From Orthocomplementations to Locality
After some background on lattices, the locality framework introduced in earlier work by the authors is extended to cover posets and lattices. We then extend the correspondence between Euclidean structures on vector spaces and orthogonal complementations to a one-to-one correspondence between a class...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Цитувати: | From Orthocomplementations to Locality. Pierre Clavier, Li Guo, Sylvie Paycha and Bin Zhang. SIGMA 17 (2021), 027, 23 pages |
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| author | Clavier, Pierre Guo, Li Paycha, Sylvie Zhang, Bin |
| author_facet | Clavier, Pierre Guo, Li Paycha, Sylvie Zhang, Bin |
| citation_txt | From Orthocomplementations to Locality. Pierre Clavier, Li Guo, Sylvie Paycha and Bin Zhang. SIGMA 17 (2021), 027, 23 pages |
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| description | After some background on lattices, the locality framework introduced in earlier work by the authors is extended to cover posets and lattices. We then extend the correspondence between Euclidean structures on vector spaces and orthogonal complementations to a one-to-one correspondence between a class of locality structures and orthocomplementations on bounded lattices. This recasts in the context of renormalisation classical results in lattice theory.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 027, 23 pages
From Orthocomplementations to Locality
Pierre CLAVIER a, Li GUO b, Sylvie PAYCHA a and Bin ZHANG c
a) Institute of Mathematics, University of Potsdam, D-14476 Potsdam, Germany
(S. Paycha on leave from Université Clermont-Auvergne, Clermont-Ferrand, France)
E-mail: clavier@math.uni-potsdam.de, paycha@math.uni-potsdam.de
b) Department of Mathematics and Computer Science, Rutgers University,
Newark, NJ 07102, USA
E-mail: liguo@rutgers.edu
c) School of Mathematics, Yangtze Center of Mathematics, Sichuan University,
Chengdu, 610064, China
E-mail: zhangbin@scu.edu.cn
Received July 06, 2020, in final form March 02, 2021; Published online March 22, 2021
https://doi.org/10.3842/SIGMA.2021.027
Abstract. After some background on lattices, the locality framework introduced in earlier
work by the authors is extended to cover posets and lattices. We then extend the correspon-
dence between Euclidean structures on vector spaces and orthogonal complementations to
a one-one correspondence between a class of locality structures and orthocomplementations
on bounded lattices. This recasts in the context of renormalisation classical results in lattice
theory.
Key words: locality; lattice; poset; orthocomplementation; renormalisation
2020 Mathematics Subject Classification: 06C15; 08A55; 81T15; 15A63
This paper is dedicated to Gianni Landi
on the occasion of his sixtieth birthday
1 Introduction
1.1 The general setup and our aims
The notion of complementation, or complement map – roughly speaking, an operation which
separates a subset M of a given set X from another subset M ′, its complement, so that the infor-
mation on X is split into the part on M and the part on the complement M ′ – provides a separat-
ing tool that is ubiquitous in mathematics. For example, the notion of (ortho)complementation
naturally arises in the context of axiomatic quantum mechanics, where various types of binary
relations are used, that are defined on the set of all questions testable for a given physical
system [4]. Also, as we shall see below, the notion of complementation plays a central role in
renormalisation procedures. Typical examples of complementations are the set complementa-
tions and the orthogonal complementations, taking respectively a subset of a reference set to its
complement in the set, a linear subspace of a reference Euclidean vector space to its orthogonal
complement in the space.
Our guiding example throughout this paper is the set (G(V ),�) of linear subspaces in a finite
dimensional vector space V equipped with the partial order � corresponding to the inclusion
of linear subspaces.
This paper is a contribution to the Special Issue on Noncommutative Manifolds and their Symmetries in
honour of Giovanni Landi. The full collection is available at https://www.emis.de/journals/SIGMA/Landi.html
mailto:clavier@math.uni-potsdam.de
mailto:paycha@math.uni-potsdam.de
mailto:liguo@rutgers.edu
mailto:zhangbin@scu.edu.cn
https://doi.org/10.3842/SIGMA.2021.027
https://www.emis.de/journals/SIGMA/Landi.html
2 P. Clavier, L. Guo, S. Paycha and B. Zhang
An inner product Q on V gives rise to a complementation on G(V ):
ΨQ : G(V ) −→ G(V ), W 7−→W⊥
Q
,
where W⊥
Q
:= {v ∈ V |Q(v, w) = 0, ∀w ∈ W} is the Q-orthogonal complement of W . It also
gives rise to a symmetric binary relation on G(V ), namely the orthogonality relation
W1 ⊥Q W2 ⇐⇒ Q(w1, w2) = 0 ∀ (w1, w2) ∈W1 ×W2, (1.1)
and we have
W1 ⊥Q W2 ⇐⇒W2 ⊆ ΨQ(W1), W2 = ΨQ(W1)⇐⇒W2 = grt
{
W � V |W ⊥Q W1
}
.
Here grt means the greatest element by inclusion. This establishes a one-one correspondence
between locality relations and orthocomplementations given by scalar products:
⊥Q←→ ΨQ. (1.2)
The symmetric binary relation ⊥Q on G(V ) is a particular instance of the more general notion
of locality relation introduced in [6]. Since the poset (G(V ),�) equipped with the intersection and
the sum (of two vector spaces) is a lattice, the one-one correspondence (1.2) serves as a motivation
to investigate the relation between complementations and locality relations on lattices.
So we ask, under what conditions one can derive on a lattice, a locality relation from a com-
plementation.
Theorem 5.16 provides an answer to this question: for any bounded lattice, there is a one-one
correspondence
orthocomplementations ←→ strongly separating locality relations. (1.3)
When applied to the lattice G(V ), this generalises the correspondence (1.2) between orthogo-
nality and orthogonal complement on vector spaces.
We were informed by a referee report on a previous version of this paper, that this one-one
correspondence was already known and proven in [3], this leading us to some substantial refor-
mulations and restructuring.1 Although the one-one correspondence (1.3) seems to be common
knowledge in the lattice community, we believe that recasting this known result in the con-
text of renormalisation is relevant for the mathematical physics community. We feel that this
exploratory and survey type article serves to promote the notion of orthocomplements beyond
the lattice theory community, in the mathematical physics community where complements are
constantly used in the context of renormalisation, as we shall now explain.
1.2 Locality and complements in renormalisation
Complementations play an essential role in renormalisation, where they arise in various disguise
and are used to separate divergent terms from convergent terms.
In A. Connes and D. Kreimer’s algebraic Birkhoff factorisation approach to renormalisation
[10, 26], the coproduct is typically built from a (relative) complementation on a poset (X,≤)
by means of ∆x =
∑
y x ⊗ (x \ y), where x \ y is a complement to an element y ≤ x. This
holds for the coproduct on rooted trees for which we can view the crown of a rooted tree as
the complement of its trunk (a sub-rooted tree) after an admissible cut (see, e.g., [14, 15]), for
the coproduct on graphs, with the contracted graph Γ\γ corresponding to the complement of
a subgraph γ inside a connected 1 particle irreducible Feynman graph Γ (see, e.g., [26]).
1In particular, we learned that a poset with locality amounts to a weak degenerate orthogonal poset in the
sense of [3, Definition 2.1]. We nevertheless use a slightly different terminology which we find well-suited for the
locality setup we have in mind.
From Orthocomplementations to Locality 3
Complementations also arise in generalised Euler–Maclaurin formulae, which relate sums
to integrals. A systematic choice of complement (called a rigid complement) of a linear subspace
of a vector space was used in [16] to interpolate between exponential sums and exponential
integrals over rational polytopes in a rational vector space. A notion of “transversal cone C\F to
a face F of a cone C” was used as a complementation F 7→ C\F by N. Berline and M. Vergne [1],
to prove a local Euler–Maclaurin formula on polytopes. Cones form a poset for the relation
“F ≤ C if F is a face of C”, and we could reinterpret the Euler–Maclaurin formula on cones, as
an algebraic Birkhoff-factorisation by means of the coproduct ∆C =
∑
F≤C(C \ F )⊗ F [20].
Renormalisation issues also underly our quest for a description of complementations on (finite
dimensional) vector spaces. To explain how this motivates our comparative study of locality
relations and complementations, let us first describe an abstract framework for a renormalisation
scheme in the context of locality structures [6, 7]:
� an (locality) algebra (A,>A,mA) (mA stands for the product and >A for the locality
relation) might it be of Feynman graphs, trees, or cones,
� an algebra of meromorphic germs at zero (M, ·) might it be the algebraM(C) of meromor-
phic germs as in A. Connes and D. Kreimer’s algebraic Birkhoff factorisation approach [10]
or the algebra M(C∞) of multi-variable meromorphic germs with linear poles studied
in [21],
� a (locality) morphism Φ: (A,>A,mA) −→ (M, ·) such as Feynman integrals [11], branched
zeta functions [8, 9] and conical zeta functions [20].
