Monoidal Categories for Formal Concept Analysis

Abstract

We investigate monoidal categories of formal contexts, in which states correspond to formal concepts. In particular we examine the category of bonds or Chu correspondences between contexts, which is known to be equivalent to the *-autonomous category of complete sup-lattices. We show that a second monoidal structure exists on both categories, corresponding to the direct product of formal contexts defined by Ganter and Wille, and discuss the use of these categories as compositional models of meaning.

Full text

arXiv:2012.08268v1 [math.CT] 15 Dec 2020
Monoidal Categories for Formal Concept Analysis
Sean Tull
Cambridge Quantum Computing
sean.tull@cambridgequantum.com
Abstract
We investigate monoidal categories of formal contexts in which states
correspond to formal concepts. In particular we examine the category
of bonds or Chu correspondences between contexts, which is known to
be equivalent to the *-autonomous category of complete sup-lattices. We
show that a second monoidal structure exists on both categories, corre-
sponding to the direct product of formal contexts defined by Ganter and
Wille, and discuss the use of these categories as compositional models of
meaning.
Introduction
Formal concept analysis (FCA) is a highly successful framework for reasoning
about collections of ob jects and their properties, initiated by Wille [Wil92].
Starting from a system described by a formal context of objects and the at-
tributes these attain, the central feature of FCA is the construction of its hier-
archy of formal concepts, which form a complete lattice known as the concept
lattice. FCA has found many successful applications in semantics, including data
mining, machine learning, the semantic web, and linguistics [GW99, GSW05].
A more recently developed framework is that of Categorical Distributional
Compositional Models of Meaning (DisCo), initiated by Coecke, Clark and
Sadrzadeh [CSC10]. Typically drawing on Lambek’s theory of pregroup gram-
mars [Lam08], this provides a structured recipe for deriving the meaning of a
sentence in terms of the meaning of its individual words when these exist in an
autonomous category. More generally, Delpeuch has extended the framework to
any monoidal category [Del19].
Though vector spaces are most commonly used, more semantic categories
have recently been explored in the DisCo framework, including the use of density
matrices for word meanings [BSC15], and convex relational spaces modelling
Gardenfors’ framework of conceptual spaces [G¨ar04, BCG+19].
In this work, we investigate monoidal categories of formal contexts, to serve
as new models of meaning in frameworks for compositional semantics such as
DisCo. Conversely, one may hope that category theory may provide new tools
for FCA, as argued by Mori [Mor08] and Pavlovic [Pav12, Pav20].
1

Since word meanings in the DisCo formalism are represented by states, we
wish to consider categories whose objects are formal contexts and states corre-
spond to their formal concepts. However, beyond this there is freedom in both
our choice of morphism and tensor product of formal contexts, and several have
been proposed for each [Wil85, GW99, KHZ05, Mor08, Ern14].
The morphisms we consider are equivalent to the notion of bond between for-
mal contexts introduced by Ganter and Wille [GW99], which Mori has studied
in detail via the equivalent notion of Chu correspondence [Mor08], and which we
show also coincide with the morphisms of contexts studied by Moshier [Mos16].
These form our category of interest Cxt. Taking the concept lattice is known to
provide an equivalence of categories between Cxt and the category SupLat of
complete sup-lattices. Since the latter is known to have a *-autonomous struc-
ture given by the tensor product ⊠of sup-lattices, this yields the *-autonomous
monoidal structure (Cxt,⊠) described by Mori in [Mor08].
However, one may prefer a tensor structure motivated by formal concepts
themselves, rather than lattices. Wille has in fact introduced a notion of di-
rect product of formal contexts [Wil85]. Here we show these provide an al-
ternative symmetric monoidal structure (Cxt,⊗). Wille has also shown the
direct product to correspond to an alternative tensor product ⊗of complete
lattices. We extend Wille’s results to merely sup-complete homomorphisms,
to show that this tensor in fact provides a second symmetric monoidal struc-
ture (SupLat,⊗) which makes the concept lattice a monoidal equivalence. In
summary then, for each of the corresponding tensors ⊗ ∈ {⊗,⊠}on Cxt and
SupLat, taking concept lattices provides an equivalence of symmetric monoidal
categories (Cxt,⊗)≃(SupLat,⊗). Here we briefly discuss the potential use
of each monoidal structure in the DisCo framework, which would be desirable
to explore in future work.
Outline In Section 1 we introduce the basics of formal concept analysis.
In Section 2 we describe the category Cxt of formal contexts, giving several
equivalent definitions of its morphisms. Section 3 introduces two symmetric
monoidal structures on Cxt. In Section 4 we describe the monoidal equiva-
lences Cxt ≃SupLat for two corresponding tensors of sup-lattices. Finally in
Section 5 we describe applications to the DisCo framework.
Related work Our definition of Cxt essentially comes from the ‘continuous
extent correspondences’ of the article [Mor08] where the equivalent category
of Chu correspondences and its relation with bonds and sup-lattices, and the
*-autonomous structure ⊠, are studied. Section 2 provides an alternative pre-
sentation of this category, and a new equivalence with the category of [Mos16].
Our main new results are the definition of the concept tensors ⊗on Cxt and
SupLat in Sections 3 and 4.2.
2

