The lattice of subspaces of a vector space over a finite field

For finite m and q we study the lattice $$\mathbf {L}(\mathbf {V})=(L(\mathbf {V}),+,\cap ,\{\vec {0}\},V)$$L(V)=(L(V),+,∩,{0→},V) of subspaces of an m-dimensional vector space $$\mathbf {V}$$V over a field $$\mathbf {K}$$K of cardinality q. We present formulas for the number of d-dimensional subspaces of $$\mathbf {V}$$V, for the number of complements of a subspace and for the number of e-dimensional subspaces including a given d-dimensional subspace. It was shown in Eckmann and Zabey (Helv Phys Acta 42:420–424, 1969) that $$\mathbf {L}(\mathbf {V})$$L(V) possesses an orthocomplementation only in case $$m=2$$m=2 and $${{\,\mathrm{char}\,}}\mathbf {K}\ne 2$$charK≠2. Hence, only in this case $$\mathbf {L}(\mathbf {V})$$L(V) can be considered as an orthomodular lattice. On the contrary, we show that a complementation $$'$$′ on $$\mathbf {L}(\mathbf {V})$$L(V) can be chosen in such a way that $$(L(\mathbf {V}),+,\cap ,{}')$$(L(V),+,∩,′) is both weakly orthomodular and dually weakly orthomodular. Moreover, we show that $$(L(\mathbf {V}),+,\cap ,{}^\perp ,\{\vec {0}\},V)$$(L(V),+,∩,⊥,{0→},V) is paraorthomodular in the sense of Giuntini et al. (Stud Log 104:1145–1177, 2016).

reflected by the proposed mathematical abstraction. Hence, alternative approaches appeared in the literature, see, e.g., the paper by Eckmann and Zabey (1969) on subspaces of a vector space over a finite field or the approach by Giuntini, Ledda and Paoli (Giuntini et al. 2016) concerning so-called paraorthomodular lattices and Kleene lattices.
The aim of the present paper is to describe the lattice L(V) of subspaces of a finite-dimensional vector space over a finite field with respect to the question of defining a suitable complementation. Similarly as in Giuntini et al. (2016), we do not restrict ourselves to orthomodular lattices, but we also consider so-called weakly orthomodular and dually weakly orthomodular lattices which were recently introduced and studied by the authors in Chajda and Länger (2018). It turns out that despite the fact that L(V) is orthomodular only in very exceptional cases, it is paraorthomodular with respect to orthogonality.
Throughout the paper let m > 1 be an integer and V = (V , +, ·) an m-dimensional vector space over some finite field K = (K , +, ·) of cardinality q. In the following, without loss of generality we identify V with K m . We denote the zero element of V by 0 and the zero element of K by 0. for every natural number n.
We want to determine |A|. For choosing x 1 we have q m − 1 possibilities. For every single one of these q m −1 possibilities for choosing x 1 we have q m −q possibilities for choosing x 2 .
Going on in this way we finally obtain Remark 2 Theorem 1 also holds in case m ∈ {0, 1}.

Lemma 4 If m is even then
Proof If m is even then

Theorem 5 |L(V)| is odd if and only if m is even and
If m is odd then according to Lemma 3 showing evenness of |L(V)|. If m is even and char K = 2 then a m is odd, and hence, |L d (V)| is odd for every d = 0, . . . , m showing oddness of |L(V)|. If, finally, m is even and char K = 2 then q is odd and Let L = (L, ∨, ∧, 0, 1) be a bounded lattice. A unary operation on L is called and an antitone involution.
A bounded lattice with an orthocomplementation is called an ortholattice.
Lemma 6 If L = (L, ∨, ∧, , 0, 1) is a non-trivial finite bounded lattice with a complementation which is an involution then |L| is even.
Proof It is easy to see that the binary relation R defined by x R y if and only if y = x or y = x (x, y ∈ L) is an equivalence relation on L consisting of two-element classes only.

Corollary 7 If m is even and char K = 2 then L(V) has no complementation which is an involution and hence no orthocomplementation.
Proof This follows from Theorem 5 and Lemma 6.
The following result is well known.

