1. Introduction
Let P be a poset. For
, we say a covers b, denoted by
; if
and there doesn’t exist
such that
. If P has the minimum (resp. maximum) element, then we denote it by 0 (resp. 1) and say that P is a poset with 0 (resp. 1). Let P be a finite poset with 0. By a rank function on P, we mean a function r from P to the set of all the integers such that
and
whenever
. Observe the rank function is unique if it exists. P is said to be ranked whenever P has a rank function.
Let P be a finite ranked poset with 0 and 1. The polynomial
is called the characteristic polynomial of P, where
is the
function on P and r is the rank function of P. A poset P is said to be a lattice if both
and
exist for any two elements
.
and
are called the join and meet of a and b, respectively. Let P be a finite lattice with 0. By an atom in P, we mean an element in P covering 0. We say P is atomic if any element in
is the join of atoms. A finite atomic lattice P is said to be a geometric lattice if P admits a rank function
satisfying
,
. Notations and terminologies about posets and lattices will be adopted from books [1] [2] .
The special lattices of rough algebras were discussed in [3] . The lattices generated by orbits of subspaces under finite (singular) classical groups were discussed in [4] [5] . Wang et al. [6] -[8] constructed some sublattices of the lattices in [4] . The subspaces of a d-bounded distance-regular have similar properties to those of a vector space. Gao et al. [9] -[11] constructed some lattices and posets by subspaces in a d-bounded distance-regular graph. In this paper, we continue this research, and construct some new sublattices of the lattices in [4] , discussing their geometricity and computing their characteristic polynomials.
Let
be a finite field with q elements, where q is a prime power. For a positive integer
, let
be the n-dimensional row vector space over
. Let
. For a fixed
-dimensional subspace
of
, let
.
If we define the partial order on
by ordinary inclusion (resp. reverse inclusion), then
is a poset, denoted by
(resp.
). In the present paper we show that both
and
are finite atomic lattices, discuss their geometricity and compute their characteristic polynomials.
2. The Lattice ![](https://www.scirp.org/html/htmlimages\10-7402050x\fdab8896-b04f-4dc7-9029-559a3b95b71c.png)
In this section we prove that the lattice
is a finite geometric lattice, and compute its characteristic polynomial. We begin with a useful proposition.
Proposition 2.1. ([12] , Lemma 9.3.2 and [13] , Corollaries 1.8 and 1.9). For
, the following hold:
1) The number of k-dimensional subspaces contained in a given m-dimensional subspace of
is
.
2) The number of m-dimensional subspaces containing a given k-dimensional subspace of
is
.
3) Let P be a fixed m-dimensional subspaces of
. Then the number of k-dimensional subspaces Q of
satisfying
is
.
Theorem 2.2.
is a geometric lattice.
Proof. For any two elements
,
![](https://www.scirp.org/html/htmlimages\10-7402050x\2614406c-d2b5-49a9-921e-43fd9178890a.png)
Therefore
is a finite lattice. Note that
is the unique minimum element. Let
be the set of all the
-dimensional subspaces of
, where
. Then
is the set of all the atoms in
. In order to prove
is atomic, it suffices to show that every element of
is a join of some atoms. The result is trivial for
. Suppose that the result is true for
. Let
. By Proposition 2.1 and
, the number of
-dimensional subspaces of
contained in
at least is
.
Therefore there exist two different l-dimensional subspaces
of
such that
. By induction
is a join of some atoms. Hence
is a finite atomic lattice. For any
, define
. It is routine to check that
is the rank function on
. For any
, we have
![](https://www.scirp.org/html/htmlimages\10-7402050x\b495ffbf-1815-428e-a891-2f20829fb648.png)
Hence
is a geometric lattice. ![](https://www.scirp.org/html/htmlimages\10-7402050x\c3acddcc-46d4-49b0-bb1b-48119123540a.png)
Lemma 2.3. For any
, suppose that
,
and
. Then the
function of
is
![](https://www.scirp.org/html/htmlimages\10-7402050x\5798a805-0ec5-48a9-a345-4b9224b67209.png)
Proof. The
function of
is
![](https://www.scirp.org/html/htmlimages\10-7402050x\0e1711b9-80b1-4c3c-9a03-6b25688ee708.png)
By Proposition 2.1, we have
.
Thus, the assertion follows. ![](https://www.scirp.org/html/htmlimages\10-7402050x\7d639c1e-719a-4507-81c6-2090f8b9e28f.png)
Theorem 2.4. The characteristic polynomial of
is
![](https://www.scirp.org/html/htmlimages\10-7402050x\55f9410b-878c-41d8-89b1-72af2ce85e92.png)
Proof. By Proposition 2.1 and Lemma 2.3, we have
![](https://www.scirp.org/html/htmlimages\10-7402050x\1b3b3c3c-ba54-4c7b-b63f-dc67ccbce50e.png)
3. The Lattice ![](https://www.scirp.org/html/htmlimages\10-7402050x\356d5d58-7cf0-44b9-9971-23760f58b9f5.png)
In this section we prove that the lattice
is a finite atomic lattice, classify its geometricity and compute its characteristic polynomial.
Theorem 3.1. The following hold:
1)
is a finite atomic lattice.
2)
is geometric if and only if
.
Proof. 1) For any two elements
,
and
![](https://www.scirp.org/html/htmlimages\10-7402050x\1b3dcf8d-dfd9-4956-872e-94e914cbd1c2.png)
Therefore
is a finite lattice. Note that
is the unique minimum element. Let
be the set of all the j-dimensional subspaces of
, where
. Then
is the set of all the atoms in
. In order to prove
is atomic, it suffices to show that every element of
is a join of some atoms. The result is trivial for
. Suppose that the result is true for
. Let
. By Proposition 2.1, the number of
subspaces of
containing
is equal to
.
Then there exist two different
subspaces
such that
. By induction
is a join of some atoms. Therefore
is a finite atomic lattice.
2) For any
, we define
. It is routine to check that
is the rank function on
. It is obvious that
is a geometric lattice. Now assume that
. Let P be a
-dimensional subspace of
and
. By Proposition 2.1, the number of 2-dimensional subspaces of
containing P is equal to
![](https://www.scirp.org/html/htmlimages\10-7402050x\3f944e91-b8ea-44ba-80d4-96c8ecc9482d.png)
Therefore, there exist two different 2-dimensional subspaces
such that
. So
,
. Hence
, which implies that
is not a geometric lattice when
. ![](https://www.scirp.org/html/htmlimages\10-7402050x\1e585da5-0ebf-42c5-afee-98c5ba550502.png)
Lemma 3.2. For any
, suppose that
,
and
. Then the
function of
is
![](https://www.scirp.org/html/htmlimages\10-7402050x\32446400-a0aa-4993-83fe-6d240e3be24a.png)
Proof. The
function of
is
![](https://www.scirp.org/html/htmlimages\10-7402050x\6766d7ac-947b-4609-aa11-538a63b09dc3.png)
Proposition 2.1 implies that
.
Theorem 3.3. The characteristic polynomial of
is
.
Proof. By Proposition 2.1, we have
![](https://www.scirp.org/html/htmlimages\10-7402050x\6d443533-472f-4beb-9df8-a7bb0a819ea9.png)