1 Formal Concept Analysis
Let us now introduce the basic ingredients of Formal Concept Analysis (FCA).
Throughout we follow the presentation of [GW99].
Definition 1. Aformal context is a tuple
K= (G, M, |=)
consisting of a set Gof objects, a set Mof attributes, and a relation |=⊆G×M.
For each g∈Gand m∈M, whenever |= (g, m) we instead write g|=mand
say that the object ghas the attribute m.
More generally, for any such context K, for any subsets A⊆Gand B⊆M
we write A|=Bwhenever a|=bfor all a∈Aand b∈B. We define
A′:= {m∈M|a|=m∀a∈A} ⊆ M
B′:= {g∈G|g|=b∀b∈B} ⊆ G
We then have A⊆B′⇐⇒ B⊆A′. This means that the mappings A7→ A′
and B7→ B′form a Galois connection between the partially ordered sets P(G)
and P(M), or in other words an adjunction
P(G)op ⊥P(M)
(−)′
(−)′
As a result we obtain (idempotent and order-preserving) closure operators on
P(G) and P(M) given by A7→ A:= A′′ and B7→ B:= B′′. For each subset A
of Gwe call Athe closure of A, and say Ais closed when A=A, and similarly
for B⊆M. For any g∈Gwe define g′={g}′and g={g}, and similarly for
m∈M. We may now define concepts themselves.
Definition 2. A(formal) concept of a context Kis a pair (A, B) where A⊆G
and B⊆M, satisfying A=B′and B=A′. We call Athe extent and Bthe
intent of the concept, respectively.
By definition, the extent of a concept is precisely the set of all objects which
satisfy all the attributes of its intent. Conversely its intent describes precisely
the attributes these objects all share. We can define an ordering on concepts by
(A1, B1)≤(A2, B2) : ⇐⇒ A1≤A2(⇐⇒ B2≤B1)
The key result of FCA is now the following.
Theorem 3 (Basic theorem of formal concept analysis).[Wil92] For any
context K, the set of concepts B(K) forms a complete lattice under ≤, with
^
i∈I
(Ai, Bi) = \
i∈I
Ai,[
i∈I
Bi!
_
i∈I
(Ai, Bi) = [
i∈I
Ai,\
i∈I
Bi!
3

Example 4. A formal context is typically depicted in terms of the cross-table
of the relation |=, and the corresponding Hasse diagram of its concept lattice,
such as the following.
juvenile
canine
feline
mature
cat dog kitten puppy
Mature Feline Canine Juvenile
Cat X X
Dog X X
Kitten X X
Puppy X X
Here G={Cat,Dog,Kitten,Puppy}while M={Mature,Feline,Canine,Juvenile}.
Example 5. Any set Adetermines a formal context
A:= (A, A, 6=)
Here every subset B⊆Ais closed, with B′=A\B, so that B(A)≃P(A), the
power set of A. In particular we define the trivial context to be I:= {⋆}.
Example 6. For any partially ordered set Pwe can define a formal context
F(P) := (P, P, ≤)
The lattice B(F(P)) is the smallest complete lattice in which Pcan be order-
embedded, known as the Dedekind-MacNeille completion of P[GW99, p 48]. In
particular, when Vis a complete lattice we have an isomorphism V≃B(F(V)).
Thus every complete lattice arises as a concept lattice.
Example 7. Any Hilbert space Hdetermines a formal context
(H,H,⊥)
where ⊥is its orthogonality relation. The concept lattice of this context is
isomorphic to the orthomodular lattice of subspaces V≤ H, via V7→ (V , V ⊥).
1.1 Notation
We will shortly describe morphisms of formal contexts based on relations and so
fix some conventions about these. For any sets A, B and any relation R:A→B,
meaning a subset R⊆A×B, we denote the converse relation by R†:B→A.
For each subset X⊆Awe set
R(X) := {b∈B|(∃x∈X)R(x, b)}
We will often equate Rwith its induced mapping A→P(B), and so define
Rby specifying the subsets R(a) := R({a})⊆Bfor each a∈A. The map
X7→ R(X) has an adjoint R•:P(B)→P(A) given by
R•(Y) := {a∈A|R(a)⊆Y}
for each Y⊆B. Finally, we also define a map R•:P(A)→P(B) by
R•(X) := {b∈B|(∀x∈X)R(x, b)}(1)
for each X⊆A.
4

2 A Category of Formal Contexts
We now introduce morphisms of contexts. In fact we will give four equivalent
ways of describing such a morphism, with most of the results of this section
being essentially due to Mori who studied these maps in [Mor08]. Throughout,
let K1= (G1, M1,|=), K2= (G2, M2,|=), ... be contexts.
Definition 8. In the category Cxt, the objects are formal contexts Kand the
morphisms K1→K2are relations R:G1→G2which are closed, meaning that
1. R(g)⊆G2is closed, for all g∈G1;
2. R•:P(G2)→P(G1) preserves closed sets.
The composition of R:K1→K2and S:K2→K3is defined by
(S◦R)(g) := (S(R(g)) (∀g∈G1)
The identity morphism on Kis the relation g7→ gfor all g∈G.
To establish that Cxt is a valid category, we will use the following.
Lemma 9. A relation R:G1→G2is closed iff R(g)⊆G2is closed for all
g∈G1, and for all subsets A⊆G1we have
R(A) = R(A) (2)
In fact for all R:K1→K2and S:K2→K3in Cxt and A⊆G1we have
(S◦R)(A) = S(R(A)) (3)
Proof. For the first point, note that for any closed relation R:G1→G2and
A⊆G1,B⊆M2we have
R(A)|=B⇐⇒ R(A)⊆B′
⇐⇒ A⊆R•(B′)
⇐⇒ A⊆R•(B′)⇐⇒ R(A)|=B
using that R•(B′) is closed since B′is. It follows that (2) holds. Conversely if
this is the case then for all A⊆G1and B⊆M2we have
A⊆R•(B′)⇐⇒ R(A)⊆B′
⇐⇒ R(A)⊆B′
⇐⇒ R(A)⊆B′⇐⇒ A⊆R•(B′)
using that B′is closed in the second step, and so R•(B′) is closed as required.
For (3) note that by definition
(S◦R)(A) = [
a∈A
S(R(a)) = [
a∈A
S(R(a)) = S(R(A))
using that in any context closure operators satisfy Si∈IAi=Si∈IAi.
5

