=Paper=
{{Paper
|id=Vol-433/paper-6
|storemode=property
|title=Connecting Many-valued Contexts to General Geometric Structures
|pdfUrl=https://ceur-ws.org/Vol-433/paper2.pdf
|volume=Vol-433
}}
==Connecting Many-valued Contexts to General Geometric Structures==
https://ceur-ws.org/Vol-433/paper2.pdf
Connecting Many-valued Contexts to General
Geometric Structures
Tim B. Kaiser
Technische Universität Dresden
Institut für Algebra
Abstract. We study the connection between certain many-valued con-
texts and general geometric structures. The known one-to-one correspon-
dence between attribute-complete many-valued contexts and complete
affine ordered sets is used to extend the investigation to π-lattices and
class geometries. The former are identified as a subclass of complete affine
ordered sets, which exhibit a close relation to concept lattices which are
closely tied to the corresponding context. The latter can be related to
complete affine ordered sets using residuated mappings and the notion
of a weak parallelism.
1 Introduction
In [5] the notion of an affine ordered set enables us to understand a many-valued
context in order-theoretic and geometric terms. In [4] affine ordered sets were
specialized to complete affine ordered sets to allow an algebraic interpretation.
Here, we relate complete affine ordered sets to two other known types of general
geometric structures, that is,
• π-lattices and
• equivalence class geometries (or short class geometries).
In [6], π-lattices were introduced as an abstraction of affine geometries over rings
and modules, yielding the possibility to study geometry in a very general setup.
They will turn out to be a well describable specialization of complete affine
ordered sets, which opens up an intimate connection to the concept lattices
arising from plain conceptual scaling of the corresponding context.
Class geometries are a generalization of congruence class geometries. They carry
a certain type of parallelism – called weak parallelism– arising naturally in the
context of coordinatizing geometric closure structures via the congruence classes
of an algebra (in the sense of universal algebra), cf. [7]. The weak parallelism of
class geometries can be related to the parallelism of complete affine ordered sets
by applying a rather abstract result – a residuated pair of mappings between
atomic complete lattices where one carries a weak parallelism induces a weak
parallelism on the other.
As a first step, we will provide the necessary basic definitions. The second step
will lead to an elaboration on the connection between complete affine ordered
c Radim Belohlavek, Sergei O. Kuznetsov (Eds.): CLA 2008, pp. 23–34,
ISBN 978–80–244–2111–7, Palacký University, Olomouc, 2008.
24 Tim B. Kaiser
sets and π-lattices. In a third step, we will show how a weak parallelism on
an atomic lattice can induce a weak parallelism on another atomic lattice via
an adjunction. The results will be applied in the concluding step to describe a
connection between complete affine ordered sets and class geometries. Finally,
we will give a summary of what we achieved.
Throughout the paper we assume that the reader is knowledgeable of the basic
concepts of order theory and formal concept analysis, as provided, for instance,
in [2] and [1].
2 Attribute-complete Many-valued Contexts and
Complete Affine Ordered Sets
We recall the relevant definitions from [5] and [4].
For a mapping f : A −→ B between sets A and B, the kernel of f is defined as
ker(f ) := {(a1 , a2 ) | f (a1 ) = f (a2 )}.
A many-valued context K := (G, M, W, I) is called attribute-complete if it
• is complete, that is, every m ∈ M can be regarded as a map m : G → W ,
• has a key attribute, that is, there exists an attribute m ∈ M with ker(m) =
∆G := {(g, g) | g ∈ G},
• is simple, that is, different attributes m1 , m2 ∈ M are not functionally equiv-
alent, that is, ker(m1 ) = ker(m2 ) implies m1 = m2 , and
• for all N ⊆ M there exists an attribute T m ∈ M such that m and N are
functionally equivalent, that is, ker(m) = n∈N ker(n).
A system of equivalence relations (SER) is a pair (G, E) where G is a set and
E is a set of equivalence relations on G which contains the identity relation. A
SER is called closed if its set of equivalence relations forms a closure system.