The locality principle translates to the partial multiplicativity of Φ:
a1>Aa2 =⇒ Φ(mA(a1, a2)) = Φ(a1) · Φ(a2).
Renormalising consists in building a (locality) character Φren : (A,>A,mA) −→ (C, ·)
a1>A a2 =⇒ Φren(mA(a1, a2)) = Φren(a1) · Φren(a2).
To build Φren, one first needs to separate the holomorphic part Φ+ from the polar part Φ−
of Φ and then to evaluate it at zero setting Φren := Φ+(0). This splitting relies on the splitting
M =M+⊕M− of the algebra of meromorphic germs at zero into a holomorphic partM+ and
a polar part M−. The map Φ+ is built:
� in one variable by means of an algebraic Birkhoff factorisation following A. Connes and
D. Kreimer [10],
� in several variables as Φ+ := π+ ◦ Φ via a projection map π+ onto M+ (see, e.g., [7]).
We adopt the second approach which can be interpreted as a minimal subtraction scheme in sev-
eral variables. To build the projection map π+, we use Laurent expansions of meromorphic germs
in several variables with linear poles (see [21]) whose construction in turn requires a (filtered)
separating device on the underlying spaces V = Ck, k ∈ N.
The resulting renormalised map Φren = π+ ◦ Φ depends on the choice of the projection π+,
which in turn is dictated by the choice of splitting M = M+ ⊕M−. The passage from one
splitting to another splitting is encoded in a renormalisation group which we hope to describe
in forthcoming work. In [21] we built Laurent expansions using the orthogonality relation Q
on V := Ck as a separating device, leading to an orthogonal projection πQ+ . The present paper
investigates alternative separating devices on V via separating devices on the lattice G(V ) which
we intend to use to extend our construction of Laurent expansions beyond the ones obtained
by orthogonal splittings. The group of transformations of meromorphic germs that preserve
holomorphic ones plays a central role in the context of renormalisation since its elements are
transformations which take one renormalised value to another and can therefore be interpreted
as elements of a hypothetical renormalised group.
4 P. Clavier, L. Guo, S. Paycha and B. Zhang
1.3 Plan of the paper
Section 2 is a review of the basics of lattice theory, such as distributivity (Proposition-Defi-
nition 2.5) and modularity (Definition 2.12) of lattices, which we spell out in order to later
generalise them to the locality setup.
Section 3 is dedicated to lattices equipped with a locality relation, which we call locality lat-
tices (Definition 3.4). We first define and characterise locality posets (Proposition-Definition 3.1)
after which we characterise locality lattices (Proposition 3.9). Whereas the subspace latticeG(V )
is not a locality lattice for the “disjointedness” locality relation W1>W2 ⇔ W1 ∩W2 = {0}, it
is for the locality relation ⊥Q of (1.1), see Example 3.7.
We then review the notion of complement (Definition 4.1) and orthocomplement (Defini-
tion 4.12) on lattices and discuss the strongly separating property (Proposition 4.13) of lattices
with orthocomplement. This is later used to classify orthocomplements on the subspace lattice
of R2 (Corollary 4.15).
Alongside the separating property of orthocomplementations singled out in Section 4, in Sec-
tion 3 we single out separating locality relations (Definition 5.4) on lattices, after which we
introduce the more stringent strongly separating locality relations (Definition 5.9). We then
prove the equivalence between orthocomplementations and strongly separating locality lattices
(Theorem 5.16). An easy consequence is the classification of strongly separating locality relations
on the lattice G
(
R2
)
in Corollary 5.17.
Section 6 is dedicated to our guiding example, the modular bounded lattice (G(V ),�).
We show (Proposition 6.1) that locality relations on the lattice G(V ) are in one-one corre-
spondence with locality relations on the vector space V introduced in [6]. Specialising to non-
degenerate locality relations on a vector space V (Definition 6.2), we show that these are in
one-one correspondence with strongly separating locality relations on G(V ) (Proposition 6.3).
Typical non-degenerate relations are orthogonality relations, and the notion of orthogonal basis
generalises to that of locality basis (Definition 6.4). A Gram–Schmidt type argument is used
to show that a strongly separating locality relation on G(V ) admits a locality basis (Proposi-
tion 6.5). The fact that a given basis can be the locality basis for multiple locality relations
suggests the richness of locality relations on a vector space.
Following a referee’s suggestion, in an appendix, we show a correspondence between another
class of locality relations and of orthocomplementations. For complete atomistic lattices, we
present a one-one correspondence between a class of locality lattice relations and certain locality
relations on the set of atoms of the lattice, which applies to the subspace lattice considered in
this paper.
We have therefore reached our goal in extending the correspondence (1.2) well beyond loca-
lity relations of the type ⊥Q of (1.1), showing the more general correspondence (1.3). This way,
we could detect strongly separating locality relations on vector spaces beyond the orthogona-
lity locality which classifies strongly separating locality relations on R2, among which lies the
orthogonality locality relations corresponding to (1.1).
2 Modular lattices
This section puts together notions and examples on lattices to provide background for our
later study of lattices in a locality setup. For references on this background material, see,
e.g., [2, 13, 17, 18, 22, 23, 24, 25, 27].
2.1 Lattices
For completeness, let us first recall that a lattice is a partially ordered set (poset) (L,≤), any two-
element subset {a, b} of which has a least upper bound (also called a join) a ∨ b, and a greatest
From Orthocomplementations to Locality 5
lower bound (also called a meet) a ∧ b such that
(a) both operations are associative and monotone with respect to the order,
(b) if a1 ≤ b1 and a2 ≤ b2, then a1 ∧ a2 ≤ b1 ∧ b2 and a1 ∨ a2 ≤ b1 ∨ b2.
We shall sometimes write (L,≤,∧,∨) for completeness.
A morphism ϕ : L→ L′ of two lattices (L,≤,∧,∨) and (L′,≤′,∧′,∨′) is a morphism ϕ : (L,≤)
→ (L′,∨′) of posets compatible with the operations
ϕ(a) ∧′ ϕ(b) = ϕ(a ∧ b)
and
ϕ(a) ∨′ ϕ(b) = ϕ(a ∨ b) ∀ (a, b) ∈ L2.
Example 2.1. A first example of lattices is the power set P(X) of a set X with inclusion as
the partial order. Then ∨ is the union and ∧ is the intersection. Another elementary example is
N with the partial order a|b⇐⇒ ∃ k ∈ N, b = ak. Then ∨ is the least common multiple and ∧ is
the greatest common divisor.
Here are central examples for our purposes.
Example 2.2.
(a) Given a finite dimensional vector space V , let G(V ) denote the set of linear subspaces of V
equipped with the partial order “to be a linear subspace of” denoted by �. The lattice
(G(V ),�), that we call the subspace lattice comes equipped with the sum ∨ = + and the
intersection ∧ = ∩ as lattice operations and we write (G(V ),�,∩,+).
(b) Given a Hilbert space V , let G(V ) denote the set of closed linear subspaces of V equipped
with the partial order “to be a closed linear subspace of” denoted by �. Since the sum
of two closed linear subspaces of V is not necessarily closed, the topological sum W +̄W ′
of two closed linear subspaces W and W ′ of V is the topological closure W +W ′ of
their algebraic sum. So an element v lies in the topological sum of W and W ′ in G(V )
if it can be written as the limit v = lim
n→∞
(wn + w′n) of a sum of elements wn ∈ W ,
w′n ∈ W ′. If W and W ′ are finite dimensional, their topological sum coincides with
their algebraic sum. The lattice
(
G(V ),�
)
with the topological sum ∨ := +̄ and the
intersection ∧ := ∩ is a lattice and we write (G(V ),�,∩, +̄) and call the closed subspace
lattice.
The study of such lattices was motivated by quantum mechanics, see, e.g., [5, 12] and the
references therein. The mathematical interpretation of quantum theory is indeed based on the
structure of the set G(V ) of a given Hilbert space V , or equivalently of projection operators,
viewed as events and on the probabilistic interpretation of Hermitean operators, viewed as
observables, see, e.g., [23, Section 16].
Remark 2.3. With the applications to renormalization in mind, we will later focus on sub-
space lattices of finite dimensional vector spaces though many results have counterparts for
sets of closed linear subspaces of topological vector spaces. We hope to investigate topological
aspects systematically in forthcoming work.
6 P. Clavier, L. Guo, S. Paycha and B. Zhang
Definition 2.4.
(a) A poset ideal of a poset (P,≤) is a subset M of P such that if a ∈ M and b ≤ a, then
b ∈M .
