**INTRODUCTION**

The notion of outer measure connected to lattice measure was introduced by
Hus (2000). Subsequent to that Khare
and Singh (2005) contributed to the concept of weakly tight functions and
their decomposition. Later on Khurana (2008) developed
the idea of lattice valued Borel measures. Hann decomposition in signed fuzzy
measure version was established by Tanaka (2008) and
further Tanaka (2009) derived a Hann decomposition for
signed lattice measure and built-up the concept of σ-algebra. Recently
the structure of gamma lattice was through by Kumar *et
al*. (2011a). Most recently Radon-Nikodym theorem and its uniqueness
of signed lattice measure was established by Kumar *et
al*. (2011b). Jordan decomposition and its uniqueness of signed lattice
measure were developed by Kumar *et al*. (2011c).

The class of positive lattice measurable sets and positive lattice measurable
functions were exposed by Pramada *et al*. (2011).
Further the class of super lattice measurable sets was successfully studied
by Pramada *et al*. (2011). Complex integrable
lattice functions and ì-free lattices were recognized by Pramada
*et al*. (2012b,c). Further Pramada
*et al*. (2012a) initiated the Boolean valued star and mega lattice
functions. Putcha and Malladi (2010) formulated a mathematical
model on litter, detritus and predators in mangrove estuarine ecosystem and
solved system by extending the Adomian’s decomposition method. Deekshitulu
*et al*. (2011) established some fundamental inequalities and comparison
results of fractional difference equation of Volterra type. Anand
*et al*. (2011) found multiple symmetric positive solutions for a system
of higher order two-point boundary-value problems on time scales by determining
growth conditions and applying a fixed point theorem in cones under suitable
conditions. Putcha (2012) constructed the approximate
analytical solutions of two species and three species ecological systems using
homotopy ananlysis and homotopy perturbation methods.

A class of measurable Borel lattices was established by Kumar
*et al*. (2011d). The concepts Boolean valued measurable functions,
function lattice, σ-lattice and lattice measurable space were contributed
by Kumar *et al*. (2011e).

This study established a general agenda for the study of characterization of
formal and conformal systems. Further, it has been noticed that measures of
theoretical concepts were generalized in terms of σ-algebra. Some elementary
characteristics of lie lattice σ-algebra has been proved and finally confirmed
that the lie lattice σ-algebra generated by formal system contained in
the lie lattice σ-algebra generated by conformal system.

**PRELIMINARIES**

In this manuscript it has been considered that the union and intersection of
set theory as the binary operations ∧ and ∨. Further, it was briefly
reviewed the well-known facts described by Birkhoff (1967),
proposed an extension lattice and investigated its properties.

The system (L, ∧, ∨) where L is a non empty set together with binary
operations ∧, ∨ called a lattice if it satisfies, for any elements x,
y, z, in L:

• |
**The commutative law (L1):** x∧y = y∧x and x∨y
= y∨x |

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**The associative law (L2):** x∧ (y∧z) = (x∧y) ∧z and
x∨ (y∨z) = (x∨y) ∨z |

• |
**The absorption law (L3):** x∨ (y∧x) = x and x∧ (y∨x)
= x |

• |
Hereafter, the lattice (L, ∧, ∨) will often be written as L for
simplicity |

A mapping h from a lattice L to another lattice L^{1} is called a lattice-homomorphism,
if it satisfies:

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h (x∧y) = h(x) ∧h(y) and h (x∨y) = h (x) ∨h
(y), for all x, yεL |

If h is a bijection, that is, h is one-to-one and onto, it is called a lattice
isomorphism and in this case, L^{1} is said to be lattice-isomorphic
to L.

A lattice (L, ∧, ∨) is called distributive if, for any x, y, z, in L.

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**The distributive law holds (L4):** x∨ (y∧z) =
(x∨y) ∧ (y∨z) and x∧ (y∨z) = (x∧y) ∨ (y∧z) |

A lattice L is called complete if, for any subset A of L, L contains the supremum
∨A and the infimum ∧A. If L is complete, then L itself includes the maximum
and minimum elements which are often denoted by 1 and 0 or I and O, respectively.