Corollary 10. Cxt is a well-defined category.
Proof. That composite of R:K1→K2and S:K2→K3is indeed a closed
relation by Lemma 9 since for all A⊆G1we have
(S◦R)(A) = S(R(A) = S(R(A)) = S(R(A)) = S(R(A)) = (S◦R)(A)
We further have R◦id = Rsince R(g) = R(g) = R(g) for all g∈Gand
id ◦R=Ragain follows from Lemma 9.
Though our definition of morphism refers only to objects, and not attributes,
we see shortly that each closed relation R:G1→G2is equivalently described by
another R∗:M2→M1in the opposite direction, related to Rin the following
manner studied by Mori.
Definition 11. [Mor08] A Chu correspondence (R, S) : K1→K2is a pair of
relations R:G1→G2and S:M2→M1for which each of the sets R(g1)⊆G2
and S(m2)⊆M1are closed and we have
R(g1)|=m2⇐⇒ g1|=S(m2)
for all g1∈G1,m1∈M2.
Another notion of morphism of contexts was put forward by Ganter and
Wille directly in the context of formal concept analysis [GW99].
Definition 12. Abond is a relation B:G1→M2for which B(g1)⊆M2and
B†(m2)⊆G1are closed, for all g1∈G1and m2∈M2.
Thus a bond is simply a relation from G1to M2whose rows and columns are
closed. We can now show that all of these notions of morphism are equivalent.
Proposition 13. For any contexts K1,K2there are bijections between:
1. Closed relations R:G1→G2;
2. Closed relations R∗:M2→M1;
3. Chu correspondences (R, R∗) : K1→K2;
4. Bonds B:G1→M2;
given by
R(g1) = R∗•(g′
1)′=B(g1)′R∗(m2) = R•(m′
2)′B(g1) = R(g1)′(4)
for each g1∈G1and m2∈M2.
6

Proof. 1⇐⇒ 2⇐⇒ 3. For any closed relation R:G1→G2define R∗:M2→
M1as above. Since each set R•(m′
2) is closed we have R•(m′
2) = R•(m′
2)′′ =
R∗(m2)′and so
R(g1)|=m2⇐⇒ R(g1)⊆m′
2
⇐⇒ g1∈R•(m′
2) = R∗(m2)′
⇐⇒ g1|=R∗(m2)
and so (R, R∗) is a Chu correspondence.
Conversely, let (R, S ) be any Chu correspondence. Then it is easy to see
that R•(B′) = S•(B)′for all B⊆M2. Then if A⊆G2is closed we have
R•(A) = R•(A′′) = S(A′)′
and so R•(A) is closed. Hence Ris a closed relation (and similarly so is S).
We now verify that S=R∗. But by definition g1|=S(m2) iff R(g1)|=m2iff
R(g1)⊆m′
2iff g1∈R•(m′
2). Similarly one may see that R(g1) = S•(g′
1)′for all
g1∈G1, i.e. R=R∗∗.
3⇐⇒ 4 Let (R, S ) be a Chu correspondence and define B⊆G1×M2by
B(g1) = R(g1)′. By construction each B(g1) is closed. Now by definition g1∈
B†(m2) whenever R(g1)|=m2. But this holds iff g1|=S(m2) iff g1∈S(m2)′.
Hence B†(m2) = S(m2)′, making it closed, so Bis a bond.
Conversely, suppose Bis a bond and define R, S as above. By construction
R(g1) and S(m2) are closed and we have
R(g1)|=m2⇐⇒ B(g1)′|=m2
⇐⇒ m2∈B(g1) = B(g1)
⇐⇒ g1∈B†(m2) = B†(m2)
⇐⇒ g1|=B†(m2)′=S(m2)
making (R, S) a Chu correspondence. Since R(g1) = R(g1)′′ ,S(m2) = S(m2)′′
and B(g1) = B(g1)′′ for any Chu correspondence (R, S ) or bond B, the assign-
ments (R, S)↔Bare inverse.
Each of the above correspondences may be made functorial. Firstly, for any
context K= (G, M, |=) define the dual context
K∗:= (M, G, |=†)
by swapping objects and attributes. Let us say that a category Cis self-dual
when it comes with an equivalence (−)∗:Cop ≃Csatisfying A∗∗ =Afor all
objects Aand f∗∗ =ffor all morphisms f.
Lemma 14. The assignment K7→ K∗and R7→ R∗defines a self-duality
(−)∗:Cxtop ≃Cxt.
7

Proof. For any R:K1→K2, Proposition 13 tells us that R∗is the unique
morphism for which (R, R∗) forms a Chu correspondence. It follows easily
that id∗
K= idKand that R∗∗ =R∗since (R∗, R) is a Chu correspondence.
Moreover if (R1, S1) and (R2, S2) are Chu correspondences one may see verify
that (R2◦R1, S1◦S2) is also, and so (−)∗preserves composition.
Proposition 13 also shows that Cxt is isomorphic to the category ChuCors
of Chu correspondences studied in the article [Mor08], where the latter is also
shown to be isomorphic to the category Bonds in which morphisms K1→K2
are bonds B:G1→M2, under the composition rule
(B2◦B1)(g) := (B2)•(B1(g)′) (∀g∈G1)
with the identity bonds being the relations |=. We verify this result ourselves.
Lemma 15. There is an isomorphism of categories Cxt ≃Bonds.
Proof. We will use the correspondence of Proposition 13.
We first establish the following fact. For any closed relation Rthe bond B
of Proposition 13 satisfies
R(A)′=\
a∈A
R(a)′=\
a∈A
B(a) = B•(A) (5)
for all A⊆G1. Now for any morphisms R1:K1→K2and R2:K2→K3with
corresponding bonds B1, B2we have
(B2◦B1)(g) := (B2)•(B1(g)′) = (B2)•(R1(g)′′) = (B2)•(R1(g)) = R2(R1(g))′
= (R2(R1(g)))′′′ = (R2◦R1)(g)′
as required.
We note also that Moshier has described a seemingly alternative relational
category of formal contexts [Mos16], further studied by Jipsen [Jip12]. In fact
this category coincides with our own.
Lemma 16. The category Cxt is identical to that of the same name in [Mos16,
Jip12]. In particular their ‘compatible relations’ are precisely bonds.
Proof. Appendix A.
3 Monoidal Structures on Formal Contexts
We will now define two distinct monoidal structures on Cxt, each sharing the
same tensor unit I, but with different tensor operations.
The first tensor has been described in the context of Chu correspondences
[Mor08], and is motivated by its close connection to the tensor product of sup-
lattices, as we see in Section 4.
8