(b) The smallest ideal of a poset (P,≤) that contains a given element a is called the principal
(poset) ideal of (P,≤) generated by a. It is given by
↓ a := {b ∈ P | b ≤ a}.
(c) A sublattice of a lattice (L,≤,∧,∨) is a subset M of L closed under the meet and join
operations in M , i.e., such that for every pair of elements a, b in M both a ∧ b and a ∨ b
are in M .
(d) An ideal of a lattice (L,≤,∧,∨) or lattice ideal is a poset ideal and a sublattice, that is,
a poset ideal such that a ∨ b lies in I for any (a, b) ∈ I2.
(e) The smallest ideal of a lattice (L,≤,∧,∨) that contains a given element a is called the
principal (lattice) ideal generated by a. It coincides with the principal poset ideal ↓ a =
{b ∈ L | b ≤ a} = {a ∧ b | b ∈ L}.
(f) Any interval
[a, b] := {c ∈ L | a ≤ c ≤ b}
in a lattice L is itself a lattice.
(g) A lattice (L,≤,∧,∨) is bounded from above (resp. from below)) if it has a greatest element
(also called top element) denoted by 1 (resp. a least element (also called bottom) denoted
by 0), which satisfies x ≤ 1 (resp. 0 ≤ x) for any x ∈ L. If it is bounded both from below
and from above in which case L = [0, 1], we call it bounded and use the short hand notation
(L,≤, 0, 1). We will typically consider bounded lattices.
2.2 Distributivity and modularity
Let L be a lattice. For a, b, c ∈ L, we have
(a ∧ b) ∨ (a ∧ c) ≤ a ∧ (b ∨ c) and a ∨ (b ∧ c) ≤ (a ∨ b) ∧ (a ∨ c),
yet the operations ∨ and ∧ are not necessarily distributive with respect to each other.
Proposition-Definition 2.5 ([17, Section 4, Lemma 10]). A lattice L is called distributive if
it fulfills one of the two equivalent properties
a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c), ∀ a, b, c ∈ L, (2.1)
or the dual identity:
a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c), ∀ a, b, c ∈ L.
Example 2.6. The power set lattice and the lattice of positive integers in Example 2.1 are
distributive.
Counterexample 2.7. The subspace latticeG(V ) introduced in Example 2.2 is not distributive.
In G
(
R2
)
, setting W1 = 〈e1〉, W2 = 〈e2〉 and W = 〈e1 + e2〉, we have W = (W1 + W2) ∩W 6=
(W1 ∩W ) + (W2 ∩W ) = {0} and W = (W1 ∩W2) +W 6= (W1 +W ) ∩ (W2 +W ) = R2.
The following examples provide simple nondistributive lattices.
From Orthocomplementations to Locality 7
Example 2.8 ([18]). The diamond lattice M3 = {0, a, b, c, 1} with a, b, c pairwise incomparable
is not distributive since (a ∧ b) ∨ c = c whereas (a ∨ c) ∧ (b ∨ c) = 1.
1
a b c
0
Figure 1. The Hasse diagram of the diamond lattice.
Example 2.9. The pentagon lattice N5 := {0, b1, b2, c, 1} with partial order defined by 0 ≤ b1 <
b2 ≤ 1 and 0 ≤ c ≤ 1 with bi and c incomparable, is not distributive since b2∧(b1∨c) = b2∧1 = b2
while (b2 ∧ b1) ∨ (b2 ∧ c) = b1 ∨ 0 = b1.
1
b2
b1
c
0
Figure 2. The Hasse diagram of the pentagon lattice.
These two examples turn out to be the cores of any non-distributive lattices.
Proposition 2.10 ([2, Chapter IX, Theorem 2], [18, Theorems 101 and 102]). A lattice is
distributive if and only if it does not contain a pentagon or a diamond as a sublattice.
Here is a useful characterisation of distributivity.
Proposition 2.11 ([2, Theorem I.10], [27, Corollary 3.1.3]). A lattice L is distributive if and
only if the following cancellation law holds:
(cancellation law) a ∧ c = b ∧ c and a ∨ c = b ∨ c⇐⇒ a = b.
We recall the following definition, which can be viewed as a relative distributivity.
Definition 2.12 ([17, Section 4, Lemma 12]). A lattice L is called modular if
(modularity) a ≥ c⇒ (a ∧ b) ∨ c = a ∧ (b ∨ c).
Example 2.13. A distributive lattice is modular since modularity is a special case of the
distributivity in (2.1). In particular, (P(X),⊆,∩,∪) and (N, |,∧,∨) are modular.
Not every modular lattice is distributive.
Example 2.14. The diamond lattice of Example 2.8 is modular and not distributive.
The following well-known example will be crucial for our applications:
Example 2.15. For any finite dimensional vector space V , the subspace lattice G(V ) introduced
in Example 2.2 is modular, yet it is not distributive when dimV > 1 (see Counterexample 2.7).
8 P. Clavier, L. Guo, S. Paycha and B. Zhang
Counterexample 2.16. The pentagon lattice of Example 2.9 is not modular.
Modularity is a hereditary property, which leads to a refinement of Proposition 2.10.
Proposition 2.17 ([18, Theorems 101 and 102]).
(a) A lattice is modular if and only if it does not contain a pentagon sublattice.
(b) A modular lattice is distributive if and only if it does not contain a diamond sublattice.
We have the following relationship between modularity and the cancellation law.
Proposition 2.18 ([27, Corollary 2.1.1]). A lattice L is modular if and only if it obeys the
following modular cancellation law:
for any (a, b, c) ∈ L3, if a ≤ b, a ∧ c = b ∧ c and a ∨ c = b ∨ c, then a = b.
To sum up we have the following correspondences:
distributivity ks +3
��
cancellation law
��
modularity ks +3 modular cancellation law
3 Locality relations on lattices
To study properties in lattices equipped with locality relations, we first equip posets with locality
relations.
As in [7], we call locality relation on a set X, any symmetric binary relation > on X and the
pair (X,>) a locality set. For A ⊆ X, we call
A> := {x′ ∈ X |x>x′, ∀x ∈ A} (3.1)
the polar set of A.
We observe that for any element a in a locality set (X,>) we have a ∈
(
a>
)>
.
Proposition-Definition 3.1. A locality poset is a poset (P,≤) equipped with a locality relation
on the set P that satisfies one of the following equivalent conditions which amount to a compa-
tibility condition with the partial order
(a) if a ≤ b, then b> ⊆ a>,
(b) for any c ∈ P , the set c> is a poset ideal of (P,≤).
(c) ↓ a ⊆
(
a>
)>
, i.e., if b ≤ a then b ∈
(
a>
)>
.
Then the relation > is called a poset locality relation.
Checking the equivalences (a)⇔ (b)⇔ (c) is an easy exercise.
Remark 3.2.
(a) A locality poset amounts to a weak degenerate orthogonal poset in the sense of [3, Defi-
nition 2.1] (see also [23, supplementary Remark 6 in Section 16], which refers to [29]).
There, the set Ker> := a> ∩ {a} = {a ∈ L, a>a} is called the kernel of >.
(b) If P has a bottom element 0 (resp. top element 1), then p> ⊆ 0> (resp. p> ⊇ 1>) for any
p ∈ P . We will later consider lattices for which 0> = P (resp. P> = {0}).
From Orthocomplementations to Locality 9
Example 3.3. Some locality posets are
(a) the power set (P(X),⊆) of Example 2.1 endowed with the locality relation
A>∩B if and only if A ∩B = ∅,
(b) the subspace poset (G(V ),�) of Example 2.2 endowed with the locality relation
W1>∩W2 if and only if W1 ∩W2 = {0}.
Just as in Proposition-Definition 3.1, we required from a locality relation > on a poset, that
the polar sets be poset ideals, from a locality relation on a lattice we require that polar sets be
lattice ideals.
Definition 3.4. A locality relation on a lattice (L ≤,∧,∨) is a locality relation > on the set L
such that, for any a ∈ L, the polar set a> (defined by (3.1)) is a lattice ideal of L.
We call the quintuple (L,≤,>,∧,∨) a locality lattice.
Remark 3.5. It is easy to see that the intersection of poset ideals is a poset ideal. Similarly,
the intersection of lattice ideals are lattice ideals. Therefore the intersection of locality relations
on a lattice is still a locality relation on the same lattice.
Since a lattice ideal is a poset ideal, a locality lattice is a locality poset.
Note that the operations ∨ and ∧ are defined on the whole cartesian product L× L.
Remark 3.6. A related notion is a partial lattice defined in [18, Section I5.4].
Example 3.7. Given a Hilbert space (V, 〈·, ·, 〉), the closed subspace lattice
(
G(V ),�, +̄,∩
)
introduced in Example 2.2 is a locality lattice for the locality relation: U1>U2 if 〈u1, u2〉 = 0,
∀ui ∈ Ui, i = 1, 2.