A distributive lattice is called a Boolean lattice if for any element x in
L, there exists a unique complement x^{c} such that:

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**The law of excluded middle (L5):** x∨x^{c}
= 1 |

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**The law of non-contradiction (L6):** x∧x^{c} = 0 |

Let L be a lattice and c: L→L be an operator. Then c is called a lattice
complement in L if the following conditions are satisfied:

• |
**L5 and L6:** for all xεL, x∨x^{c} = 1
and x∧x^{c} = 0 |

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**The law of contrapositive (L7):** for all x, yεL, x<y implies
x^{c}>y^{c} |

• |
**The law of double negation (L8):** for all xεL, (x^{c})^{c}
= x |

Throughout this study, it has been considered the lattices as complete lattices
which obey L1-L8 except for L6 the law of non-contradiction.

**LIE LATTICE SIGMA ALGEBRAS**

Unless otherwise stated, X is the entire set and F is a lattice of any subsets
of X.

**Definition 1:** If a lattice F satisfies the following conditions, then
it is called a lattice σ-algebra:

• |
For all hεF, h^{c}εF |

• |
If h_{n}εF for n = 1, 2, 3 ....., then |

Denote σ (F) is a lattice σ-algebra generated by F.

**Example 1:** (i) {φ, X} is a lattice σ-algebra and (ii) P(X)
power set is a lattice σ-algebra.

**Example 2:** Let X = ,
F = {measurable subsets of }
with usual ordering (≤). Here F is a lattice, σ (F) is a lattice σ-algebra
generated by F. Where
is an extended real number system.

**Example 3:** Let X be any non-empty set and F = {all topologies on X}.
Here F is a complete lattice but not σ-algebra.

**Example 4:** Let X =
and F = {E</E
is finite or E^{c} is finite}. Here F is lattice algebra but not lattice
σ-algebra.

**Definition 2:** The lattice σ-algebra F of all sub sets of X lies
between {φ, X}<F<P(X) is called a lie lattice σ-algebra.

**Example 5:** A partition of X is a collection of disjoint subsets of X
whose union is all of X. For simplicity, consider a partition consisting of
a finite number of sets A_{1}, A_{2},……. Thus:

A_{i}∧A_{j} = φ and A_{1}∨A_{2}
…. A_{n} = X |

Then the collection F of all unions of the sets A_{j} forms a lie lattice
σ-algebra.

**Theorem 1:** If F be a lie lattice σ-algebra of subsets of X, then
the following conditions hold good:

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XεF |

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If A_{1}, A_{2}……. A_{n}εF, then
A_{1}∨A_{2}∨…….. A_{n}εF |

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If A_{1}, A_{2}……. A_{n}εF, A_{1}∧A_{2}∧…….
A_{n}εF |

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If A_{1}, A_{2}…… is a countable collections
of sets in F then |

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If A, BεF then A-BεF |

**Proof:**

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Since φεF and X = φ^{c} it follows
that XεF |

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A_{1}∨A_{2} …. A_{n} = A_{1}∨A_{2}
…. A_{n}∨φ∨φ∨..….. εF (definition
of 2) |

• |
Since A_{1}∧A_{2} …. A_{n} = (A_{1}^{c}∧A_{2}^{c}……
A_{n}^{c})^{c} which is in F because each A_{i}^{c}εF
and F is closed under finite unions, from (2) it follows that A_{1}∧A_{2}∧…….
A_{n}εF |

• |
can be expressed as
and is in F. Since F is closed under complementation and countable unions |

• |
Since A, B^{c}εF it follows that A-B = A∧B^{c}
is in F |

**Definition 3:** Let B a non-empty collection of subsets of a set X. The
smallest lattice σ-algebra containing all the sets of B is denoted by σ(B)
and is called the indiscrete lattice σ-algebra generated by the collection
B.

**Note 1:** Any lattice σ-algebra containing the sets of B must contain
all the sets of σ(B). In the entire discussion it is assumed that the symbol
<represents the set operation proper subset (⊂).

**Observation 1:** From the definition 2 of lie lattice σ-algebra it
follows that if G is any non-empty collection of lie lattice σ-algebras
of subsets of X, then the meet ∧G is indiscrete lie lattice σ-algebra
of subsets of X. That is ∧G = {A<X | AεF for every FεG} consists
of all sets A which belong to each lie lattice σ-algebra F of G.

**Note 2:** Given a collection B of subsets of X, let G_{B} be the
collection of all lattice σ-algebras including containing all the sets
of B. Note that P(X)εG_{B} and so G_{B} is non empty. Then
∧G_{B} is a lie lattice σ-algebra, contains all the sets of
B and is minimal among such lie lattice σ-algebras. Minimally means if
F is a lie lattice σ-algebra such that B<F then ∧G_{B}<F
thus ∧G_{B} is the lie lattice σ-algebra. This lie lattice
σ-algebra is a indiscrete lie lattice σ-algebra.