Definition 17. For any contexts K1,K2we define their lattice tensor as
K1⊠K2:= (G1×G2,Cxt(K1,K∗
2),|=)
where for any morphism K1→K∗
2corresponding to a relation R:G1→G2we
set (g1, g2)|=Rwhenever R(g1)|=g2. Equivalently, we have that (g1, g2)∈B
where Bis the bond induced by R.
Another tensor of contexts has been introduced by Wille directly for FCA.
Definition 18. For any contexts K1,K2we define their concept tensor as
K1⊗K2:= (G1×G2, M1×M2,▽)
where
(g1, g2)▽(m2, m2)⇐⇒ g1|=m1or g2|=m2
In [Wil85] this is called the direct product of contexts, and denoted K1×K2.
Since both ⊠and ⊗are defined in the same way on the extent parts of a
context, we can in fact describe their bifunctors and structure isomorphisms in
the same way. We do so explicitly for ⊗. For any morphisms R1:K1→K3and
R2:K2→K4we define R1⊗R2:K1⊗K2→K3⊗K4by
(R1⊗R2)(g1, g2) = R1(g1)×R2(g2)⊆G3×G4(6)
for (g1, g2)∈G1×G2. We define the structure isomorphisms
K1⊗(K2⊗K3) (K1⊗K2)⊗K3
α
K1⊗K2K2⊗K1
σK1⊗I K1
ρ
by
α(g1,(g2, g3)) = ((g1, g2), g3)σ(g1, g2) = (g2, g1)ρ(g1, ⋆) = g1
where gi∈Gifor i= 1,2,3. In other words, the extent parts of coherence iso-
morphisms are just like those of Rel, but then followed by the closure operator.
The bifunctor and coherence isomorphisms for ⊠are given in the same way,
swapping the symbol ⊗with ⊠.
Theorem 19. [Mor08] (Cxt,⊠,I) is a symmetric monoidal category.
Let us now verify the new result that ⊗yields a monoidal structure also.
We begin with some straightforward results about the tensor.
Lemma 20. For any A⊆G1and B⊆G2, in K1⊗K2we have
1. A×B|=C×D⇐⇒ A|=Cor B|=D;
2. A×B=A×B=A×B∪(M1×M2)′
9

3. (R1⊗R2)(A×B) = R1(A)×R2(B) for all closed relations R1, R2.
Theorem 21. (Cxt,⊗,I) is a symmetric monoidal category.
Proof. Firstly, (6) forms a Chu correspondence with the relation
(R1⊗R2)(m3, m4) = R1(m3)×R2(m4)
since
R1(g1)×R2(g2)|= (m3, m4)⇐⇒ R1(g1)×R2(g2)|= (m3, m4)
⇐⇒ R1(g1)|=m3or R2(g2)|=m4
⇐⇒ g1|=R1(m3) or g2|=R2(m4)
⇐⇒ (g1, g2)|=R1(m3)×R2(m4)
To see that ⊗preserves identities, note that
id ⊗id(g1, g2) = g1×g2= (g1, g2) = id(g1, g2)
Moreover ⊗is a bifunctor since
(R3⊗R4)◦(R1⊗R2)(g1, g2) = (R3⊗R4)(R1(g1)×R2(g2))
=(R3⊗R4)(R1(g1)×R2(g2))
=(R3⊗R4)(R1(g1)×R2(g2))
=R3(R1(g1)) ×R4(R2(g2))
= (R3◦R1)⊗(R4◦R2)(g1, g2)
where in the second step we used that the result will be closed as (R3⊗R4)◦
(R1⊗R2) is a closed relation.
It is straightforward to verify that the coherence isomorphisms α, ρ, σ are
valid morphisms and are isomorphisms with inverses defined element-wise in
terms of those of Rel, followed by closure operators. We verify naturality of α,
while naturality of ρand σare simpler. Using Lemma 20 one may check that
R1⊗(R2⊗R3)(g1(g2, g3)) = R1(g1)×(R2(g2)×R3)
and also α(A×(B×C) = A×B)×Cfor all subsets Aiof Gi. It follows that
α◦(R1⊗(R2⊗R3))(g1,(g2, g3)) = (R1(g1)×R2(g2)) ×R3(g3)
which is straightforwardly seen to be equal to (R1⊗R2)⊗R3)◦α◦(g1,(g2, g3)).
Hence αis natural. The coherence equations may be verified by using Lemma 20
to reduce to the usual coherence equations in Rel, followed by applying closure
operators once at the end.
The category Cxt comes with further structure still. Recall that a symmetric
monoidal category Cis said to have discarding when each object Acomes with
a chosen morphism A:A→I, such that I= idIand A⊗B=λ◦(A⊗B).
For example, Rel has discarding with Abeing the relation with a7→ ⋆for all
a∈A.
10