To show this, let A1, A2, B be closed linear subspaces of (V, 〈·, ·〉) with Ai>B for i ∈ {1, 2}.
The fact that (A1 ∩ A2)>B is straightforward. To show the relation (A1+̄A2)>B, for any
a ∈ A1+̄A2 and b ∈ B we write a = lim
n→∞
(a1(n) + a2(n)) and b = lim
n→∞
b(n) with ai(n) ∈ Ai and
b(n) ∈ B for any n ∈ N. By the bilinearity and continuity of the inner product, we have
〈a, b〉 = lim
m,n→∞
〈a1(m) + a2(m), b(n)〉 =
2∑
i=1
lim
m,n→∞
〈ai(m), b(n)〉 = 0.
Therefore (A1+̄A2)>B and (A1+̄A2)>(B1+̄B2) for A1, A2, B1, B2 with Ai>Bj for i, j ∈ {1, 2}
as required by symmetry of >. Finally, G(V ) has a least element, the trivial vector space {0},
and a greatest element, the full vector space V . Trivially, we have {0}>G(V ) and, for any
V ∈ G(V ), if A>V then A = {0}.
In particular,
Example 3.8. The subspace lattice (G(V ),�,+,∩) on a finite dimensional Euclidean real
(resp. Hermitian complex) vector space (V, 〈·, ·〉) comes equipped with a lattice locality U1>U2
if 〈u1, u2〉 = 0 ∀ui ∈ Ui, i = 1, 2.
Proposition 3.9. A lattice (L,∧,∨,>) equipped with a poset locality relation is a locality lattice
if and only if for any finite index set I and ai ∈ L, i ∈ I, we have(∨
i∈I
ai
)>
=
⋂
i∈I
a>i . (3.2)
10 P. Clavier, L. Guo, S. Paycha and B. Zhang
Proof. If a poset locality relation > satisfies (3.2). Then from a>c, b>c, we have c ∈ a>∩ b> =
(a ∨ b)>. Hence a ∨ b is in c>. Thus c> is a lattice ideal.
Conversely, given a locality relation on L, the compatibility of the locality relation with the
partial order gives the inclusion from left to right. To show the inclusion from right to left,
let b ∈ a>i for all i ∈ I. Then ai ∈ b> for all i ∈ I and since b> is a lattice ideal, this implies
by induction on i that
∨
i∈I ai ∈ b> so that b ∈
(∨
i∈I ai
)>
. �
On the subspace lattice G(V ), Proposition 3.9 translates to the following statement.
Example 3.10. Lattice locality relations on G(V ) are poset locality relations >G such that for
any index set I and Wi ∈ G(V ), i ∈ I, we have(∑
i∈I
Wi
)>G
=
⋂
i∈I
W>G
i .
Note that sum and intersection in the above equation are operations in the lattice G(V ).
4 Orthocomplemented lattices
This section reviews the classical notion of orthcomplemented lattices, in preparation for the
forthcoming sections in which we shall equip a subclass of orthocomplemented lattices with
a locality relation. See [2, Section II.6 ] and [18, Section I.6] for background on complemented
lattices and relatively complemented lattices. We also refer the reader to [3] for a study of ortho-
complementations.
4.1 Relatively complemented lattices
We specialise to lattices (L,≤,∧,∨) bounded from below by 0, equipped with the disjointedness
locality relation >∧ : a>∧b if and only if a ∧ b = 0 so that ∧ restricted to the graph of >∧ is
identically zero. Define
a⊕ b := a ∨ b when a>∧ b.
Definition 4.1.
(a) A complemented lattice is a bounded lattice (L,≤, 0, 1) in which every element a has
a complement, i.e., an element a′ in L, such that a⊕ a′ = 1.
(b) A sectionally complemented lattice is a lattice (L,≤, 0) with bottom element 0 in which
any interval of the form [0, b] is complemented when viewed as a sublattice of L, i.e., such
that for any a ≤ b there is an element a′ in L such that a⊕ a′ = b.
(c) A relatively complemented lattice is a lattice (L,≤) in which any interval [a, b] of L is
complemented when viewed as a sublattice of L. This means that any element x ∈ [a, b]
has a relative complement, namely an element x′ in L such that x∨ x′ = b and x∧ x′ = a.
Remark 4.2. Any relatively complemented lattice which has a minimal element 0 is sectionally
complemented and any sectionally complemented lattice with a greatest element is comple-
mented. Therefore, any relatively complemented lattice with a least and a greatest element is
complemented.
Example 4.3. The power set (P(X),⊆) considered in Example 2.1 is relatively complemented.
Fix an interval [A,B] given by A ⊆ B and let C ∈ [A,B], that is, A ⊆ C ⊆ B. Let C ′ :=
A ∪ (B \ C). Then C ∪ C ′ = B and C ∩ C ′ = A.
From Orthocomplementations to Locality 11
We recall a well-known result:
Lemma 4.4 ([28, Corollary to Theorem 11.1]). Every complemented modular lattice L is rela-
tively complemented.
Example 4.5. The subspace lattice (G(V ),�,∩,+) for a finite dimensional vector space con-
sidered in Example 2.2 is relatively complemented. As can be easily proved, for U ⊆W in G(V ),
the interval [U,W ] is isomorphic to the subspace lattice of W/U under the isotone assignment
X ∈ [U,W ] to X/U ∈ G(W/U). Then the existence of relative complements in [U,W ] follows
from the existence of complements in G(W/U).
Example 4.6. The closed subspace lattice
(
G(V ),�,∩,+
)
for a Hilbert space V considered in
Example 2.2 is also relatively complemented. Indeed, for a closed subspace U of V , a closed
subspace W of U has a closed complement in U given by its orthogonal complement space W⊥U :=
W⊥ ∩U . Indeed, the latter is closed and their topological sum W +̄W⊥U = (W ⊕W⊥)∩U = U .
Counterexample 4.7. The lattice (N, |) considered in Example 2.1 is not sectionally comple-
mented, for example, 2|4, but there is no element a, such that 2⊕ a = 4.
The following example shows that the complement in a complemented lattice might not be
unique.
Example 4.8. Back to Example 4.5 in the case V = R2, and with the notations of Counter-
example 2.7, we have W ⊕W1 = V and W ⊕W2 = V .
The following lemma shows that the uniqueness of the relative complement is a strong requi-
rement.
Lemma 4.9 ([2, Corollary 1, p. 134]). On a lattice (L,≤), the uniqueness of the relative com-
plement is equivalent to the distributivity property.
4.2 Orthocomplemented lattices
We introduce the notion of orthocomplementation on a poset bounded from below, which
amounts to the (strong) orthocomplementation introduced in [3, Remark below Proposition 1.7].
Definition 4.10. A poset (P,≤, 0) with bottom 0 is called orthocomplemented if it can be
equipped with a map Ψ: P → P called the orthocomplementation, which assigns to a ∈ P its
orthocomplement2 Ψ(a) such that
(a) (Ψ antitone) b ≤ a⇒ Ψ(a) ≤ Ψ(b) ∀ (a, b) ∈ P 2,
(b) (Ψ involutive) Ψ2 = Id,
(c) (Ψ separating) for any b ∈ P , if b ≤ a and b ≤ Ψ(a), then b = 0.
Here is an elementary yet useful lemma. For a, b in a poset (P,≤), let a ∨ b (resp. a ∧ b) be
the least upper bound (resp. least lower bound) of a and b, if it exists.
Lemma 4.11. A map Ψ: P → P on a poset (P,≤) which is
(a) (antitone) ∀ (a, b) ∈ L2, b ≤ a⇒ Ψ(a) ≤ Ψ(b),
(b) (involutive) Ψ2 = Id,
2Ψ(a) is often denoted by a⊥ in the literature, a notation we avoid here not to cause any confusion when
specialising to the case of orthogonal complements on vector spaces.
12 P. Clavier, L. Guo, S. Paycha and B. Zhang
satisfies the following relations:
Ψ(a ∧ b) = Ψ(a) ∨Ψ(b) and Ψ(a ∨ b) = Ψ(a) ∧Ψ(b)
for any (a, b) ∈ P 2 for which the joins arising in these identities are well-defined (see [3, Propo-
sition 1.3(iii)]).
If moreover, P has a bottom 0, then P is bounded with top Ψ(0) (see [3, Proposition 1.3(iii)]).
Definition 4.12. A lattice (L,≤,∧,∨) bounded from below by 0 equipped with a map Ψ: L→ L
which defines an orthocomplementation on the poset (L,≤, 0), is called an orthocomplemented
lattice.
Thanks to the above lemma, the separating condition in Definition 4.10 can be replaced by
an a priori stronger condition.