Let ] be the symbol for the disjoint union. Then the lifting of an ordered
set (P, ≤) is given by (P ] {⊥}, ≤ ] ({⊥} × P ] {⊥})) and denoted as (P, ≤)⊥ .
Since the following notion is central in this paper we provide it as
Definition 1 ((atomistic, complete) affine ordered set). We call a triple
A := (Q, ≤, k)
affine ordered set, if (Q, ≤) is a partially ordered set, k is an equivalence rela-
tion (called parallelism) on Q, and the axioms (A1) - (A4) hold. Let A(Q) :=
Min(Q, ≤) denote the set of all minimal elements in (Q, ≤) and A(x) := {a ∈
A(Q) | a ≤ x}.
(A1) ∀x ∈ Q : A(x) 6= ∅
(A2) ∀x ∈ Q ∀a ∈ A(Q) ∃!t ∈ Q : a ≤ t k x
(A3) ∀x, y, x0 , y 0 ∈ Q : x0 k x ≤ y k y 0 & A(x0 ) ∩ A(y 0 ) 6= ∅ ⇒ x0 ≤ y 0
Connecting Many-valued Contexts to General Geometric Structures 25
(A4) ∀x, y ∈ Q ∃x0 , y 0 ∈ Q : x y & A(x) ⊆ A(y)
⇒ x0 k x & y 0 k y & A(x0 ) ∩ A(y 0 ) 6= ∅ & A(x0 ) * A(y 0 ).
The elements of A(Q) are called points and, in general, elements of Q are called
subspaces. We say that a subspace x is contained in a subspace y if x ≤ y.
If the lifting of (Q, ≤) forms a complete lattice L(A), the affine ordered set A
is called complete affine ordered set. We call a complete affine ordered set A
atomistic if the corresponding complete lattice L(A) is atomistic.
For a point a and a subspace x we denote by π(a|x) the subspace which con-
tains a and is parallel to x. Axiom (A2) guarantees that there is exactly one such
subspace. For every x ∈ Q we observe that θ(x) := {(a, b) ∈ A(Q)2 | π(a|x) =
π(b|x)} is an equivalence relation on the set of points.
We introduce the following condition for affine ordered sets:
(A34) ∀x, y ∈ Q : x ≤ y ⇐⇒ A(x) ⊆ A(y) & θ(x) ⊆ θ(y)
If we assume that only (A1) and (A2) hold in Definition 1 the axioms (A3)
and (A4) are equivalent to (A34).
In [4] it was shown that the notions of
• attribute-complete many-valued contexts,
• closed SERs, and
• complete affine ordered sets
with their respective morphisms form categories which are equivalent.
We recall how the objects of these equivalent categories can be translated into
each other. To an attribute-complete many-valued context K := (G, M, W, I) we
can assign a closed system of equivalence relations via
E(K) := (G, {ker(m) | m ∈ M }).
To a closed system of equivalence relations E := (G, E) we can assign a complete
affine ordered set – the ordered set of its labeled equivalence classes – via
A(E) := ({([x]θ, θ) | θ ∈ E}, ≤, k)
where ≤ is defined by
([x]θ1 , θ1 ) ≤ ([y]θ2 , θ2 ) : ⇐⇒ [x]θ1 ⊆ [y]θ2 & θ1 ⊆ θ2
and k is defined by
([x]θ1 , θ1 ) k ([y]θ2 , θ2 ) : ⇐⇒ θ1 = θ2 .
3 π-Lattices and Complete Affine Ordered Sets
The notion of a π-lattice stems from [6] where it is situated as an abstraction of
a geometry over rings. For a complete lattice L, we define
^
L+ := L \ L.
A π-lattice is defined as follows:
26 Tim B. Kaiser
Definition 2 (π-lattice). Let V be a complete atomistic lattice with set of
atoms A(V ). Then an equivalence relation k ⊆ V+ × V+ is called parallelism
if it satisfies the following axioms
(E) ∀p ∈ A(V ) ∀x ∈ V+ ∃!y ∈ V+ : p ≤ y k x
(M) ∀p ∈ A(V ) ∀x, y ∈ V+ : x ≤ y =⇒ π(p|x) ≤ π(p|y).
