Abstract: This study is an exploration on lie lattice σ-algebra, indiscrete lattice σ-algebra, formal system and co-formal systems. It has acknowledged some characterizations of formal and co-formal systems. Finally, it corroborates that the lie lattice σ-algebra generated by formal system contained in the lie lattice σ-algebra is generated by co-formal system.
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 |
| • | 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 L1 is called a lattice-homomorphism, if it satisfies:
| • | 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, L1 is said to be lattice-isomorphic to L.
A lattice (L, ∧, ∨) is called distributive if, for any x, y, z, in L.
| • | 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 xc such that:
| • | The law of excluded middle (L5): x∨xc = 1 |
| • | The law of non-contradiction (L6): x∧xc = 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∨xc = 1 and x∧xc = 0 |
| • | The law of contrapositive (L7): for all x, yεL, x<y implies xc>yc |
| • | The law of double negation (L8): for all xεL, (xc)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, hcεF |
| • | If hnεF 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 =
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 =
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 A1, A2,……. Thus:
Ai∧Aj = φ and A1∨A2
…. An = X |
Then the collection F of all unions of the sets Aj 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:
| • | XεF |
| • | If A1, A2……. AnεF, then A1∨A2∨…….. AnεF |
| • | If A1, A2……. AnεF, A1∧A2∧……. AnεF |
| • | If A1, A2…… is a countable collections
of sets in F then |
| • | If A, BεF then A-BεF |
Proof:
| • | Since φεF and X = φc it follows that XεF |
| • | A1∨A2 …. An = A1∨A2 …. An∨φ∨φ∨..….. εF (definition of 2) |
| • | Since A1∧A2 …. An = (A1c∧A2c…… Anc)c which is in F because each AicεF and F is closed under finite unions, from (2) it follows that A1∧A2∧……. AnεF |
| • | |
| • | Since A, BcεF it follows that A-B = A∧Bc 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 GB be the collection of all lattice σ-algebras including containing all the sets of B. Note that P(X)εGB and so GB is non empty. Then ∧GB 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 ∧GB<F thus ∧GB 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 A1, A2…… An are a finite number of sets in P, then their intersection A1∧A2….. An is also in P.
Definition 5: A collection L of subsets of X is called a conformal system if:
| • | L contains the empty set φ |
| • | L is closed under complementation. That is if AεL then AcεL |
| • | L is closed under countable disjoint union. That is if A1,
A2, ….. εL and Ai∧Aj = 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∨Bc 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∧Ac = (Bc∨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 A1, A2 …..….. εS. By rewriting
Thus, Bn consists of all elements of An which do not appear in all Ai, 1 = i = n-1. From the construction of Bi’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 L1 is a conformal system of X. For any set Aε L1, let SA be the set of all B<X for which A∧BεL1. Then SA is a conformal system.
Proof: The set SA contains the null set φ since A∧φ = φ and is in L1. It is also clear that SA is closed under countable disjoint unions.
Let BεSA and observe that A∧Bc = A-B = A-(A∧B) and is in L1.
Therefore SA is closed under complementation.
Result 4: The intersection l(P) of all conformal systems containing P is formal.
Proof: Let Aεl(P) and let SA be the set of all sets B<X for which A∧B is in l(P).
| • | From result 3 it follows that SA is a conformal system and P is in SA |
| • | Thus P<SA. Therefore, l(P)<SA |
| • | Similarly P<SB whenever Bεl(P) |
| • | Result 3, infers that SB is a conformal system |
| • | Therefore, l(P) <SB. 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:
| • | The line of attack is to establish the existence of a lie lattice σ-algebra between P and L |
| • | This will imply that σ(P) is the smallest lie lattice σ-algebra containing P and is contained in L |
| • | From result 4 it follows that l(P) is also a formal system |
| • | Now result 2 infers that l(P) is a lie lattice σ-algebra |
| • | From result 4 and 2 it follows, respectively that l(P) is a formal system and is a lie lattice σ-algebra |
| • | From the definition of l(P), P<l(P) <L and L is just one conformal system containing P |
| • | Thus it was shown the existence of lie lattice σ-algebra l(P) lying between P and L |
| • | 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.