Proposition 4.13 ([3, Propositions 1.1(iii) and 1.7]).
� An antitone and involutive map Ψ on a lattice (L,≤,∧,∨) induces morphisms Ψ: (L,∧)→
(L,∨) and Ψ: (L,∨)→ (L,∧) on L.
� A lattice L bounded from below equipped with an orthocomplementation Ψ is bounded from
above with top 1 = Ψ(0) and satisfies the following strongly separating property
(Ψ strongly separating) a⊕Ψ(a) = 1 ∀ a ∈ L. (4.1)
Thus an orthocomplemented lattice enjoys the strongly separating property.
Here is a class of examples of orthocomplemented lattices of direct interest to us.
Example 4.14. This is a classical example, see, e.g., [3, below Definition 1.4].
Given a Hilbert vector space (V, 〈·, ·〉), and G(V ) the closed subspace lattice of Example 2.2.
The map
Ψ〈·,·〉 : G(V ) −→ G(V ), W 7−→W⊥ := {v ∈ V | 〈v, w〉 = 0, ∀w ∈W}
defines an orthocomplementation on G(V ).
Indeed, W⊥ is closed (by the continuity of the inner product) for any linear (whether closed
or not) subspace W � V and for any closed linear subspaces W , W1, W2 of V , we have
(a) (Ψ separating) W ⊕W⊥ = W ⊕W⊥ = V ,
(b) (Ψ antitone) if W1 ≤W2, then W⊥2 �W⊥1 ,
(c) (Ψ involutive) W⊥
⊥
= W = W .
Corollary 4.15. Orthocomplementations on G
(
R2
)
are in one-one correspondence with invo-
lutive maps ψ : [0, π)→ [0, π) without fixed points.
Proof. An orthocomplementation Ψ: G
(
R2
)
→ G
(
R2
)
obeys the strongly separating condi-
tion (4.1)
U ⊕Ψ(U) = R2 ∀U ∈ G
(
R2
)
.
In particular, Ψ({0}) = R2 and Ψ
(
R2
)
= {0}. Thus Ψ is uniquely determined by its effect on
the lines Reθi in bijection with θ ∈ [0, π). Let ψ : [0, π)→ [0, π) be defined by Ψ(Reθi) = eψ(θ)i.
Then Ψ being involutive (resp. strongly separating) amounts to ψ being involutive (resp. without
fixed points). The antitonicity of Ψ is trivial. �
From Orthocomplementations to Locality 13
5 Separating locality relations and orthocomplementations
This section is dedicated to a class of locality relations on bounded lattices from which we build
orthocomplementations. We establish a one-one correspondence
{orthocomplementations} ←→ {strongly separating locality relations}
for bounded lattices. As we learned from referee reports on a previous version of this paper, this
one-one correspondence is actually known in lattice theory (see, e.g., [3]). We nevertheless use
a slightly different terminology which we find well-suited for the locality setup we have in mind.
5.1 Separating locality relations
A finite sublattice M = {a1, . . . , aN} of a lattice admits a greatest element a1 ∨ · · · ∨ aN .
In general a sublattice M of L does not have a greatest element, even if (L,≤,∧,∨) is bounded.
Counterexample 5.1. As a counterexample, take L = N ∪ {∞} with ≤ the usual order on
natural number and ∞ ≥ n, for any n ∈ N. Set n ∨m := max(n,m) and n ∧m := min(n,m).
Then (N,≤,∧,∨) is a sublattice of (L,≤,∧,∨), but does not admit a greatest element.
The following technical lemma will be useful for what follows.
Lemma 5.2. Let S be a subset of a locality poset (P,≤,>) (Definition 3.1) admitting a greatest
element grt (S). Then
grt (S)> = S>. (5.1)
Proof. Clearly, S> ⊆ grt (S)>. On the other hand, for any a ∈ S, from a ≤ grt (S) we obtain
grt (S)> ⊆ a>. Hence grt (S)> ⊆ ∩a∈Sa> = S>. �
The subsequent definition relates to that of weak and strong orthogonality introduced in [3,
Definition 2.2]. To compare our separating property with weak orthogonality, the following
elementary lemma is useful.
Lemma 5.3. In a locality lattice (L,≤,>, 0, 1), the non-degeneracy condition
a>a =⇒ a = 0 ∀ a ∈ L
(written ker> = 0 in [3]) is equivalent to the condition
a>b =⇒ a ∧ b = 0 ∀ (a, b) ∈ L2.
Proof. Taking b = a in the second condition gives the non-degeneracy condition. Conversely,
from a>b we have (a ∧ b)>(a ∧ b). Then the non-degeneracy condition gives a ∧ b = 0. �
By Proposition-Definition 3.1, on a poset with locality (P,≤) we have ↓ a ⊆
(
a>
)>
. Imposing
the converse inclusion leads to the following notion.
Definition 5.4. A locality relation > on a lattice (L,≤, 0, 1) is called separating if the following
conditions hold.
(a) a>b⇒ a ∧ b = 0 for any a and b in L,
(b) the set a> admits a greatest element grt
(
a>
)
for any a in L.
In this case, we say that (L,≤ 0, 1,>) is a separated locality lattice.
14 P. Clavier, L. Guo, S. Paycha and B. Zhang
Remark 5.5. It follows from Lemma 5.3, that a weak (non-degenerate) orthogonality lattice
of [3, Definition 2.2] satisfies the separating property.
Assumption (b) on the existence of grt
(
a>
)
which corresponds to completeness in [3, Defi-
nition 2.4] is rather strong.
Counterexample 5.6. The diamond lattice L = {0, a, b, c, 1} of Example 2.8, endowed with
the locality relation x>y ⇔ x ∧ y = 0, does not satisfy this assumption, since a> = {0, b, c},
b> = {0, a, c} and c> = {0, a, b}.
Proposition 5.7. In a separated locality lattice (L,≤,>, 0, 1), the following conditions are equi-
valent
a> = {0} ⇔ a = 1 ∀ a ∈ L, (5.2)
a⊕ grt
(
a>
)
= 1 ∀ a ∈ L (closedness condition). (5.3)
Assuming (5.3) holds, we have
a> = L⇔ a = 0 ∀ a ∈ L. (5.4)
Proof. We first prove (5.2) =⇒ (5.3). Since
(
a⊕ grt
(
a>
))> ⊆ a> ∩ (grt
(
a>
))>
= a> ∩
(
a>
)>
,
for c ∈ (a ⊕ grt (a>))> we have c>c and hence c ∧ c = c = 0 by (a) in Definition 5.4. Thus(
a⊕ grt
(
a>
))>
= {0} and a⊕ grt
(
a>
)
= 1 by assumption.
We next prove (5.2) ⇐= (5.3). From a> = {0} we have 1 = a⊕ grt
(
a>
)
= a⊕ 0 = a. From
a = 1, we have 0 = a ∧ grt
(
a>
)
= 1 ∧ grt
(
a>
)
= grt
(
a>
)
. Then a> = 0.
To prove (5.3) =⇒ (5.4), let us first notice that the implication a> = L ⇒ a = 0 holds as
a consequence of a>a, which in turn implies a ∧ a = a = 0 by (a) in Definition 5.4. It therefore
suffices to show that Assumption (5.3) implies 0> = L. This follows from 0 ⊕ grt
(
0>
)
= 1 ⇒
grt
(
0>
)
= 1. Thus 1 ∈ 0>. Since 0> is a poset ideal by Definition 3.4 and since b ≤ 1 ∀ b ∈ L,
we have b ∈ 0> ∀ b ∈ L. �
An easy counterexample shows that (5.4) does not imply (5.3).
Counterexample 5.8. We equip the bounded lattice L = {0, a, b, 1} defined by the partial
order 0 ≤ a ≤ 1 and 0 ≤ b ≤ 1 with the locality relation > defined by 0> = L, a> = b> = 1>.
Then clearly (5.4) holds but (5.3) does not.
Definition 5.9. A separating locality relation > on a lattice L is called strongly separating if
it satisfies
(c) grt
((
a>
)>)
= a for any a ∈ L,
(c′) or equivalently ↓ a ⊇
(
a>
)>
,
in which case the lattice endowed with the locality relation is called strongly separated. We also
say that (L,≤, 0, 1,>) is a strongly separated locality lattice.
Not all separating lattices are strongly separating as the next example shows.
Example 5.10. Let R3 be equipped with the canonical basis {e1, e2, e3}. We consider the
locality relation > on R3 defined by 0R3>R3, ei>ei+1 for i ∈ {1, 2} extended by symmetry and
linearity. By construction,
(
R3,>
)
is a locality vector space and > endows G
(
R3
)
with a lattice
locality structure.
