=Paper=
{{Paper
|id=None
|storemode=property
|title=CryptoLat - a Pedagogical Software on Lattice Cryptomorphisms and
Lattice Properties
|pdfUrl=https://ceur-ws.org/Vol-1062/paper8.pdf
|volume=Vol-1062
|dblpUrl=https://dblp.org/rec/conf/cla/Domenach13
}}
==CryptoLat - a Pedagogical Software on Lattice Cryptomorphisms and
Lattice Properties==
https://ceur-ws.org/Vol-1062/paper8.pdf
CryptoLat - a Pedagogical Software on Lattice
Cryptomorphisms and Lattice Properties
Florent Domenach
Computer Science Department, University of Nicosia
46 Makedonitissas Ave., 1700 Nicosia, Cyprus
domenach.f@unic.ac.cy
Abstract. Although lattice theory is a rich field dating from Dedekind,
Birkhoff and Öre, few studies in FCA use lattice properties to enhance
their results. Moreover, out of the many cryptomorphisms associated
with lattices, only the ones associating context, lattice and implicational
system are effectively studied.
CryptoLat is a software implemented in C] which provides an intuitive
view on different cryptomorphisms of lattices as well as on different prop-
erties of lattices. Its purpose is pedagogical, for students and researchers
likewise, and work by showing incremental changes in the lattice and the
associated cryptomorphisms.
Keywords: Lattice Cryptomorphisms, Lattice Properties, Pedagogy
1 Introduction
This article, together with the software, originates from an observation about the
current state of research in the lattice and the FCA communities. Lattice theory
dates back to the works of Dedekind, Birkhoff [7] and Öre [19], and is a rich field
in which FCA is deeply and fundamentally rooted. Despite that fact, relatively
few studies in FCA takes advantage of lattice properties to enhance their results.
This dichotomy of the two communities is apparent in the research interests:
on the one hand, researchers from the graph and ordered set communities are
focusing on equivalence theorem and characterizations, while on the other hand
data mining researchers are more focused on practical or practically oriented
results.
Although it is natural that this dichotomy exists because of the nature of each
community’s focus of interest, it is the author’s suggestion that improvement of
the communication between the two communities may produce a synergistic
effect. The study of lattice cryptomorphisms and lattice properties is important
for many reasons: other than giving a better understanding of the tools being
used, it was shown to lead to the discovery of efficient algorithms in counting
the number of Moore families [16] or calculating the stem basis [6] for example.
There are two main purposes for this software: first, to promote and to give
a better understanding of the many shapes and forms of lattices thanks to an
incremental approach - any change made in any equivalent cryptomorphism is
c paper author(s), 2013. Published in Manuel Ojeda-Aciego, Jan Outrata (Eds.): CLA
2013, pp. 93–103, ISBN 978–2–7466–6566–8, Laboratory L3i, University of La
Rochelle, 2013. Copying permitted only for private and academic purposes.
94 Florent Domenach
transcribed to the other cryptomorphisms. Second, to help familiarize users to
fundamental latticial properties that are recalculated after any modification. Its
purpose is pedagogical for both students and researchers. Students can learn
and get a better understanding on what is happening during the construction
of the lattice and the implicational basis, while researchers can test hypothe-
ses and find small illustrating examples. The program is written in C] and is
freely available (executable and source code) at http://www.cs.unic.ac.cy/
florent/software.htm
This paper is organized as follows: we start in Sec. 2 by describing the soft-
ware itself. Sec 3 will recall the well-known definitions of lattices and some of
the existing cryptomorphisms implemented in CryptoLat, together with a short
explanation on how the equivalence is done. Finally, we compare in Sec. 5 Cryp-
toLat with already existing software.
2 Description of CryptoLat
CryptoLat software is structured around a model view controller type of software
architecture: each cryptomorphism described in Sec. 3 is a view with which the
user can interact. The model is the reduced context associated with a Moore
family on which all the calculations are based. When a view is updated, by adding
or deleting a Sperner family for example, then the Moore family associated is
calculated, the associated concept lattice, together with all the properties, is
computed, and the other views get updated.
Depending on which view getting updated, the process may involve different
steps. If the overhanging list or the Sperner village change, then first the Moore
family associated with the new list, and the reduced context is then calculated.