Proposition 22.
1. For all contexts K1,K2we have
(K1⊗K2)∗:= K∗
1⊗K∗
2
Hence the equivalence (−)∗is strong monoidal with respect to ⊗.
2. (Cxt,⊗) is a symmetric monoidal category with discarding.
3. There is a full and faithful strong monoidal functor (Rel,×)֒→(Cxt,⊗)
which preserves discarding and maps (−)†to (−)∗.
Proof. 1 is immediate from the definitions. For 2, on each object Kwe set K
to have intent relation Rsatisfying R(⋆) = M. Then I= id, and we have
K1⊗K2(⋆) = M1×M2=M1×M2.
For 3, via Example 5 we define the embedding by A7→ Aand viewing each
relation R:A→Bas a closed relation. By construction A=A∗, A ×B=A⊗B
and one may verify that R∗=R†for each relation R.
However, ⊠is even more well-behaved, in the following sense. Recall that a
self-dual symmetric monoidal category Cis ∗-autonomous when it comes with
natural isomorphisms C(A⊗B, C∗)≃C(A, (B⊗C)∗).
Theorem 23. [Mor08] (Cxt,⊠,I) is a ∗-autonomous category.
On the other hand, since (Cxt,⊠) is not compact closed, it follows that
K∗
1⊠K∗
26≃ (K1⊠K2)∗.
4 Categories of Lattices
We now study how our category Cxt and its monoidal structures relate to those
of complete lattices, via the concept lattice construction.
Throughout, we write SupLat for the category of complete sup-lattices.
That is, the ob jects are complete lattices (with 0,1) and the morphisms are
mappings f:V1→V2which preserve arbitrary suprema. Similarly we write
InfLat for the category of complete inf-semilattices.
There is an isomorphism of categories SupLat ≃InfLat given by simply
switching ≤with ≥. Moreover, both categories are self-dual, with
(−)∗:SupLatop ≃SupLat
sending each lattice V= (V, ≤) to the opposite lattice V∗= (V, ≥) and f:V→
Wto its adjoint f∗:W∗→V∗. Since fpreserves suprema, f∗preserves infima
W→Vand hence suprema W∗→V∗. We denote the 2-element complete
lattice by 2:= {0≤1}.
Recall that any context Kdefines its concept lattice lattice B(K) and any
complete lattice Vdefines a context F(V) via Example 6. A key fact is the
following, which is essentially from [GW99], and more explicitly in [Mor08].
11

Theorem 24. There is a (−)∗-preserving equivalence of categories
Cxt ≃SupLat
B(−)
F
(7)
Proof. For each morphism R:K1→K2we define a join preserving map B(R): B(K1)→
B(K2) by
B(R)(A, A′) := (R(A), R(A)′) (8)
for each (A, A′)∈B(K1). Conversely, for any complete sup-lattice morphism
f:V1→V2we define F(f): F(V1)→F(V2) to have by F(f) = fwhich is
indeed a closed relation, forming a Chu correspondence with its order adjoint
f∗.
Noting that R(A)′=B•(A) via (5), the assignment (8) is a bijection on
homsets by [GW99, Theorem 53, Corollary 112], and is functorial by [GW99,
Proposition 113], as is F. Every complete lattice Vis readily shown to satisfy
V≃B(F(V)), making this an equivalence [Mor08, Theorem 73]. It is easy to
check that B(−)∗=B((−)∗), ensuring that Fpreserves (−)∗also.
In particular it follows that the category Cxt is complete and co-complete.
The following result captures the fact that ‘states in Cxt are concepts’, giving
another description of the functor B(−), also from [Mor08].
Lemma 25. Each homset Cxt(K1,K2) forms a complete lattice under inclusion
⊆of relations and there are natural isomorphisms
Cxt(I,−)≃B(−) (9)
Cxt(−,I)≃B(−)∗(10)
Proof. The first statement follows from the point-wise ordering on maps in
SupLat. Now (9) sends each closed relation R:{⋆} → Gto the concept
(R(⋆), R(⋆)′), and for (10) send each closed relation S:{⋆} → Mto the concept
(S(⋆)′, S(⋆)). The details may be checked directly, or using the equivalence (7)
and that SupLat(2, V )≃Vby sending each f:2→Vto f(1).
Thanks to the equivalence (7) each of our monoidal structures ⊠,⊗on Cxt
corresponds to a monoidal structure on SupLat, and we now describe each.
4.1 The Lattice Tensor
As our naming suggests, the lattice tensor on Cxt corresponds to the most well-
known monoidal structure on SupLat, which in fact makes it a *-autonomous
category. For any complete lattices V1, V2we define (V1⊸V2) := SupLat(V1, V2),
which forms a complete lattice under the point-wise ordering of maps, and then
V1⊠V2:= (V1⊸V∗
2)∗
Alternatively, V1⊠V2may be represented as the collection of bi-ideals in V1, V2.
The following is well-known.
12

Theorem 26. (SupLat,⊠,2,∗) is a *-autonomous category.
Moreover, Mori has established the following.
Theorem 27. [Mor08] The functors (B, F ) yield a *-autonomous equivalence
(Cxt,⊠,I,∗)≃(SupLat,⊠,2,∗)
4.2 The Concept Tensor
Less well-known is the monoidal structure on SupLat corresponding to the
tensor ⊗on Cxt, suggested by Wille.
Definition 28. [Wil85] For a pair of complete lattices V1, V2we define their
concept tensor to be the complete lattice
V1⊗V2:= B(F(V1)×F(V2))
Explicitly, it is the concept lattice of the context with G=M=V1×V2and
relation ▽with (x, y)▽(w, z ) whenever x≤wor y≤z.
The tensor V1⊗V2has a representation in terms of closed bi-ideals of V1, V2.
However, it can also be worked with directly by making use of a pair of complete
lattice embeddings
V1V1⊗V2V2
ε1ε2(11)
From these we define a pair of tensorial operations >,?:V1×V2→V1⊗V2by
x1?x2=ε1(x1)∧ε2(x2)
x1>x2=ε1(x1)∨ε2(x2)
By construction we have
ε1(x1) = x1?1 = x1>0ε2(x2) = 1 ?x2= 0 >x2(12)
The operations ?and >satisfy a number of axioms that allow one to perform
calculations in V1⊗V2, see [Wil85, p.83]. Most notably, the subsets ε(V1) and
ε(V2) generate V1⊗V2as a complete lattice, and are ‘mutually distributive’, in
the following sense.
Definition 29. [GW99] We call a pair of subsets Xand Yof a complete lattice
mutually distributive when for all indexed sets of elements (xi)i∈I⊆Xand
(yi)i∈I⊆Ywe have
_
i∈I
(xi∧yi) = ^
J⊆I
_
j∈J
xj∨_
k∈I\J
yk