In G
(
R3
)
, the only non trivial polar sets are 0> = G
(
R3
)
, 〈e1〉> = {0, 〈e2〉}, 〈e2〉> =
{0, 〈e1〉, 〈e3〉, 〈e1, e3〉}, 〈e3〉> = {0, 〈e2〉}, 〈e1, e3〉> = {0, 〈e2〉}. In particularly > is a separating
locality relation on G(V ). However it is not strongly separating since
grt
((
〈e1〉>
)>)
= grt
(
{0, 〈e2〉}>
)
= 〈e1, e3〉.
From Orthocomplementations to Locality 15
Remark 5.11.
(a) Under the additional assumption (b) of a separating locality, the mere strong separation
property (c) actually amounts to the strong orthogonality of [3, Definition 2.3].
(b) Note that the existence of grt
((
a>
)>)
in (c) follows from the existence of grt
(
a>
)
for
any a ∈ L. Indeed, we can then apply (5.1) grt (S)> = S> to S := a>, which yields
grt
(
a>
)>
=
(
a>
)>
.
Example 5.12. The locality relation in Example 3.3 is strongly separating. It is clearly sepa-
rating with grt
(
A>
)
= X\A. Then by Lemma 5.2,
grt
((
A>
)>)
= grt
(
grt
(
A>
)>)
= X\(X\A) = A.
Example 5.13. The locality lattice
(
G(V ),�, {0}, V,>
)
in Example 3.7, is strongly separated
by the same argument, noting that for any U ∈ G(V ), the linear space grt
(
U>
)
is the orthogonal
complement of U for the inner product.
Corollary 5.14. A strongly separated locality lattice (L,≤,>, 0, 1) satisfies the closedness con-
dition (5.3).
Proof. We show that a strongly separated locality lattices satisfies (5.2). To show the im-
plication from right to left, if b ∈ 1>, then b>1, so b>b which means b = 0, so 1> = {0}.
The converse implication follows from (c) in Definition 5.9. Indeed, since 0> = L, we have
a> = {0} ⇒
(
a>
)>
= L which implies that a = grtL = 1. �
The subsequent example shows that the separating property is not hereditary.
Example 5.15. Consider the lattice L = {0, a, b, 1}, 0 ≤ a ≤ 1 and 0 ≤ b ≤ 1 and no relation
between a and b. On L endowed with the disjointedness locality relation > := >∧, we have
0> = L, 1> = 0, a> = {0, b} and b> = {0, a}. So L is (strongly) separating, but not its
restriction to L̃ = {0, a, 1}, since a> = {0}.
5.2 One-one correspondence
It turns out to be a specialisation of [3, Theorem 3.1], which relates weak degenerate orthog-
onalities and weak degenerate orthocomplementations on posets, to a one-one correspondence
between strong (non degenerate) orthogonalities and strong (non degenerate) orthocomplemen-
tations on bounded lattices. The latter is what we need in view of the applications we have
in mind.
Proposition 5.7 provides sufficient conditions on a locality poset for the existence of a can-
didate orthocomplement to any element a, namely grt
(
a>
)
arising in (5.3). The subsequent
statement confirms and strenghtens this fact.
Theorem 5.16. Let (L,≤, 0, 1) be a bounded lattice.
(a) Let > be a locality relation on the set L for which (L,≤, 0, 1,>) is a lattice with strongly
separated locality. The assignment
Ψ := Ψ> : L→ L, a 7→ grt
(
a>
)
is an orthocomplementation.
16 P. Clavier, L. Guo, S. Paycha and B. Zhang
(b) Conversely, given an orthocomplementation Ψ on L, the locality relation > := >Ψ defi-
ned by
a>b⇐⇒ b ≤ Ψ(a)
yields a strongly separating locality relation.
(c) The maps defined this way:
F : {strongly separating locality relations on L} → {orthocomplementations on L}
and
G : {orthocomplementations on L} → {strongly separating locality relations on L}
are inverse to each other.
Proof. (a) It follows from Proposition 5.7, that on a lattice (L,>) with strongly separated
locality, the map a 7→ Ψ>(a) := grt (a>), which is well-defined, satisfies a ⊕ Ψ>(a) = 1. This
is (4.1) in Proposition 4.13. To show that Ψ is an orthocomplementation, we need to check that
it is involutive and isotone. From a ≤ b we have b> ⊆ a> which yields grt
(
b>
)
≤ grt
(
a>
)
.
So Ψ> is antitone. Further we have(
Ψ>(a)
)>
= grt
(
a>
)>
=
(
a>
)>
as a consequence of (5.1) applied to S = a>. So
Ψ>
(
Ψ>(a)
)
= grt
(
Ψ>(a)
)>
= grt
(
a>
)>
.
The strongly separating condition (c′) in Definition 5.9 tells us that grt
((
a>
)>)
= a so that
Ψ>
(
Ψ>(a)
)
= a, which ends the proof of the involutivity.
(b) Let Ψ: L→ L be an orthocomplementation. We show that
a>b⇐⇒ b ≤ Ψ(a)
is a strongly separating locality relation on L in several steps. Note that > is equivalently defined
by a> :=↓ Ψ(a) for all a ∈ L.
(1) > is symmetric since the monotonicity and involutivity of Ψ yield that if b ≤ Ψ(a), then
Ψ(b) ≥ Ψ(Ψ(a)) = a.
(2) > equips the poset (L,≤) with a locality lattice structure (see Definition 3.4) since a> :=↓
Ψ(a) is a (principal) lattice ideal (see Definition 2.4).
(3) We have 0> =↓ Ψ(0) =↓ 1 = L and 1> =↓ Ψ(1) =↓ 0 = {0}.
(4) By definition, a ∧ Ψ(a) = 0. Now if a>b, then b ≤ Ψ(a), so a ∧ b = 0. Furthermore,
a> =↓ Ψ(a) means that grt
(
a>
)
= Ψ(a). Hence (L,≤, 0, 1,>) is a lattice with separated
locality.
(5) Finally,
grt
((
a>
)>)
= grt
(
(↓ Ψ(a))>
)
= grt
(
Ψ(a)>
)
= grt
(
↓ Ψ2(a)
)
= grt (↓ a) = a.
This proves that > is strongly separating.
From Orthocomplementations to Locality 17
(c) For a lattice L strongly separating locality relation >, by definition
(a, b) ∈ G(F (>))⇔ b ≤ F (>)(a)⇔ b ≤ grt
(
a>
)
⇔ (a, b) ∈ >.
So
G(F (>)) = >.
For an orthocomplementation Ψ on L,
F (G(Ψ))(a) = grt
(
aG(Ψ)
)
= Ψ(a).
Hence
F (G(Ψ)) = Ψ. �
Combining Corollary 4.15 with Theorem 5.16 yields the following characterisation of strongly
locality relations on G
(
R2
)
.
Corollary 5.17. Strongly separating locality relations on R2 are in one-one correspondence with
involutive maps ψ : [0, π)→ [0, π) without fixed points.
6 Locality relations on vector spaces
In the previous section, we established a one-one correspondence between the set of strongly
separating locality relations on a bounded lattice and the set of orthocomplementations on the
lattice. In this section, we apply the correspondence to the subspace lattice of a finite dimensional
vector space, a case of direct interest for the applications to renormalisation we have in mind,
as explained in the introduction.
6.1 Correspondence between locality relations
From a set locality >V on a vector space V , we build a poset locality relation >G := >V,G on the
lattice
(
G(V ),�
)
:
U>V,GW if u>V w ∀u ∈ U, w ∈W. (6.1)
On the other hand, a set locality relation >G on G(V ) (ignoring its lattice structure) induces
a locality relation >V := >G,V on V by
v1>G,V v2 if Kv1>GKv2 ∀ v1, v2 ∈ V. (6.2)
As in [6], we call locality vector space, a vector space V equipped with a (set) locality rela-
tion > such that the polar set X> of any subset X ⊆ V is a vector subspace of V .
Proposition 6.1. Let LVR(V ) denote the set of vector space locality relations >V on V and
let LGR(V ) denote the set of lattice locality relations >G (see Definition 3.4) on the set G(V )
such that {0}>G = G(V ). The equations (6.1) and (6.2) gives a one-one correspondence between
LVR(V ) and LGR(V ).
Proof. Let >V be a vector locality relation on V . Then for any W ∈ G(V ), W>V is a subspace
and hence contains 0. Thus we have {0}>V,G = G(V ). Clearly, for any U ∈ G(V ),
U>V >V,G U
18 P. Clavier, L. Guo, S. Paycha and B. Zhang
and, for W ∈ G(V ), if W>V,G U , then W ⊆ U>V . Therefore,
U>V,G =↓
(
U>V
)
, (6.3)
which is a lattice ideal. Hence >V,G is in LGR(V ).