We call an atomistic complete lattice with parallelism π-lattice.
It turns out that complete affine ordered sets are a natural generalization of
π-lattices.
Proposition 1. Let V be a π-lattice. Then (V+ , ≤V , k) forms a complete affine
ordered set.
Proof. Since V is a π-lattice, k is an equivalence relation. We have to show
that (A1) - (A4) hold for (V \ {0}, ≤V , k). Since V is atomistic (A1) holds. (E)
directly implies (A2). For showing (A34), let x ≤ y for x, y ∈ V+ . Obviously
A(x) ⊆ A(y) follows directly from x ≤ y. Furthermore, θ(x) ⊆ θ(y) by (M). For
the other direction, already A(x) ⊆ A(y) implies x ≤ y since we have
_ _
x= A(x) ≤ A(y) = y
because V is atomistic. By construction (V+ )⊥ = V is a complete lattice. t
u
The notion of parallelism for affine ordered sets fulfills the criteria of a par-
allelism from the definition of π-lattices without problems.
Proposition 2. Let A := (Q, ≤, k) be an affine ordered set. Then (E) and (M)
hold in L(A).
Proof. (E) follows directly from (A2). To show (M) let x ≤ y for x, y ∈ L(A)+
and p ∈ A(L(A)). We have to show that π(p | x) ≤ π(p | y). By (A34) we have
θ(π(p | x)) = θ(x) ⊆ θ(y) = θ(π(p | y)).
Additionally, it follows directly that
A(π(p | x)) = [p]θ(x) ⊆ [p]θ(y) = A(π(p | y))
and therefore by applying the equivalence in (A34) from right to left we get
π(p | x) ≤ π(p | y) which shows (M). t
u
Propositions 1 and 2 yield the following characterization of π-lattices in terms
of complete affine ordered sets:
Theorem 1. The atomistic complete affine ordered sets are in one-to-one cor-
respondence with π-lattices. More precisely, moving between the two structures
requires only attaching or respectively removing the bottom element while the
parallelism can be reused.
Connecting Many-valued Contexts to General Geometric Structures 27
We will illuminate what it means for a complete affine ordered set to be
atomistic. We call a system of equivalence relations E := (D, E) regular if its set
of equivalence relations E is regular, that is, if there do not exist two different
equivalence relations sharing an equivalence class.
Proposition 3. Let A be a complete affine ordered set and let E be a closed
system of equivalence relations with
A∼
= A(E) & E(A) ∼
= E.
Then A is atomistic if and only if E is regular.
Proof. “⇒”: Let E := (D, E) be a regular closed system of equivalence relations
and let A(E) be the associated complete affine
W ordered set. Then we have to
show for a subspace x from A(E) that x = A(x). But since the subspaces of
A(E) are the labeled equivalence classes of E we know that x = (X, θ) for an
equivalence class X of an equivalence relation θ ∈ E. Then we have
_ _
A(x) = ({p}, ∆) | p ∈ X} = (X, θ(X)).
W
But θ(X) = θ since E is regular. Therefore, A(x) = (X, θ(X)) = (X, θ) and
hence A(E) is atomistic.
“⇐”: Let A := (Q, ≤, k) be an atomistic complete affine ordered set and let E(A)
be the associated closed system of equivalence relations. We have to show that
E(A) is regular, that is, for a point p ∈ A(Q) where [p]θ(x) = [p]θ(y) it follows
that θ(x) = θ(y). Since A is atomistic we have
_ _
x= [p]θ(x) = [p]θ(y) = y.
Hence E(A) is regular. t
u
The subclass of atomistic complete affine ordered sets can be related to con-
cept lattices arising in a certain fashion from the many-valued context corre-
sponding to the affine ordered set. To be able to formulate this connection, we
need the following
Definition 3 (derived context via nominal scaling). Let K := (G, M, W, I)
be a complete many-valued context. Then the formal context Knom := (G, N, J)
is called derived context via nominal scaling of K if
N := {(m, w) ∈ M × W | ∃g ∈ G : m(g) = w} and
J := {(g, (m, w)) ∈ G × N | (g, m, w) ∈ I}.