Similarly, if the input is an arbitrary set system, then the closure system is cal-
culated as in Sec. 3.2. If the input is on the set of implications, since implications
are usually on the set of attributes, the corresponding Moore family will be dual
to the previous ones. From the reduced context, we are using Closed by One
algorithm to calculate the concept lattice.
Fig. 1 shows a screen shot of the software with the five different views: the
Moore family on top, the context as a datagrid, the list of implications, the list
of overhanging and the list of Sperner families. The drawing of the lattice is done
with a simple algorithm that optimise the position of each concept depending
on the number (and position) of concepts covered.
3 Cryptomorphisms of Lattices
In this section, we present the different cryptomorphisms of lattices implemented
in CryptoLat in details; we do not claim to be exhaustive, so we refer the readers
to [9] for a survey on those cryptomorphisms. Since we are describing implemen-
tations, the following descriptions needs only to be for the finite case.
CryptoLat - a Pedag. Soft. on Lattice Cryptomorphisms and Lattice Prop. 95
Fig. 1. CryptoLat software with the different views displaying our running example
3.1 Lattice and Irreducible Elements
Different yet equivalent definitions for lattices exist: here, we will define a lat-
tice (L, ∧, ∨) as a set L with two binary operators satisfying the following four
properties:
– Associativity: x ∧ (y ∧ z) = (x ∧ y) ∧ z and x ∨ (y ∨ z) = (x ∨ y) ∨ z
– Commutativity: x ∧ y = y ∧ x and x ∨ y = y ∨ x
– Idempotence: x ∧ x = x and x ∨ x = x
– Absorption: x ∧ (y ∨ x) = x and x ∨ (y ∧ x) = x
One can associate an order on the elements of L to these operators as follow:
∀x, y ∈ L, x ≤ y if x ∧ y = x or, equivalently, x ∨ y = y. Fig. 2 shows a
lattice having six elements using its Hasse diagram representation: the elements
of L are depicted by nodes in the plane, and a line segment connects the nodes
corresponding to a covering pair x ≺ y (i.e. x ≤ y and ∀z, x ≤ z ≤ y implies
x = z or y = z), where the node representing y is placed ’higher’ in the plane
than the node representing x.
Some elements of lattices are of particular interest: in any lattice, we have
a minimal element, usually denoted by 0L or ⊥, and a maximal element 1L or
>. Irreducible elements of the lattice are also the subject of studies since they
constitute minimal generating sets, and are often used to characterize classes
of lattices. An element j of L is join-irreducible if j = ∨X then j ∈ X, or,
96 Florent Domenach
f
e
b c d
a
Fig. 2. Example of a lattice with 6 elements
equivalently, j covers only one element in L. Dually, an element m of L is meet-
irreducible if m = ∧X then m ∈ X, or m is covered by only one element of L.
Let JL and ML be the sets of join and meet-irreducible of L.
3.2 Moore Family and Closure Operator
Let S be a finite set. A closure system (or Moore family) C on S is a family of
subsets of S satisfying S ∈ C and, for all A, B ∈ C, A ∩ B ∈ C. A closure system
is a lattice with the following operations:
– C1 ∧ C2 = C1 ∩ C2
– C1 ∨ C2 = ∩{C ∈ C : C1 ∪ C2 ⊆ C}
Let A be an arbitrary set system on a set S. Then one can associate a closure
system CA to A:
CA = {∩{Ai : Ai ∈ A}} ∪ {S}
Example 1 (continued). The family {∅, {G0 }, {G1 }, {G2 }, {G1 , G2 }, {G0 , G1 , G2 }},
with the operators previously defined, constitute a lattice isomorphic to the lat-
tice of Fig. 2.
Another fundamental cryptomorphism of lattice concerns closure operators.
A closure operator on a set S is a map σ : P(S) → P(S) satisfying ∀X, Y ⊆ S:
– Isotone: X ⊆ Y ⇒ σ(X) ⊆ σ(Y )
– Extensive: X ⊆ σ(X)
– Idempotent: σ(σ(X)) = σ(X)
These three properties together are equivalent to extensivity and path indepen-
dence property [20]: σ(X ∪ Y ) = σ(σ((X) ∪ σ((Y )).