Conversely, give a locality relation >G ∈ LGR(V ), then {0}>G = G(V ). So for any v ∈ V ,
{0}>GKv, yielding 0>G,V v. Next let X ⊆ V . Then the lattice ideal (KX)>G has a greatest
element grt (KX)>G =
∑
W>GKXW . If W>GKX, then W ⊆ X>G,V , that is grt (KX)>G ⊆
X>G,V . On the other hand, if y ∈ X>G,V , then Ky>GKX, so y ∈ Ky ⊆ grt (KX)>G . Thus
X>G,V = grt (KX)>G (6.4)
is a subspace. That is, >G,V ∈ LGR(V ).
Thus we obtain maps
φ : LVR(V ) −→ LGR(V ), >V 7→ >V,G,
ψ : LGR(V ) −→ LVR(V ), >G 7→ >G,V .
We set
>V,G,V := ψ(φ(>V )) = ψ(>V,G).
Then for X ⊆ V , by (6.3) and (6.4) we have
X>V,G,V = grt
(
(KX)>V,G
)
= grt
(
↓ (KX)>V
)
= (KX)>V = X>V .
Similarly, let us set
>G,V,G := φ(ψ(>G)) = ψ(>G,V ).
For a principal poset ideal I =↓ a, we have ↓ grt (I) =↓ grt (↓ a) =↓ a = I. Thus for U ∈ G(V ),
by (6.3) and (6.4) we obtain
U>G,V,G =↓
(
U>G,V
)
=↓
(
grt
(
U>G
))
= U>G .
We have proved that the maps φ and ψ are mutual inverses. This proves the proposition. �
6.2 Non-degenerate locality relations on vector spaces
Definition 6.2. A locality relation > on a vector space V is called non-degenerate if v>v ⇒
v = 0 for any v ∈ V , it is called strongly non-degenerate if moreover for any subspace U $ V ,
the polar space U> is nonzero.
Lemma 1. In a strongly non-degenerate locality vector space (V,>V ), for any U 5 V ,
V = U ⊕ U>V .
Proof. By strong non-degeneracy, we have
V = U + U>V ,
otherwise we can find 0 6= v ∈
(
U+U>V
)>V which means v ∈ U>V and v ∈
(
U>V
)>V , so v>V v.
Then v = 0, a contradiction. Further, the non-degeneracy gives
U ∩ U>V = {0},
yielding the conclusion. �
From Orthocomplementations to Locality 19
Proposition 6.3. A locality >V on a vector space V is
(a) non-degenerate if and only if (G(V ),>V,G) is a locality lattice which has the separating
property: U>V,G W ⇒ U ∩W = 0,
(b) strongly non-degenerate if and only if (G(V ),>V,G) is a strongly separating locality lattice.
In this case, the map U 7→ U>V defines a orthocomplementation on G(V ).
Proof. (a) Let (V,>V ) be a non-degenerate locality vector space. Then by Proposition 6.1,
>V,G equips G(V ) with a locality lattice. If U>V,GW , then for v ∈ U ∩W , v>V v, so v = 0.
Thus U ∩W = {0}.
Now suppose that (G(V ),>V,G) is a locality lattice with the separating property. We already
know (V,>V ) is a locality vector space. If v>V v, then Kv>V,GKv. So from the separating
condition we have Kv = Kv ∩Kv = {0}. So v = 0, which means (V,>V ) is non-degenerate.
(b) If (V,>V ) is a strongly non-degenerate locality vector space, then for any U ⊆ V , by (6.4),
grt
(
U>V,G
)
= U>V . Further, we have U ∈
(
U>V,G
)>V,G by definition. By non-degeneracy of
>V , V = U ⊕ U>V and U =
(
U>V
)>V . Thus
(
U>V,G
)>V,G =
(
grt
(
U>V,G
))>V,G =
(
U>V
)>V,G
and hence
grt
(
U>V,G
)>V,G = grt
(
U>V
)>V,G =
(
U>V
)>V = U.
Now assume that (G(V ),>V,G) is a strongly separating locality lattice. If U $ V , then
U>V,G 6= {0}, otherwise
(
U>V,G
)>V,G = G(V ), we have U = V by taking the greatest element.
Now take 0 6= w ∈ U>V,G . Then Kw ≤ U>V,G . Hence Kw>V,GU , and Kw>V,GKu for any u ∈ U ,
that is, w ∈ U>V . Thus U>V 6= {0}.
The last assertion is a consequence of Theorem 5.16. �
6.3 Locality bases
By Proposition 6.3, we can construct locality relations on subspace lattice from locality relations
on the underlying vector space so that we now focus on vector spaces. In much the same way
as a Euclidean vector space can be equipped with an orthogonal basis, we show that a vector
space with a strongly separating locality relation possesses a basis that is compatible with the
relation. However, the relation is not uniquely determined by this basis.
Definition 6.4. For locality vector space (V,>), a basis B = {eα}α∈Γ of V is called a locality
basis for > if the basis vectors are mutually independent for >, i.e.,
if α 6= β then eα>eβ.
We now study locality relations on a vector space V with a countable basis. We start with
a generalisation to arbitrary strongly separating locality relations (instead of orthogonality) of
[19, Chapter II, Theorem 1].
Proposition 6.5. Let (V,>) be a strongly separating locality vector space of countable dimen-
sion. Then V has a locality basis for >.
20 P. Clavier, L. Guo, S. Paycha and B. Zhang
Proof. This process is similar to the Gram–Schmidt process in linear algebra. By assumption,
V admits a countable basis B = {en |n ∈ N}. We apply the induction on n to construct a locality
basis {u1, . . . , un} for the subspace Wn of V spanned by {e1, . . . , en}.
First take u1 = e1 and then K{u1} = K{e1}. Assume that a locality basis {u1, . . . , un}
of Wn := K{e1, . . . , en} has been constructed. Let Ψ> be the polar map induced by >:
Ψ>(W ) := grt
(
W>
)
. By Example 2.15, G(V ) is modular. So we have Wn+1 = Wn⊕
(
Ψ>(Wn)∩
Wn+1
)
. Let un+1 be a nonzero element in Ψ>(Wn). Then {u1, . . . , un+1} is a locality basis
of Wn+1. This completes the induction. Then {ui}i≥1 is a locality basis of V . �
We now provide an example that shows that, unlike a basis which uniquely determines a vector
space, a locality basis is not enough to determine the locality vector space, suggesting the richness
of locality relations on a vector space.
Remark 6.6. A strongly non-degenerate locality relation on R2 ' C therefore has infinitely
many locality bases
{
eθi, eψ(θ)i
}
parametrised by θ ∈ [0, π).
Proof. By Proposition 6.3, strongly non-degenerate locality relations on R2 are uniquely deter-
mined by strongly separating relations on G
(
R2
)
. By Corollary 5.17, these in turn are in
one-one correspondence with involutive maps ψ : [0, π) → [0, π) without fixed points, which
give a strongly separating orthocomplement Ψ: Reθi 7→ Reψ(θ)i. Thus strongly non-degenerate
locality relations on R2 are determined by involutive maps ψ : [0, π) → [0, π) without fixed
points. �
A An alternative correspondence
Following the suggestion by a referee, we present here another class of symmetric binary relations
which can be put in one-one correspondence with a weak form of orthocomplementations. This is
carried out on the class of complete atomistic lattices to give a one-one correspondence between
a class of locality lattice relations and certain locality relations on the set of atoms of L.
Definition A.1. Let L be a lattice bounded from below by 0.
(a) An atom in L is an element p in L that is minimal among the non-zero elements, i.e., such
that for x ∈ L, x < p if and only if x = 0.
(b) L is an atomic lattice whose elements a are all bounded from below by some atom, i.e.,
there is an atom p in L such that p ≤ a.
(c) A lattice is atomistic if it is atomic and every element is a join of some finite set of atoms.
Example A.2. The lattice of divisors of 4, with the partial ordering “is divisor of”, is atomic,
with 2 being the only atom. It is not atomistic, since 4 cannot be obtained as least common
multiple of atoms.
Definition A.3. A lattice is complete if all subsets have both a supremum (join) and an infimum
(meet).
Example A.4. Every non-empty finite lattice is complete.
Example A.5. Let V be a finite dimensional vector space. The subspace lattice G(V ) conside-
red previously is an atomistic complete lattice with atom set the set P (V ) := {Kv, v ∈ V \ {0}}
of one dimensional subspaces of V .
We first give a useful preparation lemma on posets.
From Orthocomplementations to Locality 21
Lemma A.6. Given a poset (L,≤) bounded from below by 0, there is a one-one correspondence
between
(1) poset locality relations > on L (cf. Definition 3.1) such that 0>x for all x ∈ L with the
separating property: for any x ∈ L, the set x> admits a greatest element grt
(
x>
)
.
