The lattice of subspaces of a vector space over a finite field
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Abstract
For finite m and q we study the lattice \(\mathbf {L}(\mathbf {V})=(L(\mathbf {V}),+,\cap ,\{\vec {0}\},V)\) of subspaces of an mdimensional vector space \(\mathbf {V}\) over a field \(\mathbf {K}\) of cardinality q. We present formulas for the number of ddimensional subspaces of \(\mathbf {V}\), for the number of complements of a subspace and for the number of edimensional subspaces including a given ddimensional subspace. It was shown in Eckmann and Zabey (Helv Phys Acta 42:420–424, 1969) that \(\mathbf {L}(\mathbf {V})\) possesses an orthocomplementation only in case \(m=2\) and \({{\,\mathrm{char}\,}}\mathbf {K}\ne 2\). Hence, only in this case \(\mathbf {L}(\mathbf {V})\) can be considered as an orthomodular lattice. On the contrary, we show that a complementation \('\) on \(\mathbf {L}(\mathbf {V})\) can be chosen in such a way that \((L(\mathbf {V}),+,\cap ,{}')\) is both weakly orthomodular and dually weakly orthomodular. Moreover, we show that \((L(\mathbf {V}),+,\cap ,{}^\perp ,\{\vec {0}\},V)\) is paraorthomodular in the sense of Giuntini et al. (Stud Log 104:1145–1177, 2016).
Keywords
Vector space Finite field Lattice of subspaces Antitone Involution Complementation Orthocomplementation Ortholattice Orthomodular lattice Weakly orthomodular lattice Dually weakly orthomodular lattice Paraorthomodular latticeIt is well known that in a Hilbert space \(\mathbf {H}\) there exists a onetoone correspondence between the set of projection operators and the set of closed subspaces. These subspaces form an orthomodular lattice \((L(\mathbf {H}),\vee ,\cap ,{}^\perp ,\{\vec {0}\},H)\) where for \(M,N\in L(\mathbf {H})\) we have \(M\vee N=\overline{M+N}\).
Some doubts concerning the relevance of such an approach for an algebraic treatment of quantum mechanics arose when it was discovered that the class of orthomodular lattices arising from projections on Hilbert spaces does not generate the variety of orthomodular lattices showing that there are equational properties of eventstate systems that are not correctly 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 socalled paraorthomodular lattices and Kleene lattices.
The aim of the present paper is to describe the lattice \(\mathbf {L}(\mathbf {V})\) of subspaces of a finitedimensional 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 socalled 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 \(\mathbf {L}(\mathbf {V})\) is orthomodular only in very exceptional cases, it is paraorthomodular with respect to orthogonality.
Theorem 1
Proof
Remark 2
Theorem 1 also holds in case \(m\in \{0,1\}\).
Lemma 3
\(L_d(\mathbf {V})=L_{md}(\mathbf {V})\) for all \(d=0,\ldots ,m\).
Proof
Lemma 4
If m is even then \(L_{m/2}(\mathbf {V})=(q^{m/2}+1)L_{m/2}(\mathbf {K}^{m1})\).
Proof
Theorem 5
\(L(\mathbf {V})\) is odd if and only if m is even and \({{\,\mathrm{char}\,}}\mathbf {K}=2\).
Proof

antitone if \(x\le y\) implies \(y'\le x'\) (\(x,y\in L\)),

an involution if it satisfies the identity \((x')'\approx x\),

a complementation if it satisfies the identities \(x\vee x'\approx 1\) and \(x\wedge x'\approx 0\),