Now we explain the connection between atomicity of complete affine ordered
sets and conceptual scaling.
28 Tim B. Kaiser
Proposition 4. Let K := (G, M, W, I) be a simple many-valued context with
key attribute, let A := A(K) be the associated affine ordered set and let Knom be
the derived context of K via plain nominal scaling. Let ϕ : A → B(Knom ) be a
mapping where (C, θ) 7→ (C, C J ). Then ϕ is an order-preserving mapping which
is
• surjective if and only if A is complete and
• injective if and only if A is atomistic.
Proof. To see that (C, C J ) ∈ B(Knom ) we have to show that C = C JJ , that
is that C is an extent of a formal concept of the concept lattice of Knom . It is
obvious that C ⊆ C JJ since ·JJ is a closure operator. By construction of A we
know that there exists a h ∈ G and a m ∈ M such that
C = [h]ker(m) = {g ∈ G | m(g) = m(h)} = {g ∈ G | (g, (m, m(h))) ∈ J}.
Hence, (m, m(h)) ∈ C J . But then for all g ∈ C JJ we have gJ(m, m(h)) which
shows that if g ∈ C JJ then g ∈ C. Therefore C = C JJ . It is obvious that ϕ is
order-preserving.
“⇒”: Let A be complete. We show that ϕ is surjective. Since the extents of
B(Knom ) are exactly the meets of equivalence classes induced by K, and the set
of equivalence classes induced by K is already meet-closed it is immediate that
ϕ is surjective.
Let A be atomistic. We show that ϕ is injective. Let ϕ(C1 , θ1 ) = ϕ(C2 , θ2 ). Then
(C1 , C1 J ) = (C2 , C2 J ) which implies C1 = C2 . But since we know by Proposition
3 that E(A) is regular we have θ1 = θ2 .
“⇐”: Let ϕ be surjective. Then every extent of B(Knom ) is an image of ϕ. But
since the extents of B(Knom ) are exactly the meets of equivalence classes induced
by K, we know that the set of equivalence classes is meet-closed and therefore
A is complete.
Let ϕ be injective. Then whenever (C1 , C1 J ) = ϕ(C1 , θ1 ) = ϕ(C2 , θ2 ) = (C2 , C2 J )
which is equivalent to C1 = C2 we have θ1 = θ2 . That means, E(A) is regular.
Again, by Proposition 3 we know that A is atomistic. t
u
The proof of the previous proposition yields the following
Corollary 1. Let K := (G, M, W, I) be an attribute-complete many-valued con-
text, let A := A(K) be the associated complete affine ordered set and let Knom
be the derived context of K via plain nominal scaling. Then
B(Knom ) ∼
= L(A)
if and only if A is atomistic. t
u
The combination of Propositions 2 and 3 and Corollary 1 can be cast as:
Theorem 2. Let K := (G, M, W, I) be an attribute-complete many-valued con-
text. Then the following conditions are equivalent:
Connecting Many-valued Contexts to General Geometric Structures 29
• E(K) is regular
• A(K) is atomistic
• A(K) induces a π-lattice
• L(A(K)) ∼= B(Knom )
t
u
Example 1. We get a nice example of an attribute-complete many-valued context
if we consider a K-vector space V. Let
K(V) := (V, End(V), V, I)
where V is the set of vectors of V, End(V) is the set of endomorphisms of V,
and I is defined as
(v, ϕ, w) ∈ I : ⇐⇒ ϕ(v) = w.
Since for vector spaces, every congruence relation is already representable as
the kernel of an endomorphism (and the kernels of endomorphisms are always
congruence relations), we know that E(K(V)) is closed. The lattice of the cor-
responding complete affine ordered set is isomorphic to the lattice of affine sub-
spaces of the vector space V. Since E(K(V)) is regular we know by Theorem 2
that A(K(V)) is atomistic, its lattice is isomorphic to the concept lattice derived
by nominal scaling, and it induces a π-lattice.