The well-known equivalence between closure operators and closure systems
is the following: given the closure system C, the associated closure operator is
σC (X) = ∩{C ∈ C : X ⊆ C}. Conversely, given a closure operator σ, the
associated closure system is the set of fixed points of σ, i.e. Cσ = {X ⊆ S : X =
σ(X)}.
CryptoLat - a Pedag. Soft. on Lattice Cryptomorphisms and Lattice Prop. 97
3.3 Context Associated with a Lattice
One of the most celebrated cryptomorphism in the FCA community is the one
between a lattice and a binary relation, also known as the basic theorem of
Formal Concept Analysis [3, 13].
Formally, a context (G, M, I) is a binary relation I between a set of objects
G and a set of attributes M . One can associate the Galois connection between
P(G) and P(M ) to this binary relation as ∀A ⊆ G, B ⊆ M ,
A0 = {m ∈ M : ∀g ∈ A, (g, m) ∈ I}
B 0 = {g ∈ G : ∀m ∈ B, (g, m) ∈ I}
The lattice associated with the context (G, M, I), also called the concept
lattice or Galois lattice, is the set of all concepts (A, B), with A0 = B and
B 0 = A, together with the order (A1 , B1 ) ≤ (A2 , B2 ) ⇐⇒ A1 ⊆ A2 ( ⇐⇒
B1 ⊇ B2 ). Conversely, any lattice L is isomorphic to the Galois lattice of the
context (JL , ML , ≤), also called the reduced context.
Two other binary relations, the arrows relations, on P(J) × P(M ) are often
used for characterization theorems. Let j ∈ J, m ∈ M , we define:
– j ↑ m ⇐⇒ j 6≤ m and j < m+
– j ↓ m ⇐⇒ j 6≤ m and j − < m
– j l m ⇐⇒ j ↑ m and j ↓ m
Example 1 (continued). The lattice associated with the context of Fig. 3 is iso-
morphic to the lattice of Fig. 2.
Fig. 3. The reduced context, together with the arrows relations, associated with our
running example.
3.4 Implications
An implicational system I on S is a binary relation on P(S), and if (A, B) ∈ I,
we denote it A →I B or A → B if no confusion is possible. We say that A
implies B or A → B is an implication (of I). A complete implicational system I
is an implicational system on P(S) satisfying, ∀A, B, C, D ⊆ S:
– B ⊆ A implies A → B;
98 Florent Domenach
– A → B and B → C imply A → C;
– A → B and C → D imply (A ∪ C) → (B ∪ D).
[2] showed that a one-to-one correspondence between closure operators and
complete implicational system exists. The closure system associated to a com-
plete implicational system is CI = {C ⊆ S : ∀X ⊆ S, [X ⊆ C and X →
Y ] ⇒ Y ⊆ C}, while the implicational system associated to a closure mapping
is {X → Y : Y ⊆ σ(X)}.
Example 1 (continued). The complete implicational system associated with the
lattice in Fig. 2 is the following:
c → e; d → e; b, c → d;
b, c → e; bc → d, e; b, d → c;
b, d → e; b, d → c, e; b, e → c;
b, e → d; b, e → c, d; c, d → b;
c, d → e; c, d → b, e; c, d, e → b;
b, d, e → c; b, c, e → d; b, c, d → e;
As the above example shows, many implications in a complete implicational
system can be redundant: in our example, it is evident that if we have c →
e then we have b, c → e. So smaller implicational system allowing a smaller
representation of a complete implicational system is of interest. We say that
an implicational system Σ is a generating system for I if any implication of I
can be obtained from Σ by applying recursively the rules defining a complete
implicational system. A minimal generating system for I is called a basis for I,
and a minimum basis is called a canonical basis.
One particular basis, called the stem basis [15], can be directly defined: the
stem basis is made of all implications X → σ(X) − X, with X a critical set. A
set Q ⊆ S is called a quasi-closed set if Q 6∈ C and C + {Q} is a closure system,
and Q is a F -quasi-closed set if Q is quasi-closed and σ(Q) = F . So a set C ⊆ S
is critical if there exists F ∈ C such that C is a minimal F -quasi-closed set.
