On the Convergence of Monotone Lattice Matrices

Copyright © 2013 Jing Jiang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT Since lattice matrices are useful tools in various domains like automata theory, design of switching circuits, logic of binary relations, medical diagnosis, markov chains, computer network, traffic control and so on, the study of the properties of lattice matrices is valuable. A lattice matrix A is called monotone if A is transitive or A is monotone increasing. In this paper, the convergence of monotone matrices is studied. The results obtained here develop the corresponding ones on lattice matrices shown in the references.


Introduction
In the field of applications, lattice matrices play major role in various areas such as automata theory, design of switching circuits, logic of binary relations, medical diagnosis, markov chains, computer network, traffic control (see e.g.[1]).Since several classical lattice matrices, for example transitive matrix, monotone increasing matrix, nilpotent matrix, have special applications, many authors have studied these types of matrices.In fact, a transitive matrix can be used in clustering, information retrieval, preference, and so on (see e.g.[2,3]); a nilpotent matrix represents an acyclic graph that is used to represent consistent systems and is important in the representation of precedence relations (see e.g.[4]).Recently, the transitive closure of lattice matrix has been used to analyze the maximum road of network.In this paper, we continue to study transitive lattice matrices and monotone increasing matrices.The main results obtained in this paper develop the previous results on transitive lattice matrices [5] and monotone increasing matrices [6].

Definitions and Preliminaries
At this section, we shall give some definitions and lemmas.Let be a partially ordered set (simply denoted by poset) and is said to be distributive if the operations and " " "  are distributive with respect to each other.A matrix is called a lattice matrix if its entries belong to a distributive lattice.In this paper, the lattice

 
, , , L    is always supposed to be a distributive lattice with the least and greatest elements 0 and the powers of A are defined as follows: 1 , , the set of all positive integers.The entry of In this paper, A lattice matrix A is called monotone if A is transitive or A is monotone increasing.
For any A is said to be almost periodic if there exist positive integers k and d such that .
The least positive integers k and d are called the index and the period of A, and denoted by and

Convergence of Monotone Lattice Matrices
In this section, we shall discuss the convergence of Monotone Lattice Matrices.In [5,6], Tan studied the convergence index of transitive matrices and monotone increasing matrices.In the following, we continue to study the convergence index of these matrices which discussed by Tan [5,6], and the convergence index of these discussed matrices is smaller than previous considered index.
Since is the sum of some term in we have .
In the following, we shall prove that By the result of 2), we only need to show that for .
From above, we can get .
Since the number of indices in T is greater than , there must be two indices and such that Now delete the term in T , thus we can get a new term Since is a term of we have .But by the property of the operation On the other hand, by the hypothesis .
From above, we can get and so This completes the proof.
3) A converges to a a A with .
Thus , and so .
On the other hand, by the result in 1), we have 3) It follows from Theorem 3.2.This completes the proof.
Theorem 3.4.If A is transitive and Where .
 Now, we consider any term T of .Since the number of indices in T is greater than n, there must be two indices and such that .Then Since A is transitive, we have k A A  for all , and so is a term of , we have Then , and so .Therefore .On the other hand, since We have , then .From above, we can get , and so . 2) By the proof of 1), we have .In the following we shall prove that .
) for some and j N  From above, we have , and so 3) The proof of 3) is similar to that of 2).This completes the proof.
As a special of Theorem 3.4, we obtain the following Corollary.  .