4 Weak Parallelisms and Affine Ordered Sets
In this section, we will derive insights about trace parallelisms induced by resid-
uated mappings between atomic lattices. An application of these abstract results
in the next section will lead to a better understanding of the connection between
affine ordered sets and class geometries.
In the following let L and M denote complete lattices. For a lattice L, let A(L)
denote the set of the atoms of L and for s ∈ L let A(s) := {p ∈ A(L) | p ≤ s}
denote the atoms less than or equal to s. A lattice L is called atomic if for every
s ∈ L+ we have A(s) 6= ∅.
Definition 4 (residuated maps). A map ϕ : L −→ M is called residuated if
a residuated map, there exists a map ϕ+ : M −→ L,
W
it is – preserving. For V
called residual, which is – preserving with
ϕm ≤ l ⇔ m ≤ ϕ+ l
The maps uniquely determine each other. If one of the maps is surjective, the
other is injective, and vice versa. The maps are called adjoint to each other. We
call (ϕ, ϕ+ ) a residuated pair or an adjunction (this is sometimes also called a
covariant Galois connection).
Note, that for a residuated pair (ϕ, ϕ+ ) where ϕ is injective, we have ϕ+ ϕ =
∆, since ϕϕ+ ϕ = ϕ. In general ϕ+ ϕ is a closure operator and ϕϕ+ is a kernel
operator.
30 Tim B. Kaiser
Definition 5 (weak parallelism). Let L be an atomic complete lattice. We
call a relation k on L+ weak parallelism if the following holds for arbitrary
r, s, t, u ∈ L+ and arbitrary p ∈ A(L).
(P1) r k r
(P2) r k s ≥ t k u ⇒ r k≥ u
(P3) r k s ≥ p ⇒ r ∨ p ≥ s
(P4) ∃!s : r k s ≥ p
We say for r, s ∈ L+ with r k≥ s that s is part-parallel to r. If k is an
equivalence relation the weak parallelism is called pre-parallelism.
We will investigate the connection between affine ordered sets and the intro-
duced weak parallelism.
Proposition 5. Let A be a complete affine ordered set. Then L(A) is an atomic
complete lattice whith pre-parallelism.
Proof. Obviously, L(A) is atomic (by (A1)) and complete. It remains to verify the
axioms (P1)–(P4) for L(A). Axiom (P1) follows from the fact that the parallelism
of A is an equivalence relation. Axiom (A2) grants us that (P4) holds.
To show (P2), let r k s ≥ t k u. Let p ≤ u be a point. By (A3) we know that
from u k t ≤ s k π(p|r) we get u ≤ π(p|r)). Therefore we have r k π(p|r) ≥ u.
To show (P3), let r k s ≥ p. Let q ≤ s be an arbitrary point of s. By Proposition
2 we know that (M) holds. Therefore r ≤ r ∨ p yields s = π(q|r) ≤ π(q|r ∨ p).
But p ≤ s ≤ π(q|r ∨ p) implies π(q|r ∨ p) = r ∨ p. Hence, s ≤ r ∨ p. t
u
Example 2. The converse of the x xkx
previous proposition does not hold: In
a
general, atomic complete lattices with
pre-parallelism do not induce a
aka
complete affine ordered set. If we
remove the bottom element of the
lattice in Figure 1 we can not consider
the resulting structure as an affine
ordered set since θ(a) = θ(x) would Fig. 1. Complete atomic lattice with
imply a = x. trivial pre-parallelism
As in the case of π-lattices – where it was enough to require an affine ordered
set to be atomistic to let the concepts coincide – for atomistic lattices a pre-
parallelism is already a parallelism.
We will show that a residuated pair between two complete atomic lattices
where the latter carries a weak parallelism yields a weak parallelism on the
former. This parallelism will also be called trace parallelism.
