Discrete Mathematics

1. A lattice is a poset where every pair of elements has both a
supremum and an infimum.
Definition Lattice: A poset (P,v) is called a lattice, if for all
x, y 2 P the subset {x, y} of P has a supremum and an
infimum. The supremum of x and y is denoted by x t y and
the infimum as x u y.
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2.  Supremum: We say that A is bounded above if there
is b∈R such that ∀x∈A (x⩽b). The number b is called
Supremum for A.
 Infimum : We say that A is bounded below if there
is c∈R such that ∀x∈A (x⩾c). The number c is called
an Infimum for A.
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3.  (R,) is a lattice. If x, y 2 R, then sup{x, y} = max{x, y}
and inf{x, y} =min{x, y}.
 If S is a set and P = P(S) the poset of all subsets of S
with relation , then P is a lattice with u = and t = [.
 The poset (N, |) of natural numbers with order
relation | is a lattice with the least common multiple as
t and the greatest common divisor as u.
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5.  Definition : The dual of a lattice Λ is the set Λˆ of all
vectors x ∈ span(Λ) such that hx, yi is an integer for all
y ∈ Λ.
Ex-
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6.  a set of elements of a lattice, in which each subset of
two elements has a least upper bound and a greatest
lower bound contained in the given set.
 A lattice (L,∨,∧) is distributive if the following
additional identity holds for all x, y, and z in L:
- x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z).
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7. Modular Lattice: A modular lattice is
a lattice L=⟨L,∨,∧⟩ that satisfies the modular identity.
identity: ((x∧z)∨y)∧z=(x∧z)∨(y∧z)
Bounded Lattice: A bounded lattice is an algebraic
structure , such that is a lattice, and the
constants satisfy
The element 1 is called the upper bound, or top of and
the element 0 is called the lower bound or bottom of .
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8. Definition : A complemented lattice is a
bounded latticee (with least elementt 0 and greatest
elementt 1), in which every element a has
a complement, i.e. an element b satisfying a ∨ b = 1
and a ∧ b = 0. Complements need not be unique.
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9. Partial order and lattice theory now play an important role in
many disciplines of computer science and engineering.
For example-> they have applications in distributed
computing (vector clocks, global predicate detection),
concurrency theory (pomsets, occurrence nets),
programming language semantics (fixed-point semantics),
and data mining (concept analysis). They are also useful in
other disciplines of mathematics such as combinatorics,
number theory and group theory. In this book, I introduce
important results in partial order theory along with their
applications in computer science. The bias of the book is
on computational aspects of lattice theory (algorithms)
and on applications (esp. distributed systems).
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