Example 1 (continued). Canonical basis associated with the lattice of Fig.2.
b, e → c, d; c → e; d → e; c, d, e → b;
3.5 Overhanging Relation
An overhanging relation O is a binary relation on P(S) that was originated in
[1] in consensus theory on trees under the term of nesting, and was generalized
in [12] to any lattice. We will indifferently write (A, B) ∈ O or A O B to signify
that a set A is overhanged in B. An overhanging relation satisfies the following
properties:
– for all A, B ∈ S, A O B implies A ⊂ B;
– for all A, B, C ∈ S, A ⊂ B ⊂ C implies [A O C ⇐⇒ A O B or B O C];
– for all A, B ∈ S, A O (A ∪ B) implies (A ∩ B) O B.
CryptoLat - a Pedag. Soft. on Lattice Cryptomorphisms and Lattice Prop. 99
We showed in [12] that overhanging relations and closure operators (and
so lattices) are in a one-to-one correspondence by associating the overhanging
relation defined as A O B ⇐⇒ A ⊂ B and σ(A) ⊂ σ(B) to any closure operator,
and, conversely, associating the closure operator σ(X) = {x ∈ S : x Oc X ∪ {x}}
to any overhanging relation. In fact, an overhanging relation can be seen as a
kind of negative implication, as given by the equivalence that A → B if and only
if (A, A ∪ B) 6∈ O
Example 1 (continued). The overhanging relation associated with the lattice of
Fig.2 is as follows:
∅Ob ∅Oc ∅Od
∅ O b, c ∅ O b, d ∅ O c, d
∅ O b, c, d b O b, c b O b, d
b O b, c, d c O b, c c O c, d
c O b, c, d d O b, d d O c, d
d O b, c, d c, d O b, c, d
3.6 Sperner Family
A Sperner family F on S is a family s.t. ∀X, Y ∈ F, X and Y are incomparable
for set inclusion. A Sperner village V = (F1 , ..., Fn ) on S is a set of n Sperner
families. The equivalence between a closure operator and a Sperner village is
obtained as follows: given σ closure operator, for any x ∈ S we create the Sperner
family Cx = {X ⊆ S : x ∈ σ(X) and X minimal for that property}. Conversely,
the closure operator associated with V is defined as σ(X) = {x ∈ S : X contains
a set of Cx }.
Example 1 (continued). The Sperner village equivalent with our running exam-
ple is the following Fig. 4.
When we add to the existing Sperner village the Sperner family {{e}, {b, c}},
Fig. 4. Sperner village associated with our running example.
the software computes the smallest Sperner village possible from the existing
village together with the new family, and from this Sperner village finds the
associated concept lattice. Fig. 5 shows the results.
100 Florent Domenach
Fig. 5. Screenshot of the software after adding a new Sperner family
4 Properties of lattices
The main focus of the software concerns the dependency between the different
cryptomorphisms associated with lattices. In the same pedagogical spirit, we
also implemented different properties of lattices and elements of lattices, briefly
described in Tab. 1 and in Tab. 2 for completeness sake. We refer the readers to
[11, 14, 22] for more information on lattices and lattice properties.
5 Existing Software
All of the software mentioned in Table 3 have different goals. In terms of ease of
use, Concept Explorer is the closest to our goal, but offers limited functionalities.