L
Theorem 3. Let M and L be complete atomic latticesW and let k be a weak
parallelism on L. Furthermore, let ϕ : M ,→ L be a – preserving, injective
Connecting Many-valued Contexts to General Geometric Structures 31
mapping with ϕA(M ) ⊆ A(L) and let (ϕ, ϕ+ ) form a residuated pair. Then we
define a relation on M+ as follows
r kM s :⇔ ∃y ∈ L : ϕr kL y & ϕ+ y = s.
The relation kM is a weak parallelism.
Proof. In the following, let r, s, t, u ∈ M and p ∈ A(M ).
For (P1), we have to show that kM is reflexive. Since ϕϕ+ ϕr = ϕr and ϕ
is injective, we have ϕ+ ϕr = r. Since ϕr kL ϕr we have r kM r via setting
y := ϕ+ ϕr in the definition of kM .
For (P2), let us assume that r kM s ≥ t kM u. We have to show the existence of
an element v ∈ M with v ≥ u and r kM v. From r kM s we know there exists
y ∈ L such that ϕr kL y and ϕ+ y = s. From t kM u we know there exists z ∈ L
such that ϕt kL z and ϕ+ z = u. But since ϕ+ y ≥ t implies y ≥ ϕt we have
ϕr kL y ≥ ϕt kL z. Applying (P2) yields the existence of an element q ∈ L with
q ≥ z and ϕr kL q. We have v := ϕ+ q ≥ ϕ+ z = u and r kM v.
For (P3), let us assume r kM s ≥ p. From r kM s we know there exists y ∈ L
such that ϕr kL y and ϕ+ y = s. Since s = ϕ+ y ≥ p implies y ≥ ϕp and ϕ maps
atoms to atoms we can apply (P3) in M . This yields y ≤ ϕr ∨ ϕp = ϕ(r ∨ p)
which implies s = ϕ+ y ≤ ϕ+ ϕ(r ∨ p) = r ∨ p as required.
For (P4), we have an atom p ∈ A(M ) and an arbitrary element r ∈ M+ . We
have to show that there exists exactly one s ∈ M+ with r kM s ≥ p. We can
apply (P4) for ϕp and ϕr which yields the existence of exactly one y ∈ L+ with
ϕr kL y ≥ ϕp. We set s := ϕ+ y. Since y ≥ ϕp implies s = ϕ+ y ≥ p and by
construction of s we have r kM s it remains to show that s is unique. Assume
we have an element s0 ∈ M+ with r kM s0 ≥ p. This means that there exists an
element y 0 ∈ L+ with ϕr kL y 0 and ϕ+ y 0 = s0 . But since ϕr kL y 0 ≥ ϕp (P4)
yields y 0 = y we have s = s0 . t
u
In the following theorem we characterize relations which arise from weak par-
allelisms in the manner described in Theorem 3 by ”part-parallelity”. This result
can be used to see how the two weak parallelisms in Theorem 3 are connected.
Theorem 4. Let M and L be complete atomic lattices and let kL be a weak
parallelism on L, furthermore, let (ϕ, ϕ+ ) be a residuated pair for M and L and
let kM be defined as in the previous theorem. Then we have
r kM ≥ s ⇔ ϕr kL ≥ ϕs.
Proof. Since r kM ≥ s there exists an u ∈ M+ with r kM u ≥ s. By definition
of kM we have the existence of an element y ∈ L with ϕr kL y and ϕ+ y = s.
Since ϕϕ+ is a kernel operator we have ϕs = ϕϕ+ y ≤ y which yields that ϕr is
part-parallel to ϕs. The proof is finished since the argument is symmetric. t
u
5 Class Geometries and Affine Ordered Sets
Throughout this section, let E := (D, E) be a closed system of equivalence
relations. We know that we can assign a complete affine ordered set, denoted
32 Tim B. Kaiser
by A(E), to E. Alternatively, we can also assign the ordered set of equivalence
classes ({[x]θ | θ ∈ E}, ⊆) to E. It is convenient to attach a bottom element to
get a lattice
G(E) := (S ∪ {∅}, ⊆)
which we call class geometry of E. If the equivalence relations can be regarded
as the congruence relations of an algebra (in the sense of universal algebra) we
call their class geometry congruence class geometry. Congruence class geometries
were introduced and characterized geometrically via their closure operators in [7].
