] by adding a [Formula: see text]-matrix.

Gorenstein tiled orders

The aim of this article is to describe exponent matrices of Gorenstein tiled orders. The necessary and sufficient condition for possibility such to construct a Gorenstein tiled order which has given orders over some discrete valuation ring (the unique for the whole main diagonal) on the main diagonal and its permutation be a product of correspond cycles with given cyclic Gorenstein tiled order is considered.

Higher algebraic K-groups and $${\mathcal D}$$ -split sequences

In this paper, we use $\mathcal D$-split sequences and derived equivalences to provide formulas for calculation of higher algebraic $K$-groups (or mod-$p$ $K$-groups) of certain matrix subrings which cover tiled orders, rings related to chains of Glaz-Vasconcelos ideals, and some other classes of rings. In our results, we do not assume any homological requirements on rings and ideals under investigation, and therefore extend sharply many existing results of this type in the algebraic $K$-theory literature to a more general context.

Global dimension of Noetherian serial rings

The global dimension of Noetherian serial rings is studied. It is proved that if an indecomposable serial ring has infinite global dimension then it is Artinian and its quiver is a simple cycle. Using methods of the theory of right serial quivers, we give an upper estimate on the Loewy length of Artinian rings of finite global dimension. Applications to the calculation of the global dimension of tiled orders of width 2 are given.

Locally nilpotent groups of units in tiled rings

We consider locally nilpotent subgroups of units in basic tiled rings A , over local rings O which satisfy a weak commutativity condition. Tiled rings are generalizations of both tiled orders and incidence rings. If, in addition, O is Artinian then we give a complete description of the maximal locally nilpotent subgroups of the unit group of A up to conjugacy. All of them are both nilpotent and maximal Engel. This generalizes our description of such subgroups of upper-triangular matrices over O given in M. Dokuchaev, V. Kirichenko, and C. Polcino Milies (2005) [3] .

On rigid quivers

We consider quivers that appear in the theory of tiled orders, in particular, rigid quivers. We prove that a quiver having a loop at each vertex is not rigid, and the quiver associated with a finite partially ordered set having one minimal element is rigid.

An elementary exact sequence of modules with an application to tiled orders

Let $m \ge 2$ be an integer. By using $m$ submodules of a given module, we construct a certain exact sequence, which is a well known short exact sequence when $m=2$. As an application, we compute a minimal projective resolution of the Jacobson radical of

Quivers of semi-maximal rings

In this paper, the set of quivers of semi-maximal rings is investigated. It is proved that the elements of this set are formed by the elements of the set of quivers of tiled orders and that the set of quivers of tiled orders with n vertices is determined by the integer points of a convex polyhedral domain that lie in the nonnegative part of the space \(\mathbb{R}^{n^2 - n} \). It is also proved that the set of quivers of tiled orders with n vertices contains all simple, oriented, strongly connected graphs with n vertices and n loops, does not contain any graphs with n vertices and n − 1 loops, and contains only a part of the graphs with n vertices and m (m < n − 1) loops.

Frobenius Full Matrix Algebras and Gorenstein Tiled Orders

ABSTRACT For integers n ≥ 2, n × n full matrix algebras with structure systems are introduced in Fujita (2003) as a framework to study certain factor algebras of tiled orders. Such factor algebras of Gorenstein tiled orders are Frobenius full matrix algebras. In this article, we verify that for every integer n ≤ 7, the converse is true. For every integer n ≥ 8, we show that the converse is not true, that is, there are Frobenius n × n full matrix algebras having no corresponding Gorenstein tiled orders.

EXPONENT MATRICES AND TILED ORDERS OVER DISCRETE VALUATION RINGS

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.

Neat idempotents and tiled orders having large global dimension

We study neat primitive idempotents in a semiperfect Noetherian ring, and as an application, we improve an example of a tiled order having large global dimension given by Jansen and Odenthal. Moreover, another two tiled orders having large global dimension are added and two questions on tiled orders of finite global dimension are posed.

Tameness criterion for posets with zero-relations and three-partite subamalgams of tiled orders

A criterion for tame prinjective type for a class of posets with zero-relations is given in terms of the associated prinjective Tits quadratic form and a list of hypercritical posets. A consequence of this result is that if ${\mit\Lambda }^{\bullet }$ is

Automorphisms of tiled orders

Let Λ be a tiled R-order. We give a description of Aut R ( Λ) as the semidirect product of Inn( Λ) and a certain subgroup of Aut( Q( Λ)), where Q( Λ) is the link graph of Λ. Additionally, we give criteria for determining when an element of Aut( Q( Λ)) belongs to this subgroup in terms of the exponent matrix for Λ.

GORENSTEIN TILED ORDERS

Let Λ = {O, E(Λ)} be a reduced tiled Gorenstein order with Jacobson radical R and J a two-sided ideal of Λ such that Λ ⊃ R 2 ⊃ J ⊃ Rn (n ≥ 2). The quotient ring Λ/J is quasi-Frobenius (QF) if and only if there exists p ∈ R 2 such that J = pΛ = Λp. We prove that an adjacency matrix of a quiver of a cyclic Gorenstein tiled order is a multiple of a double stochastic matrix. A requirement for a Gorenstein tiled order to be a cyclic order cannot be omitted. It is proved that a Cayley table of a finite group G is an exponent matrix of a reduced Gorenstein tiled order if and only if G = Gk = (2) × ⃛ × (2). Commutative Gorenstein rings appeared at first in the paper [3]. Torsion-free modules over commutative Gorenstein domains were investigated in [1]. Noncommutative Gorenstein orders were considered in [2] and [10]. Relations between Gorenstein orders and quasi-Frobenius rings were studied in [5]. Arbitrary tiled orders were considered in [4], 11-14.

Two-point differentiation for general orders

We establish a Zavadskij-type reduction for orders Λ in a finite-dimensional algebra over a complete field K with respect to a discrete valuation. For a suitable monomorphism u of Λ-lattices, we define a derived order δ u Λ, and a functor ∂ u between Λ- and δ u Λ-lattices which yields an equivalence modulo finitely many indecomposables. The known versions of Zavadskij's differentiation algorithm (for tiled orders [15], representations of posets [14], and vector space categories [10,11]) are unified, and extended in this way to a part of representation theory of general orders.