OpenFCA is similar to Concept Explorer, offering an interesting, web based and
intuitive approach to context creation, lattice visualization and attribute explo-
ration. FCAstone’s main purpose is file formats conversion to improve interoper-
ability between FCA and graph editing. The other software (Galicia, ToscanaJ,
Lattice Miner, FCART) were designed to push the boundaries of the scalability
of analysis and lattice drawing. Research wise, Formal Concept Analysis Re-
search Toolbox (FCART) is a particularly interesting approach of a universal
CryptoLat - a Pedag. Soft. on Lattice Cryptomorphisms and Lattice Prop. 101
Table 1. List of lattice properties
Types of Lattices
Atomistic Every join-irreducible element of L is an atom
Co-atomistic Every meet-irreducible element is covered by 1L
Ranked Tthere is a ranking function r : L → N such that ∀x, y ∈ L, x ≺ y
implies r(y) = r(x) + 1
Upper semi-modular ∀x, y ∈ L: x ∧ y ≺ x ⇒ y ≺ x ∨ y
Lower semi-modular ∀x, y ∈ L: y ≺ x ∨ y ⇒ x ∧ y ≺ x
Modular Upper and lower semi-modular
Distributive ∀x, y, z ∈ L, x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)
Semi-distributive ∀x, y, z ∈ L, we have both x ∧ y = x ∧ z ⇒ x ∧ y = x ∧ (y ∨ z) and
x ∨ y = x ∨ z ⇒ x ∨ y = x ∨ (y ∧ z)
Families of Lattices
Chain ∀x ∈ L, x 6= 0L , x ∈ JL
Hierarchy ∀x ∈ S, {x} ∈ L and ∀A, B ∈ L, A ∩ B ∈ {∅, A, B}
Weak hierarchy ∀x ∈ S, {x} ∈ L and ∀A, B, C ∈ L, A∩B ∩C ∈ {A∩B, A∩C, B ∩C}
Boolean Isomorphic to the power set of some set
Complemented Lattices
Complemented ∀x ∈ L, ∃y ∈ L: x ∧ y = 0L and x ∨ y = 1L
Uniquely complemented The complement is unique
Pseudo-complemented ∀x ∈ L, ∃y ∈ L: x ∧ y = 0L and ∀z ∈ L, x ∧ z = 0L ⇒ z ≤ y
Relatively comple- Any [u, v] is relatively complemented, i.e. ∀x in [u, v], ∃y such that
mented x ∧ y = u and x ∨ y = v
Uniquely relatively com- The complement in [u, v] is unique
plemented
Sectionally comple- Any [0L , v] is complemented
mented
Other Lattices Properties
δ-hard ∀j, j 0 ∈ JL , jδj 0 , i.e. there exists x ∈ L such that j 6≤ x, j 0 6≤ x and
j < j0 ∨ x
δ-strong The transitive graph of the δ relation is complete
Sperner poset L is ranked, and the size of the maximal antichain is equal to the
maximal size rank
Table 2. List of elements properties
x is distributive ∀y, z ∈ L, x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z)
x is standard ∀y, z ∈ L, y ∧ (x ∨ z) = (y ∧ x) ∨ (y ∧ z)
x is neutral ∀y, z ∈ L, (x ∧ y) ∨ (y ∧ z) ∨ (x ∧ z) = (x ∨ y) ∧ (x ∨ z) ∧ (y ∨ z)
x is an articulation point ∀y ∈ L, either x ≤ y or y ≤ x
x is join-prime x ≤ a ∨ b implies x ≤ a or x ≤ b
x is meet-prime a ∧ b ≤ x implies a ≤ x or b ≤ x
102 Florent Domenach
integrated environment to lattice construction and Formal Concept Analysis
techniques, where one can test new algorithms for drawing lattice for example.
Table 3. Some FCA software tools (taken from [18])
Program title Authors Web-site
Concept Explorer S.A. Yevtushenko [25] conexp.sourceforge.net
Galicia P. Valtchev et al. [23] www.iro.umontreal.ca/~galicia
ToscanaJ U. of Queensland, TU Darmstadt [24] toscanaj.sourceforge.net
FcaStone U. Priss et al. [21] fcastone.sourceforge.net
Lattice Miner B. Lahcen et al. [17] lattice-miner.sourceforge.net
Conexp-clj TU-Dresden, D. Brochman daniel.kxpq.de/math/conexp-clj
OpenFCA P. Borza, O. Sabou et al. [8] code.google.com/p/openfca
FCART A. Neznanov and D. Ilvovsky [18]
6 Concluding Remarks
CryptoLat is a pedagogical and research oriented software in active development
phase, and it is open source and freely available. It is still in alpha testing, and
the next milestone of development is to make it cloud based with a web interface.
Many other cryptomorphisms of concept lattices exist but are not imple-
mented yet. For example, [4] proved that we can associate a minimal separator
of the co-bipartite graph to every concept of the concept lattice, and [16] showed
a bijection between Moore families and ideal colour sets of the coloured poset.
Similarly we are planning to increase the number of properties available, adding
for example the dismantlable property (one can repeatedly remove a doubly ir-
reducible element until the lattice becomes a chain) or being a convex geometry
(∅ ∈ L and ∀x ∈ L, there is a unique minimal (for inclusion) generator of x). the
lattice drawing component is also under consideration, to be enhance and more
user friendly.
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