^
i∈I
(xi∨yi) = _
J⊆I
^
j∈J
xj∧^
k∈I\J
yk

13

Wille has studied ⊗as a tensor for lattices which does not favour suprema
over infima (or vice versa), characterising it with respect to complete homomor-
phisms [Wil85, Theorem 2]. However, we will now see that this characterisation
may be extended to completely join-preserving maps, yielding a monoidal struc-
ture on SupLat. The following results are new.
Proposition 30. Let f:V1→Mand g:V2→Mbe complete sup-lattice
morphisms and suppose that f(V1) and g(V2) are mutually distributive in M.
Then there exists a unique complete sup-lattice morphism h:V1⊗V2→Mwith
h(x?y) = f(x)∧g(y) (13)
for all x∈V1, y ∈V2. Moreover, when fand gare complete lattice morphisms,
so is h.
Proof. The result and proof is similar to [GW99, Theorem 37] which, though
stated for complete morphisms, in many places only uses preservation of joins.
See the Appendix for details.
As as a consequence we obtain Wille’s characterisation of this tensor.
Corollary 31. [Wil85, Thm 2] For any complete lattice morphisms f:V1→M
and g:V2→Mwhose images f(V1) and g(V2) are mutually distributive, there
is a unique complete lattice morphism h:V1⊗V2→Mwith
h◦ε1=f h ◦ε2=g(14)
We can also now make this tensor into a bifunctor, thanks to the following.
Lemma 32. For any complete sup-lattice morphisms f:V1→W1and g:V2→
W2there is a unique such morphism
(f⊗g): V1⊗V2→W1⊗W2
satisfying
(f⊗g)(x?y) = f(x)?g(y)
for all (x, y)∈V1×V2. If fand gare complete lattice morphisms, so is f⊗g.
Proof. Apply Proposition 30 to ε1◦fand ε2◦g, with M=W1⊗W2.
We are now ready to establish the following. We equip SupLat with dis-
carding morphisms with V:V→2sending xto 1 iff x6= 0. Let us also write
CLat for the wide subcategory of SupLat given by the completely join and
meet preserving maps.
Theorem 33. (SupLat,⊗,2) is a symmetric monoidal category with discard-
ing, with CLat as a symmetric monoidal subcategory. Moreover (B, F ) provide
a symmetric monoidal equivalence
(Cxt,⊗,I)≃(SupLat,⊗,2) (15)
which preserves discarding.
14

A consequence is the following, which we could have verified directly.
Corollary 34. The self-duality (−)∗: (SupLat,⊗)op ≃(SupLat,⊗) is strong
monoidal. In particular (V⊗W)∗≃V∗⊗W∗for all complete lattices V , W .
5 Outlook: Applications
We close by briefly discussing potential applications of the category Cxt as a
compositional model of natural language meaning, which we hope to expand on
in future work.
Before considering each of our tensors, we note that Cxt itself naturally
models order relations on words. In detail, following [CSC10], we choose a
formal context Kto represent each basic word type, e.g. nouns. The semantics
of a noun wis then a state JwK:I→K, which by Lemma 25 corresponds to a
concept of K. The ordering on concepts allows one to capture entailment, as is
treated using density matrices in [BSC15].
5.1 DisCo in Closed Categories
The DisCo framework is typically applied to autonomous (i.e. rigid) monoidal
categories C, in which each object comes with a dual object with a ‘cup’ and
‘cap’ [CSC10]. Any pregroup forms such a category, allowing one to interpret
pregroup grammars in C.
Though neither of our monoidal structures on Cxt is autonomous, a known
result due to Lambek tells us that in fact a simpler structure than pregroups is
required in practice.
Definition 35. [Lam97] A protogroup is a partially ordered monoid (P, ≤,·)
such that for every a∈Pthere are chosen elements al, ar∈Pwith
al·a≤1a·ar≤1 (contractions) (16)
Pis a pregroup when additionally for all a∈Awe have
1≤a·al1≤ar·a(expansions) (17)
Lemma 36 (Switching Lemma, [Lam08]).For any terms t, t′in the free pre-
group PBgenerated by basic types B, if t≤t′then there exists t′′ ∈ PBsuch
that t≤t′′ without expansions and t′′ ≤t′without contractions.
In particular each inequality t≤sfor the sentence type s, determining that
a phrase of type tis a valid sentence, requires contractions only, e.g.
N
S
Alice
Nr
likes
NlN
Bob
15

