1. 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].
2. Definitions and Preliminaries
At this section, we shall give some definitions and lemmas. Let 
be a partially ordered set (simply denoted by poset) and
. If 
or
then 
and 
are called comparable. Otherwise, 
and 
are called incomparable, noted by
. If for any
, 
and 
are comparable, then P is called a chain. An unordered poset is a poset in which 
for all
. A chain c in a poset P is a nonempty subset of P, which, as a subposet, is a chain. An antichain C in a poset P is a nonempty subset which, as a subposet, is unordered. A lattice is a poset in which every two elements have a unique least upper bound and a unique greatest lower bound. For any x and y in L, the least upper bound and the greatest lower bound will be denoted by 
and
, respectively. It is clear that any chain is a lattice, which is called a linear lattice. It is obvious that if 
is a linear lattice (especially, the fuzzy algebra [0,1] or the binary Boolean algebra
) then
and 
for all x and y in L. Let 
be a lattice and
. X is called a sublattice of L if for any 
and 
A lattice 
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 
is always supposed to be a distributive lattice with the least and greatest elements 0 and 1, respectively. Let 
be all 
matrices over L. For any A in
, we shall denote by 
or 
the element of L which stands in the 
entry of A. For convenience, we shall use the set N to denote the set 
For any A, B, C in
, we define:
iff 
for 
in N;
iff 
for 
in N;
iff 
for 
in N;
iff 
for 
in N and 
iff
;
where 
if 
and 
if 
for 
For any A in 
the powers of A are defined as follows: 
where Z+ denotes the set of all positive integers. The 
entry of 
is denoted by 
and
Let 
A is called transitive if 
A is called monotone increasing if 
A is called reflexive if 
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 
respectively. In particular, if 
then A is said to converges in a finite number of steps.
3. 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.
Theorem 3.1. Let 
if 
holds for all 
then 1) 
2) 
3) 
converges to 
with 
Proof. 1) Let
By the hypothesis 
it follows that
Since 
is the sum of some term in 
we have
Thus 
2) By 
we have
Then
Therefore, 
3) By
, it follows that
. Hence, 
In the following, we shall prove that 
By the result of 2), we only need to show that 
for 
Let
Since the number of indices in 
is
, there must be two indices 
and 
such that
. Then
Since 
is a term of 
we have
Thus 
then
(since 
for
).
From above, we can get 
This completes the proof.
Corollary 3.1. Let
if 
then 1) 
2) 
for all 
3) A converges to 
with 
Proof. It follows from Theorem 3.1.
Theorem 3.2. Let 
If 
and
holds for all
, then A converges to
with
Proof. Since
we have 
Then
Let 
be any term of 
Since the number of indices in T is greater than
, there must be two indices 
and 
such that 
. Then
Now delete the term 
in
, thus we can get a new term
Since 
is a term of 
we have
. But by the property of the operation
, we have
Thus 
On the other hand, by the hypothesis 
we have
From above, we can get 
Since
, we have
and so
This completes the proof.
Theorem 3.3. Let
. If for any
, 
or
, then 1)
;
2)
;
3) A converges to 
with
.
Proof. 1) Let
.
If
, then
If
, then
Thus
, and so
. Therefore
2) for any
, 
,
On the other hand, by the result 
in 1), we have
.
3) It follows from Theorem 3.2. This completes the proof.
Corollary 3.2. Let
. If for any 
and
, then 1)
;
2) A converges to 
with
.
Proof. 1) By
, we can get
Since
We have
. On the other hand, since
, we have
. Therefore
2) It follows from Theorem 3.2. This completes the proof.
Theorem 3.4. If A is transitive and
. Where
, with 
and
, then 1) 
converges to 
with
;
2) If A satisfies 
(or
) for some
, then B converges to 
with
;
3) If B satisfies 
(or
) for some
, then B converges to 
with
.
Proof. First by
, we have 
.
1) Let
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
And
Since 
is transitive, we have 
for all
, and so
. Thus
Since 
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
.
Let
Now consider any term 
of
.
a) If 
for some 
and
, then
And so
Then
b) Suppose that 
for all
. By the hypothesis, 
(or
) for some 
and
, we can get 
Thus
From above, we have
, and so
. Therefore
.
3) The proof of 3) is similar to that of 2). This completes the proof.
Theorem 3.4 is an improvement of Theorem 4.1 [6].
As a special of Theorem 3.4, we obtain the following Corollary.
Corollary 3.3. If 
is transitive, then 1) 
converges to 
with
;
2) If A satisfies 
(or
) for some
, then A converges to 
with
.
Corollary 3.3 is an improvement of Corollary 4.1 [6].
NOTES