(2) antitone maps Ψ: L→ L, x 7→ x′ (i.e., x ≤ y ⇒ y′ ≤ x′) such that x ≤ (x′)′.3
The correspondence is given by (cf. Theorem 5.16)
� Ψ>(x) := x′ := grt
(
x>
)
,
� x>y if and only if y ≤ x′.
Moreover, let Ψ be an antitone as in (2). For any subset X ⊂ L, if the supremum supX of X
in L exists, then so does the infimum inf Y of the set Y = {x′ |x ∈ X}, and (supX)′ = inf Y .
Proof. Given a poset locality > on L in (1) and let Ψ>(x) := grt
(
x>
)
. Then from x ≤ y,
we obtain x> ⊇ y> and hence grt
(
x>
)
≥ grt
(
y>
)
. Further, since x>grt
(
x>
)
, we have x ∈(
grt
(
x>
))>
and thus x ≤ grt
((
grt
(
x>
))>)
.
Given an antitone map Ψ: L → L in (2) and define x>Ψy if y ≤ x′ = Ψ(x). The symmetry
of > follows from the antitone property: y ≤ x′ implies x ≤ (x′)′ ≤ y′. Further 0>Ψ x for
any x ∈ L since 0 is the smallest element. It also follows from the definition of >Ψ that
grt
(
x>Ψ
)
= Ψ(x).
Now let Ψ be an antitone as in (2) and X ⊆ L. If a = supX exists, then x′ ≥ a′ for all x ∈ X
and hence a′ is a lower bound of Y . Further, for any lower bound z of Y , we have z ≤ x′ for all
x ∈ X. Then x ≤ (x′)′ ≤ z′. Thus a = supX ≤ z′ and then z ≤ a′. Therefore a′ = inf Y . �
Remark A.7. If a poset (L,≤) is a lattice, then a poset locality relation on L satisfying (1)
is also a lattice locality relation, since the separating property means that x> =↓ grt
(
x>
)
and
hence is a lattice ideal.
Proposition A.8. Given an atomistic complete lattice L with set P of atoms. For a ∈ L, define
Pa := {p ∈ P | p ≤ a} =↓ a ∩ P .
There is a one-one correspondence between lattice locality relations >L on L as in (1) of
Lemma A.6 and
(3) symmetric binary relations >P on P such that for any p ∈ P there is (unique) a ∈ L such
that p>P = Pa.
The correspondence is given by
� >P is the restriction of >L to P ,
� >L is defined in terms of >P by: a>Lb if and only if
(a) a = 0 or b = 0, or
(b) a 6= 0, b 6= 0, and p>P q for all p, q ∈ P with p ≤ a and q ≤ b.
Proof. Given >L as in (1) and let >P be as defined. For any p ∈ P , p>L has the greatest
element a := grt
(
p>L
)
. Then p>L =↓ a. Then p>P = p>L ∩ P =↓ a ∩ P = Pa. Thus >P
satisfies (3).
Given >P as in (3) and let >L be as defined. Then >L is obviously symmetric. Also 0 ∈ L>L
by definition.
3Note that this is implied by the involutivity required of an orthocomplementation.
22 P. Clavier, L. Guo, S. Paycha and B. Zhang
Now we prove that a>L is a lattice ideal for any a ∈ L. If a = 0, then 0>L = L which is
a lattice ideal. First consider a = p ∈ P . Then p>P = Pb for some b ∈ L. We just need to show
p>L =↓ b.
First let c ∈↓ b, that is, c ≤ b. Then for any q ≤ c with q ∈ P , we have q ∈ Pb. So q>P p,
thus c>L p, that is, c ∈ p>L . This proves ↓ b ⊆ p>L . On the other hand, let c ∈ p>L , that is,
c>L p. Then for any q ≤ c with q ∈ P , we have q>P p. Thus q ∈ Pb, that is, q ∈↓ b, implying
c ∈↓ b since L is atomistic. This proves p>L ⊆↓ b.
As a consequence, we have p>Lb and p>Lc implying p>L(b ∨ c).
We now show that a>L is closed under ∨. For a>Lb, a>Lc and p ∈ P with p ≤ a, we have
p>Lb and p>L c. Thus p>L (b ∨ c) and hence p>P q for any q ∈ P with q ≤ b ∨ c. Therefore
a>L(b∨ c). Further, if b ∈ a>L and c ≤ b, then for p ∈ Pa and q ∈ Pc, we have q ∈ Pb and hence
p>P q. This gives c ∈ a>L . Thus a>L is also a poset ideal and hence a lattice ideal.
We finally show
a>L =↓
(
∨
{
p ∈ P | p ∈ a>L
})
.
The right hand side is defined since L is complete. The inclusion ⊇ holds since a>L is a lattice
ideal. Further, for b ∈ a>L , write b = q1 ∨ · · · ∨ qk with q1, . . . , qk ∈ P . Since q1, . . . , qk ∈ a>L ,
we have b = q1 ∨ · · · ∨ qk ≤ ∨
{
p ∈ P | p ∈ a>L
}
. This completes the proof of the equality.
In summary, the locality relation >L from >P satisfies (1).
The one-one correspondence results from the assignment
>P 7→ >L := >P,L 7→ >P,L,P ,
which is clearly the identity since >P,L extends >P and the assignment
>L 7→ >P := >L,P 7→ >L,P,L,
using the conditions on TL. �
Remark A.9. Applying the above proposition to the subspace lattice L := G(V ) of a finite
dimensional K-vector space V whose atom set P (V ) = {Kv | v ∈ V \ {0}} consists of one
dimensional subspaces gives another proof of Proposition 6.1.
Acknowledgements
This work is supported by Natural Science Foundation of China (Grant Nos. 11771190, 11821001,
11890663). The first and third authors are grateful to the Perimeter Institute where part of this
paper was written. They also thank Tobias Fritz for inspiring discussions and the third author
is grateful to Daniel Bennequin for his very useful comments at a preliminary stage of the prepa-
ration of the paper. We greatly appreciate very helpful suggestions of the referees on previous
versions of the paper.
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1 Introduction
1.1 The general setup and our aims
1.2 Locality and complements in renormalisation
1.3 Plan of the paper
2 Modular lattices
2.1 Lattices
2.2 Distributivity and modularity
3 Locality relations on lattices
4 Orthocomplemented lattices
4.1 Relatively complemented lattices
4.2 Orthocomplemented lattices
5 Separating locality relations and orthocomplementations
5.1 Separating locality relations
5.2 One-one correspondence
6 Locality relations on vector spaces
6.1 Correspondence between locality relations
6.2 Non-degenerate locality relations on vector spaces
6.3 Locality bases
A An alternative correspondence
References
|
| id | nasplib_isofts_kiev_ua-123456789-211322 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-18T05:01:35Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Clavier, Pierre Guo, Li Paycha, Sylvie Zhang, Bin 2025-12-29T11:11:28Z 2021 From Orthocomplementations to Locality. Pierre Clavier, Li Guo, Sylvie Paycha and Bin Zhang. SIGMA 17 (2021), 027, 23 pages 1815-0659 2020 Mathematics Subject Classification: 06C15; 08A55; 81T15; 15A63 arXiv:2007.03003 https://nasplib.isofts.kiev.ua/handle/123456789/211322 https://doi.org/10.3842/SIGMA.2021.027 After some background on lattices, the locality framework introduced in earlier work by the authors is extended to cover posets and lattices. We then extend the correspondence between Euclidean structures on vector spaces and orthogonal complementations to a one-to-one correspondence between a class of locality structures and orthocomplementations on bounded lattices. This recasts in the context of renormalisation classical results in lattice theory. This work is supported by the Natural Science Foundation of China (Grant Nos. 11771190, 11821001, 11890663). The first and third authors are grateful to the Perimeter Institute, where part of this paper was written. They also thank Tobias Fritz for inspiring discussions, and the third author is grateful to Daniel Bennequin for his very useful comments at a preliminary stage of the preparation of the paper. We greatly appreciate the very helpful suggestions of the referees on previous versions of the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications From Orthocomplementations to Locality Article published earlier |
| spellingShingle | From Orthocomplementations to Locality Clavier, Pierre Guo, Li Paycha, Sylvie Zhang, Bin |
| title | From Orthocomplementations to Locality |
| title_full | From Orthocomplementations to Locality |
| title_fullStr | From Orthocomplementations to Locality |
| title_full_unstemmed | From Orthocomplementations to Locality |
| title_short | From Orthocomplementations to Locality |
| title_sort | from orthocomplementations to locality |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211322 |
| work_keys_str_mv | AT clavierpierre fromorthocomplementationstolocality AT guoli fromorthocomplementationstolocality AT paychasylvie fromorthocomplementationstolocality AT zhangbin fromorthocomplementationstolocality |