Hence for such applications protogroup grammars are sufficient, requiring our
category to only have ‘cups’ (and not ‘caps’).
Any closed symmetric monoidal category (C,⊗,⊸) can model protogroup
grammars as follows. For each object Awe set Al=Ar=A∗:= (A⊸I), and
we interpret al·a≤1 as the canonical morphism A∗⊗A→Igiven by
A∗A
:= eval
A⊸I A
(18)
and a·ar≤1 by applying the symmetry to the above.
5.2 The Lattice Tensor
Since the category (Cxt,⊠) is *-autonomous, it is in particular closed. Hence as
above it can model protogroup grammars and all the sentence-parsing aspects
of the DisCo framework. We may use the cups defined in (18) to interpret any
valid sentence as a single concept.
5.3 The Concept Tensor
In contrast (Cxt,⊗) is merely a symmetric monoidal category, with no apparent
canonical cups or caps. However, Antonin Delpeuch has shown how the DisCo
framework may be extended even to bare monoidal categories, thanks to the
following result.
Theorem 37. [Del19] For any monoidal category Cthere is a free autonomous
category L(C) with a strong monoidal full and faithful embedding C֒→L(C).
Since the embedding is full, this means that any interpretation of a sentence
as a state I→Sinvolving formal cups and caps can be re-arranged to a valid
morphism in C, and so be rewritten without them.
In future it would be desirable to fully explore the usefulness of both tensors
on Cxt when modelling sentence meanings.
References
[BCG+19] Joe Bolt, Bob Coecke, Fabrizio Genovese, Martha Lewis, Dan Mars-
den, and Robin Piedeleu. Interacting conceptual spaces i: Grammat-
ical composition of concepts. In Conceptual Spaces: Elaborations and
Applications, pages 151–181. Springer, 2019.
[BSC15] Esma Balkir, Mehrnoosh Sadrzadeh, and Bob Coecke. Distribu-
tional sentence entailment using density matrices. In International
Conference on Topics in Theoretical Computer Science, pages 1–22.
Springer, 2015.
16

[CSC10] Bob Coecke, Mehrnoosh Sadrzadeh, and Stephen Clark. Mathemati-
cal foundations for a compositional distributional model of meaning.
arXiv preprint arXiv:1003.4394, 2010.
[Del19] Antonin Delpeuch. Autonomization of monoidal categories. Proceed-
ings of Applied Category Theory 2019. arXiv:1411.3827, 2019.
[Ern14] Marcel Ern´e. Categories of contexts. arXiv preprint
arXiv:1407.0512, 2014.
[G¨ar04] Peter G¨ardenfors. Conceptual spaces: The geometry of thought. MIT
press, 2004.
[GSW05] Bernhard Ganter, Gerd Stumme, and Rudolf Wille. Formal concept
analysis: foundations and applications, volume 3626. springer, 2005.
[GW99] Bernhard Ganter and Rudolf Wille. Formal concept analysis: math-
ematical foundations. Springer Science & Business Media, 1999.
[Jip12] Peter Jipsen. Categories of algebraic contexts equivalent to idempo-
tent semirings and domain semirings. In International Conference
on Relational and Algebraic Methods in Computer Science, pages
195–206. Springer, 2012.
[KHZ05] Markus Kr¨otzsch, Pascal Hitzler, and Guo-Qiang Zhang. Morphisms
in context. In International Conference on Conceptual Structures,
pages 223–237. Springer, 2005.
[Lam97] Joachim Lambek. Type grammar revisited. In International con-
ference on logical aspects of computational linguistics, pages 1–27.
Springer, 1997.
[Lam08] Joachim Lambek. From Word to Sentence: a computational alge-
braic approach to grammar. Polimetrica sas, 2008.
[Mor08] Hideo Mori. Chu correspondences. Hokkaido Mathematical Journal,
37(1):147–214, 2008.
[Mos16] MA Moshier. A relational category of formal contexts. Preprint,
2016.
[Pav12] Dusko Pavlovic. Quantitative concept analysis. In International
Conference on Formal Concept Analysis, pages 260–277. Springer,
2012.
[Pav20] Dusko Pavlovic. The nucleus of an adjunction and the street monad
on monads. Journal of Computer Research Repository, 2020.
[Wil85] Rudolf Wille. Tensorial decomposition of concept lattices. Order,
2(1):81–95, 1985.
[Wil92] Rudolf Wille. Concept lattices and conceptual knowledge systems.
Computers & mathematics with applications, 23(6-9):493–515, 1992.
17

A Proofs
Proof of Lemma 16. The morphisms K1→K2in [Jip12] are relations B⊆
G1×M2for which C=B†satisfies
C•(Y) = C•(Y) = C•(Y) (19)
for all Y⊆M2. We will show that Bis a bond. By the above each set
B†(m2) = C•({m2}) is closed. For any X⊆G1by definition one may see that
X⊆C•(Y)⇐⇒ Y⊆B•(X) (20)
with either holding iff B(x, y) holds for all x∈X, y ∈Y. Hence by (19) we
have Y⊆B•(X)⇐⇒ Y⊆B•(X), and so each set B•(X) is closed, making
each set B(g) = B•({g}) closed as required.
Conversely, given a bond Bdefine C=B†and R:M2→M1its intent
relation. Then for all Y⊆M2just as in (5) one may see that C•(Y) = R(Y)′
making each such set closed. Moreover this satisfies (19) by Lemma 9. The
composition •of B1:G1→M2and B2:G2→M3in [Jip12] is
(B2•B1)†(m) := (C1)•((C2)•({m})′) = (R1)•(R2(m))′=R(m)′= (B2◦B1)†(m)
for each m∈M3, where Ris the intent relation for (B2◦B1). Hence both
categories coincide.
Proof of Proposition 30. The first part of the proof of [GW99, Theorem 37]
shows that for any A⊆V1×V2we have
_
(x,y)∈A
f(x)∧g(y) = ^
(w,z)∈A′
f(w)∨g(z) (21)
We use this to define h:V1⊗V2→Mby
h(A, B) = _
(x,y)∈A
f(x)∧g(y) = ^
(w,z)∈B
f(w)∨g(z) (22)
for each concept (A, B)∈V1⊗V2. The verification that hpreserves suprema is
just as in [GW99, Theorem 37].
We now check that (13) is indeed satisfied. From the explicit definition of ?
in [Wil85] we have that x?y= (A, B) where (w, z)∈Awhenever w≤xand
z≤yor w= 0 or z= 0. Hence we have
h(x?y) = _
w≤x,z≤y
f(w)∧g(z)∨_
w∈V1
f(w)∧g(0) ∨_
z∈V2
f(0) ∧g(z)
= (f(x)∧g(y)) ∨0∨0 = f(x)∧g(y)
It remains for us to verify that his unique. Firstly, let us consider when f
and gare complete lattice morphisms, and so preserve infima. In this case, by
18