**Definition 4:** Let X be a set, A collection P of subsets of X is called
a formal system. If it is closed under finite intersections that is if A_{1},
A_{2}…… A_{n}
are a finite number of sets in P, then their intersection A_{1}∧A_{2}…..
A_{n} is also in P.

**Definition 5:** A collection L of subsets of X is called a conformal system
if:

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L contains the empty set φ |

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L is closed under complementation. That is if AεL then A^{c}εL |

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L is closed under countable disjoint union. That is if A_{1},
A_{2}, ….. εL and A_{i}∧A_{j} = φ
for every i≠j, then |

**Result 1:** Every conformal system is closed under proper differences,
that is, if A, BεL, where L is a conformal system and A<B then the difference
B-A is also in L.

**Proof:** Since B-A can be expressed as A∨B^{c }when ever A<B
and L being a conformal system it follows that B-AεL. The same thing can
also be realized from the fact that B-A = B∧A^{c} = (B^{c}∨A)^{c}.

**Result 2:** A family which is both formal and conformal system is a lie
lattice σ-algebra.

**Proof:** Let S be a collection of subsets of X which is both formal system
and conformal system. To prove that S is a lie lattice σ-algebra it is
sufficient to show that S is closed under countable union (not just disjoint
countable unions).

Let A_{1}, A_{2} …..….. εS. By rewriting
as a countable union of disjoint sets
, where B_{1} = A_{1} and B_{n} = A_{n}-(A_{1}∨A_{2}∨…..
A_{n-1}) = A_{n}∧A_{1}^{c}∧A_{2}^{c}∧……
∧A_{n}^{c}, for n = 1.

Thus, B_{n} consists of all elements of A_{n} which do not
appear in all A_{i}, 1 = i = n-1. From the construction of B_{i}’s
(i=1,2,3,...), it follows that they are mutually disjoint. Since S is conformal
and formal it follows that S is closed under complementation and finite intersection,
respectively.

**Result 3:** Suppose L^{1} is a conformal system of X. For any
set Aε L^{1}, let S_{A} be the set of all B<X for which
A∧BεL^{1}. Then S_{A} is a conformal system.

**Proof:** The set S_{A} contains the null set φ since A∧φ
= φ and is in L^{1}. It is also clear that S_{A} is closed
under countable disjoint unions.

Let BεS_{A} and observe that A∧B^{c} = A-B = A-(A∧B)
and is in L^{1}.

Therefore S_{A} is closed under complementation.

**Result 4:** The intersection l(P) of all conformal systems containing
P is formal.

**Proof:** Let Aεl(P) and let S_{A} be the set of all sets
B<X for which A∧B is in l(P).

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From result 3 it follows that S_{A} is a conformal
system and P is in S_{A} |

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Thus P<S_{A}. Therefore, l(P)<S_{A} |

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Similarly P<S_{B} whenever Bεl(P) |

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Result 3, infers that S_{B} is a conformal system |

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Therefore, l(P) <S_{B}. Thus, l(P) is a formal system |

**Theorem 3:** The lie lattice σ-algebra generated by a formal system
P and a conformal system generated by L is contained in L.

**Proof:** Let P is a formal system and L is a conformal system, with P<L:

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The line of attack is to establish the existence of a lie
lattice σ-algebra between P and L |

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This will imply that σ(P) is the smallest lie lattice σ-algebra
containing P and is contained in L |

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From result 4 it follows that l(P) is also a formal system |

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Now result 2 infers that l(P) is a lie lattice σ-algebra |

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From result 4 and 2 it follows, respectively that l(P) is a formal system
and is a lie lattice σ-algebra |

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From the definition of l(P), P<l(P) <L and L is just one conformal
system containing P |

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Thus it was shown the existence of lie lattice σ-algebra l(P) lying
between P and L |

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Therefore, P<σ(P) <l(P) <L, where σ(P) is the intersection
of all lie lattice σ-algebras which contain P |

**CONCLUSION**

This study illustrates the notions of lie lattice σ-algebra, indiscrete
lattice σ-algebra, formal system and conformal system. Also it establishes
some characterizations of formal and conformal systems. Finally it confirms
that, the lie lattice σ-algebra generated by formal system is contained
in the lie lattice σ-algebra generated by conformal system.

**ACKNOWLEDGMENT**

Venkata Sundaranand Putcha is supported by project Lr. No. SR/S4/MS: 516/07
and Dt.21-04-2008 from the DST-CMS.