Now, we want to relate the class geometry G := G(E) and the lattice of the
affine ordered set L := L(A(E)) of a closed system of equivalence relations to
each other. Let ϕ+ : L → G be defined by ϕ+ (C, θ) := C. Since
^ \ \
(Ci , θi ) = ( Ci , θi ),
i∈I i∈I i∈I
we have ^ \ ^
ϕ+ si = Ci = ϕ+ si
i∈I i∈I i∈I
+
for si = (Ci , θi ). Note that ϕ is surjective.
From Proposition 9 in [2], p. 14, we know that for any residual map its
residuated is given by ^
ϕs := {l | s ≤ ϕ+ l}.
If we define for a closed system of equivalence relations (D, E) the smallest
relation containg M ⊆ D as
\
θ(M ) := {θ ∈ E | M × M ⊆ θ}
the above definition of the residual yields in our context that ϕ : S ,→ L is
defined by
ϕC := (C, θ(C)).
Since ϕ+ is surjective, it follows that ϕ is injective. This implies that ϕS is a
kernel system in L. We summarize the results of the argumentation in
Theorem 5. Let E := (D, E) be a closed system of equivalence relations. Let
G := G(E) be its class geometry and let L := L(A(E)) be the lattice of its affine
ordered set. Then (ϕ, ϕ+ ) (as defined above) forms an adjunction between G and
L, where ϕ is injective and ϕ+ is surjective. This implies that G is embedded in
L as a kernel system via ϕ.
As an illustration of the previous theorem we provide
Example 3. Figure 2 shows the well-known non-modular lattice N5 . Figure 3
shows the lattice of congruence relations of N5 . Figure 4 shows the congruence
class geometry of N5 embedded as a kernel system into the lattice of the affine
ordered set of (the congruence relations of) N5 . The kernel system is marked by
black dots in Figure 4.
Connecting Many-valued Contexts to General Geometric Structures 33
1 ? ∇ ?
???
?
???
??
?
?? ?? θ({a,0})
? c θ({a,1})
??
??
??
a b
?
??
??
θ({b,c})
??
??
0 ∆
Fig. 2. N5 Fig. 3. The congruence lattice of N5
It is easily observable that both, the class geometry G and the lattice L of
the complete affine ordered set, form atomic lattices. By Proposition 5 we know
that the parallelism of the affine ordered set constitutes a weak parallelism (even
a pre-parallelism) in the sense of Definition 5. We use the residuated pair (ϕ, ϕ+ )
to apply Theorem 3. Since ϕ maps atoms to atoms, Theorem 3 yields that
r kS s :⇔ ∃l ∈ L : ϕr k l & ϕ+ l = s
defines a weak parallelism on S+ .
What does it mean for two equivalence classes C, D to be weakly parallel in
S in terms of their equivalence relations? Expanding the definition we get
C kS D
⇔ ∃(P, ψ) ∈ L : θ(C) = ψ & P = D
⇔ D is a class of θ(C).
Surprisingly, this is exactly the same weak parallelism as is used in [7] on the
closed sets of a closure operator to be able to characterize this closure operator as
assigning to a set M the smallest congruence class of a suitable algebra containing
M.
6 Conclusion
Studying the connection between complete affine ordered sets and π-lattices
yielded the fruitful characterization of π-lattices as atomistic affine ordered sets
and opened up the possibility to interpret these structures as concept lattices.
Through an adjunction between a complete affine ordered set and its correspond-
ing class geometry we could view the class geometry as a kernel system in the
affine ordered set and were able to recognize the induced parallelism as known
from congruence class spaces, where it is used to coordinatize geometric spaces.
We conclude that the findings in this paper support the thesis that affine ordered
sets are a conceptually useful paradigm to connect different notions arising when
studying geometric structures abstractly.
34 Tim B. Kaiser
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Fig. 4. Congruence class geometry of N5 embedded as kernel system in the lattice of
the labeled congruence classes of N5
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