the symmetry of the definition (22), hdoes also, making it a complete lattice
morphism as stated. Moreover since the subsets ε1(V1) and ε2(V2) generate
V1⊗V2,his fully determined by (13) and (12), making it unique.
In particular, taking f=ε1and g=ε2we must have h= idV1⊗V2since
x?y=ε1(x)∧ε2(y). Now (22) tells us that for any (A, B) in V1⊗V2we have
(A, B) = _
(x,y)∈A
x?y(23)
Hence any sup-preserving map V1⊗V2→Mis determined entirely by its action
on elements of the form x?y, making hunique in the general case.
Proof of Theorem 33. From the uniqueness in Lemma 32, ⊗preserves identities
and composition, making it a bifunctor on SupLat. Using Corollary 31, we
define α=αU,V,W : (U⊗V)⊗W→U⊗(V⊗W) as the unique complete
homomorphism with
α◦εU⊗V◦εU=εU
α◦εU⊗V◦εV=εV⊗W◦εM
α◦εW=εV⊗W◦εW
Similarly, we define α′:U⊗(V⊗W)→(U⊗V)⊗Win the analogous way,
and then since α′◦αpreserves each of εU⊗V◦εU,εU⊗V◦εVand εWit is the
identity by uniqueness. Similarly α◦α′= id, making αan isomorphism. By
construction we have
α((x?y)?z) = x?(y?z)
for all x∈U, y ∈V, z ∈W. A quick calculation shows that for any fi:Vi→Wi
for i= 1,2,3 we have that (αW◦(f1⊗f2)⊗f3) and (f1⊗(f2⊗f3)◦αV) are
equal on elements of the form ((x?y)?z). But such elements are sup-dense
in (V1⊗V2)⊗V3, since elements of the form x?yare sup-dense in V1⊗V2by
(23). Hence the αare natural. We define the right unitor
ρ:V⊗2→V
to be the unique complete lattice homomorphism with ρ◦εV= idVand ρ◦
ε2(0) = 0 and ρ◦ε2(1) = 1. Then we have ρ(x?1) = xwhile ρ(x?0) = ρ(0) = 0
for all x∈V. Then one may check that ρis an isomorphism, and using elements
again that f◦ρV=ρW◦(f⊗id2) for any f:V→Win SupLat, establishing
naturality. The left unitor is defined similarly. The symmetry
σ:V⊗W→W⊗V
is the unique complete homomorphism with σ◦ε1=ε2and σ◦ε2=ε1. Equiva-
lently this means that σ(x?y) = y?xfor all x∈V,y∈W, and then naturality
19

from the definition of f⊗g. Moreover we have σW,V ◦σV,W = idV⊗Wdue to
preservation of ε1and ε2and so σis a symmetry.
Verifying the coherence conditions is straightforward using elements and that
maps from a tensor are determined by elements of the form x1?x2?...?xn
(after bracketing). For example, the triangle law follows from the fact that
(ρ⊗id) ◦α(x?(y?z)) = (ρ⊗id)((x?y)?z) = ρ(x?y)?z
= (x∧y)?z=x?(y∧z) = (id ⊗λ)(x?(y?z))
whenever y∈ {0,1}. Hence SupLat is a symmetric monoidal category. By
construction the coherence maps belong to CLat, and the bifunctor restricts
there by Lemma 32, making CLat a symmetric monoidal subcategory.
We now wish to establish the monoidal equivalence (15). For any pair of
contexts K1,K2, by definition, the elements of B(K1)⊗B(K2) are subsets C⊆
B(K1)×B(K2) which are closed in the appropriate sense. By (the proof of )
[GW99, Theorem 26] there is a canonical isomorphism φ=φK1,K2:B(K1)⊗
B(K2)→B(K1×K2) defined by
φ(C) := _
(A1,A′
1),(A2,A′
2)∈C
(A1×A2,(A1×A2)′) (24)
for each such C⊆B(K1)×B(K2), where (−)′and (−) are taken in K1×K2.
Now, for any concepts (A, A′)∈B(K1) and (B, B ′)∈B(K2), from the
explicit definition of ?, we have that ((A1, A′
1),(A2, A′
2)) ∈(A, A′)?(B, B′)
whenever A1≤Aand A2≤B, or (A1, A2)∈(M1×M2)′. It follows that we
have
φ((A, A′)?(B, B′)) = (A×B, (A×B)′)
Hence for all morphisms f:V1→W1and g:V2→W2in SupLat we have
(B(f⊗g)◦φV1,V2)((A, A′)?(B, B′)) = B(f⊗g)(A×B , (A×B)′)
=(f⊗g)(A×B)
=f(A)×g(B)
=φ(f(A)?g(B))
=φW1,W2◦(B(f)⊗B(g))((A, A′)?(B, B′))
where we used Lemma 20 several times in the third step. Since elements of the
form (A, A′)?(B, B ′) are sup-dense by (23) it follows that the isomorphisms
φare natural. We have B(I)≃P({⋆})≃2. We omit the verification of the
monoidal coherence equations, from which it follows that B(−) is a monoidal
functor, and hence the equivalence a monoidal one.
Finally, on any context K, from (8) and the definition of K=Rwe have
that B(K) maps a concept (A, A′) to 0 iff R(A)′=R∗• (A′) = 0, which holds
iff A′=M, that is iff (A, A′) = 0 in B(K). Hence Bpreserves discarding.
20