an orthocomplementation if it is both a complementation and an antitone involution.
Lemma 6
If \(\mathbf {L}=(L,\vee ,\wedge ,{}',0,1)\) is a nontrivial finite bounded lattice with a complementation which is an involution then L is even.
Proof
It is easy to see that the binary relation \({{\,\mathrm{R}\,}}\) defined by \(x{{\,\mathrm{R}\,}}y\) if and only if \(y=x\) or \(y=x'\) (\(x,y\in L\)) is an equivalence relation on L consisting of twoelement classes only. \(\square \)
Corollary 7
If m is even and \({{\,\mathrm{char}\,}}\mathbf {K}=2\) then \(\mathbf {L}(\mathbf {V})\) has no complementation which is an involution and hence no orthocomplementation.
A lattice \(\mathbf {L}=(L,\vee ,\wedge )\) is called modular if \((x\vee y)\wedge z=x\vee (y\wedge z)\) or all \(x,y,z\in L\) with \(x\le z\).
The following result is well known.
Proposition 8
The lattice \(\mathbf {L}(\mathbf {V})\) is modular.
Definition 9
(cf. Chajda and Länger 2018) Let \(\mathbf {L}=(L,\vee ,\wedge ,{}')\) be a lattice with a unary operation \('\). \(\mathbf {L}\) is called weakly orthomodular if \(y=x\vee (y\wedge x')\) for all \(x,y\in L\) with \(x\le y\), and it is called dually weakly orthomodular if \(x=y\wedge (x\vee y')\) for all \(x,y\in L\) with \(x\le y\). Now assume \(\mathbf {L}\) to be bounded. The element b of L is called a complement of the element a of L if both \(a\vee b=1\) and \(a\wedge b=0\). 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.
Lemma 10
(cf. Chajda and Länger 2018) Every bounded modular lattice \(\mathbf {L}=(L,\vee ,\wedge ,{}',0,1)\) equipped with a complementation \('\) is both weakly orthomodular and dually weakly orthomodular. Hence every modular ortholattice is orthomodular.
Proof
Theorem 11
Proof
Corollary 12
For any complementation \('\) on \(\mathbf {L}(\mathbf {V})\), \((L(\mathbf {V}),+,\cap ,{}')\) is both weakly orthomodular and dually weakly orthomodular.
As pointed out in Eckmann and Zabey (1969), the fact that a complementation on \(\mathbf {L}(\mathbf {V})\) is an orthocomplementation is very exceptional:
Theorem 13
The lattice \(\mathbf {L}(\mathbf {V})\) has an orthocomplementation if and only if \(m=2\) and \({{\,\mathrm{char}\,}}\mathbf {K}\ne 2\).
Hence by defining a unary operation on \(L(\mathbf {V})\) in a suitable way, \(\mathbf {L}(\mathbf {V})\) can be transformed into an orthomodular lattice if and only if \(m=2\) and \({{\,\mathrm{char}\,}}\mathbf {K}\ne 2\). The cases \(m=2\) and \({{\,\mathrm{char}\,}}\mathbf {K}=2\) as well as \(m=2\) and \({{\,\mathrm{char}\,}}\mathbf {K}\ne 2\) will be shown in the next examples. At first, we recall some concepts from lattice theory.
The situation described by Theorems 1 and 5, Corollary 7, Proposition 8 and Theorem 11 is illustrated by the following examples.
Example 14
More generally, we have
Theorem 15
If \(m=2\) and \({{\,\mathrm{char}\,}}\mathbf {K}=2\) then \(\mathbf {L}(\mathbf {V})\cong \mathrm{M}_{q+1}\) and any complementation on \(\mathbf {L}(\mathbf {V})\) is antitone, but none of them is an orthocomplementation.
Proof
Assume \(m=2\) and \({{\,\mathrm{char}\,}}\mathbf {K}=2\). Since \(L_1(\mathbf {V})=q+1\) according to Theorem 1 we have \(\mathbf {L}(\mathbf {V})\cong \mathrm{M}_{q+1}\). Clearly, any complementation on \(\mathbf {L}(\mathbf {V})\) is antitone. That \(\mathbf {L}(\mathbf {V})\) has no orthocomplementation follows from Corollary 7 and it follows from Theorem 13. \(\square \)
Now let us introduce the concept of orthogonality in \(\mathbf {V}\).
Let \(\vec {a}=(a_1,\ldots ,a_m),\vec {b}=(b_1,\ldots ,b_m)\in V\). By \(\vec {a}\vec {b}\) we denote the inner or scalar product\(a_1b_1+\cdots +a_mb_m\) of \(\vec {a}\) and \(\vec {b}\). Define \(\vec {a}\perp \vec {b}\) if \(\vec {a}\vec {b}=0\), and for any subset A of V put \(A^\perp :=\{\vec {x}\in V\mid \vec {x}\vec {y}=0\text { for all }\vec {y}\in A\}\).
Lemma 16
The mapping \(^\perp :U\mapsto U^\perp \) is an antitone involution on \(\mathbf {L}(\mathbf {V})\).
Proof
Example 17
More generally, we have
Theorem 18
Proof
As mentioned in the introduction, another approach to the lattice \(\mathbf {L}(\mathbf {V})\) was developed in Giuntini et al. (2016). We recall the following definition:
Definition 19
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.
Proposition 20
Every bounded modular lattice with an antitone involution is paraorthomodular.
Proof
If \((L,\vee ,\wedge ,{}',0,1)\) is a bounded modular lattice with an antitone involution, \(a,b\in L\), \(a\le b\) and \(a'\wedge b=0\) then \(a\vee a'\ge a\vee b'=(a'\wedge b)'=0'=1\) and hence \(a\vee a'=1\) whence \(a=a\vee 0=a\vee (a'\wedge b)=(a\vee a')\wedge b=1\wedge b=b\).
\(\square \)
Corollary 21
The lattice \((L(\mathbf {V}),+,\cap ,{}^\perp ,\{\vec {0}\},V)\) is paraorthomodular.
Proof
Example 22
Finally, we want to present a new proof of the fact that \(\mathbf {L}(\mathbf {V})\) has no orthocomplementation in case \({{\,\mathrm{char}\,}}\mathbf {K}=2\).
Theorem 23
Proof
Theorem 24
If \({{\,\mathrm{char}\,}}\mathbf {K}=2\) then \(\mathbf {L}(\mathbf {V})\) has no orthocomplementation.
Proof
We can summarize our results as follows: Despite the fact that \((L(\mathbf {V}),+,\cap ,{}',\{\vec {0}\},V)\) 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.
Notes
Acknowledgements
Open access funding provided by TU Wien (TUW). The authors are grateful to the anonymous referee for his/her valuable suggestions. Support of the research by ÖAD, Project CZ 02/2019, and support of the research of the first author by IGA, Project PřF 2019 015, are gratefully acknowledged
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The authors declare that they have no conflict of interest.
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References
 Chajda I, Länger H (2018) Weakly orthomodular and dually weakly orthomodular lattices. Order 35:541–555MathSciNetCrossRefzbMATHGoogle Scholar
 Eckmann JP, Zabey PhCh (1969) Impossibility of quantum mechanics in a Hilbert space over a finite field. Helv Phys Acta 42:420–424MathSciNetzbMATHGoogle Scholar
 Giuntini R, Ledda A, Paoli F (2016) A new view of effects in a Hilbert space. Stud Log 104:1145–1177MathSciNetCrossRefzbMATHGoogle Scholar
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