# Ordered sets and Lattices

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```					Ordered sets and Lattices
CS 5303 – Logical Foundations of
Computer Science
Orders and pre-orders
• These relations are very important in
Computer Science as they give a mean to
compare objects
• Definition: an order relation is a reflexive,
antisymmetric and transitive relation. A
strict order is an irreflexive and transitive
relation
• Example: >, <, , , order in a tree
Total order and partial order
• Definition: if R is an order relation such that for
a,a’ and aa’, then aRa’ or a’Ra, R is called a
total order.
• Example:  and  are total orders over ℕ or over
the real numbers etc…
Exercise: show that over the natural numbers, |
(the division) is a partial order. Show that if E is
a set, then  is a partial order over P(E) (show
that both orders are not total).
Pre-order

• Definition: a pre-order relation is a
transitive relation
Exercise: Let E be a set and IdE the relation
defined by (a,a’)IdE iff a=a’. Let R be q
pre-order over E. Then show that
IdE  (R  R-1) is an equivalence relation
Ordered sets

• An ordered set (E,) is a set E endowed
with an order .
• Example: (ℕ,) is an ordered set and so is
(ℕ,|).
Min and Max 1
• Definition: if E’ is a subset of and ordered
set (E,), then an element of E is a
majoring element of E’ if yx for all y in E’
• Minoring elements are defined similarly.
• Maj(E’) and Min(E’) denotes the set of
majoring and minoring elements.
Exercise: show that Maj(E’)E’ and
Min(E’)E’ have at most one element
Min and Max 2
• Definition: if Maj(E’)E’ and Min(E’)E’ are non
empty then their elements are called the
maximum of E’ and minimum of E’ respectively.
•   Definition: an element x is the least greatest
bound (sup) of E’ in E if
for any y in E’ yx and if z in E is such that for
any y in e’ yz then xz.
The inf is defined similarly.
Lattices

• Definition: if (E,) is an ordered set, then
it is a lattice if every pair of elements x,y
admits an sup and an inf.
• Notations: sup(x,y) is sometimes written
x∨y and inf(x,y) is written x∧y
Exercise: show that (P(E),) is a lattice.
Properties
• The inf and sup are
1. idempotent
2. commutative
3. associate
4. absorbing x∧(x∨y)=x = (y∧x)∨x
• If inf and sup are distributive wrt one
another, the lattice is called a distributive
lattice
Exercise: distributive lattice?

⊤

a       b     c

⊥
Complemented lattice
• Definition: a lattice is complemented iff
1. There is a minimum ⊥ and a maximum ⊤ that
are distinct.
2. There exists an application :EE such that
for all x in E,
- x∧(x)= ⊥
- x∨(x)= ⊤
Exercise: Show that (P(E), ) is a complemented
lattice
Complete lattices

• Definition: a lattice is called a complete
lattice if every subset A of A admits a sup
and an inf.
Exercise: show that (P(E),) is a complete
lattice.
Application: formal concept analysis

• FCA is a model to formalize the notions of
concepts and conceptual hierarchies
•   Critical for conceptual data analysis and
knowledge processing
•   Successfully used in decision engines,
classification engines (e-mail), but also formal
methods and patterns in software engineering,
analysis of flight movements (Frankfurt),
analysis of diabetic children (mcgill U.)
Basics of FCA
• Definition: a formal context K=(G,M,I)
consists of a set G of objects, a set M of
attributes and a binary relation I over GxM
where gIm (or (g,m)I) reads object g
has property m
• I is called the incidence relation
• Small contexts can be represented by
cross tables
Formal concepts
• Definition: a formal concept of the context
(G,M,I) is a pair (A,B) with AG, BM and
A’=B, B’=A with
• A’={m in M| gIm for all g in A}
• B’={g in G | gIm for all m in B}
• If (A1,B1) and (A2,B2) are concepts then
A1 A2 defines an order on the concepts
making B(G,M,I) the concept lattice
Example (R. Godin – UQAM)

Ref:http://www.info.uqam.ca/~godin/
Formal concept analysis based normal forms
Software development
Example 2 (Godin)

• Designing and maintaining good class
hierarchy large solution space,
conflicting criteria, evolution
• FCA  minimize redundancy, number of
classes, multiple inheritance only if
necessary, subclasses as specialization
Example 3 (Godin)

• G classes, objects
• M instance variables, methods
• I comes from analyst, code, patterns

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