Proposition 8 The lattice L(V) is modular.
Definition 9 (cf. Chajda and Länger 2018) Let L = (L, ∨, ∧, ) be a lattice with a unary operation . L is called weakly for all x, y ∈ L with x ≤ y. Now assume L to be bounded. The element b of L is called a complement of the element a of L if both a ∨ b = 1 and a ∧ b = 0. An ortholattice is called an orthomodular lattice if it is weakly orthomodular or, equivalently, if it is dually weakly orthomodular. The corresponding condition is then called the orthomodular law.
Proof Let a, b ∈ L and assume a ≤ b. Then, using modularity, complementations.
We want to determine |A|. For choosing x d+1 we have q m − q d possibilities. For every single one of these q m − q d possibilities for choosing Going on in this way we finally obtain As in the proof of Theorem 1 we see that there are complements. Together with Theorem 1 we conclude that complementations.
As pointed out in Eckmann and Zabey (1969), the fact that a complementation on L(V) is an orthocomplementation is very exceptional:

Theorem 13
The lattice L(V) has an orthocomplementation if and only if m = 2 and char K = 2.
Hence by defining a unary operation on L(V) in a suitable way, L(V) can be transformed into an orthomodular lattice if and only if m = 2 and char K = 2. The cases m = 2 and char K = 2 as well as m = 2 and char K = 2 will be shown in the next examples. At first, we recall some concepts from lattice theory.
In the following, for n ≥ 3 let M n denote the modular lattice whose Hasse diagram is visualized in Fig. 1 0 a 1 a 2 a n−1 a n 1 Fig. 1 and for n ≥ 2 let MO n denote the modular ortholattice whose Hasse diagram is visualized in Fig. 2. 0 = 1 a 1 a 1 a n a n 1 = 0  Hence L(V) ∼ = M 3 . It is easy to see that there are the following eight possibilities for defining a complementation on L(V):

A B C B A A B A B B C A B C B C A A C A B C C A C C B
This is in accordance with Theorem 11. Every single of these complementations is antitone, but none of them is an orthocomplementation.
More generally, we have

Lemma 16 The mapping ⊥ : U → U ⊥ is an antitone involution on L(V).
Proof Let U ∈ L(V). The definition of U ⊥ implies that ⊥ is antitone and U ⊆ U ⊥⊥ . From the theory concerning the solving of systems of linear equations (Gaussian elimination method) one easily obtains that dim U +dim U ⊥ = m. Hence also dim U ⊥ + dim U ⊥⊥ = m, and we obtain showing that U = U ⊥⊥ , i.e., ⊥ is an involution on L(V).
In general, ⊥ is not an orthocomplementation on L(V). For example, in Example 14 we have (m, q) = (2, 3), i.e., char K = 2. Then the Hasse diagram of L(V) looks as follows (see Fig. 4):  Proof Assume m = 2 and char K = 2. It is clear that there is a one-to-one correspondence between the set of all orthocomplementations on L(V) and the set of all partitions of L 1 (V) into two-element classes. It is easy to see by induction on n that for an arbitrary positive integer n there are exactly (2n − 1)(2n − 3) · . . . · 1 different partitions of a 2n-element set into two-element classes. Now we have
As mentioned in the introduction, another approach to the lattice L(V) was developed in Giuntini et al. (2016). We recall the following definition: Definition 19 A bounded lattice L = (L, ∨, ∧, , 0, 1) with an antitone involution is called paraorthomodular if x = y for all x, y ∈ L satisfying both x ≤ y and x ∧y = 0.
(1) It was shown in Giuntini et al. (2016) that for ortholattices (1) is equivalent to the orthomodular law. Note that in Definition 19 we do not ask to be a complementation, and we only ask to be an antitone involution.
The following result is taken from Giuntini et al. (2016). For the reader's convenience we provide a proof.  where It is easy to see that the following table defines a complementation on L(V) which is an involution: x A B C D E F G x K I J H M L N but is not an orthocomplementation on L(V) since A ⊆ J , but J = C K = A . Moreover, ⊥ is given by Going on in this way we finally obtain We can summarize our results as follows: Despite the fact that (L(V), +, ∩, , { 0}, V ) with an appropriate is an orthomodular lattice in exceptional cases only, we have shown that this lattice is weakly orthomodular, dually weakly orthomodular and paraorthomodular when is chosen in a appropriate way. This motivates further study of these structures.