INTRODUCTION
In the context of Complexity Reduction in LatticeBased Information Retrieval (Ming and Wang, 2005), the reduction process must preserve the algebraic structure of a lattice (Kalinin and Spatzier, 2005; Karen and Vogel, 2005). The SK (SharmaKaushik)Lattices are known to be of high applicational value. Hence the present study is aimed at detailing the algebraic structure of SKlattice. In the context of InformationTheoretic approach to coding, the ntuples of integers are relevant. Therefore, the Algebraic properties are obtained for lattices of SKpartitions, which are characterized as ntuples of integers.
Lattices of sets have been studied to a significant extent. Lachlan (1968) investigated the lattice of recursively enumerable sets and characterized the hhsimple sets as coinfinite r.e. sets whose r.e. sets form a Boolean algebra. The lattice of partitions of a set is also a topic of interest. Haiman (1994) presented a construction realising a continuous partition lattice as a lattice of measurable partitions.
SKpartitions were introduced by Sharma and Kaushik (1977), in connection with
metrics in Coding Theory. SKpartitions can be characterized as ntuples of
integers and so a lattice of SKpartitions can hence be considered to be a lattice
of sets of ntuples. We show that a lattice of SKpartitions satisfy the wellknown
Modular Identity and JordanDedekind Chain Condition (Stern, 1999) and is a
clsemigroup. We also obtain conditions satisfied by the rightresidual and
deficit of SKpartitions and present a condition for the existence of a certain
right residual. Most of these results are obtained directly by considering the
structure of SKpartitions.
DEFINITIONS AND NOTATION
The following definitions and notations are found in Lattice Theory by Birkhoff (1967).
Definition: If a and b are elements of a partially ordered set P, then a is said to cover b if a < b and a > x > b is not satisfied by any x in P.
Notation: 0 denotes the least element and 1 the greatest element of P.
Definition: An element which covers 0 is called an atom.
Definition: If a ≥ b in a partially ordered set P, the set of x satisfying a ≥ x ≥ b is called the closed interval [b, a]. The elements x satisfying a ≥ x ≥ b are said to be between a and b.
Definition: Intervals which can be written as [x ∧ y, x] and [y, x ∨ y] are called transposes.
Definition: A lattice having the property that every nonempty bounded set has a greatest lower bound and a least upper bound is called conditionally complete.
Definition: The modular identity is: If x ≤ z, x ∨ (y ∧ x) = (x ∨ y) ∧ x for elements x, y, z of a lattice L.
Definition: JordanDedekind chain condition: All finite connected chains between fixed endpoints have the same length.
Definition: An element of a lattice L is called join irreducible if x ∨ y = a ⇒ x = a or y = a ∀ x, y ∈ L.
Definition: A multiplicative lattice or mlattice is a lattice L with a binary operation satisfying a(b∨ c) = ab ∨ ac and (a ∨ b) c = ac ∨ bc.
• 
A zero of an mlattice L is an element 0 satisfying 0 ∧
x= 0x = x0 = 0∀ x ∈ L. 
• 
A unity of L is an element e satisfying ex = xe = x∀
x ∈ L. 
• 
An infinity of L is an element I satisfying I ∨ x = Ix
= xI = I∀ x ∈ L. 
• 
L is called commutative if xy = yx ∀ x, y ∈ L.
L is called associative if x(yz) = (xy) z ∀ x, y, z ∈ L. 
If L is conditionally complete and satisfies the unrestricted distributive laws a ∨ b_{a} = ∨ (ab_{a}) and (∨a_{a})b = ∨(a_{a}b), it is called a complete mlattice, or cmlattice. An associative lattice with unity is called a latticeordered semigroup or lsemigroup and if complete it is called a clsemigroup.
Definition: Let G be any mlattice. The rightresidual h:k of h by k is the largest x (if it exists) satisfying xk ≤ h; the leftresidual h:k of h by k is the largest y satisfying ky ≤ h.
We will also require the following notations.
Notation: We denote the SKpartition P = {B_{0}, B_{1},…, B_{m1}} by ((1,b_{1},b_{2},…,b_{m1})), where b_{i} = B_{i} = number of elements of B_{i}; i = 1,2,…, m1.
Notation: The set of all SKpartitions will be denoted by F_{P} and the set of SKpartitions with m classes will be denoted by F_{P,m}.
Definition: The dimension function d is defined on F_{p} by
THE MODULAR IDENTITY AND JORDANDEDEKIND CHAIN CONDITION
Lemma 1: Let x, y, a ∈ F_{P,m} ý x and y cover a and x ≠ y. Then x ∨ y covers x and y.
Proof: Let
a = ((1,a_{1},a_{2},…,a_{m1})).
x = ((1,x_{1},x_{2},…,x_{m1})).
y = ((1,y_{1},y_{2},…,y_{m1})).
and
x_{i}
= a_{i} = z_{i}, i ≠ u, v ; x_{i} = a_{i}+
2 z_{i}, i = u; x_{i} + 2 = a_{i} + 2 = zi_{,}
i = v. Hence x ∨ y covers x. Similarly, x ∨ y covers y.
Lemma 2: Let x, y, a p F_{P,m} ^ a cover x and y and x ≠ y. Then x and y cover x ∧ y.
Proof: Let
x_{i}
= a_{i} = z_{i}, i ≠ u, v ; x_{i} = a_{i}
2 = z_{i}, i = u; x_{i} = a_{i} = z_{i} + 2,_{,}
i = v. Hence x covers x ∧ y. Similarly, y covers x ∧ y.
Theorem 1: (F_{P,m}, ≤_{s}) satisfies the modular identity and the JordanDedekind chain condition.
Proof: This follows immediately from Theorem 1 of Lattice Theory by Birkhoff.
TRANSPOSES
Theorem 2: Let [l, x] and [y, n] be intervales ∋
And
Where, a_{1},a_{2},…,a_{r} are fixed elements of
{1,2,…,m1}. Then, [1,x] and [y,u] are transposes.
Proof: For i ∈ {1,2,…,m1} ý i ≠ a_{1},a_{2},…,a_{r}, l_{j} = x_{i} ≤ y_{i}.

(1) 
For i = a_{1},a_{2},…a_{m1}; 1_{i} = y_{i} ≤ u_{i} = x_{i}

(2) 
From Eq. 1 and 2, l = x ∧ y.
For i ∈ {1,2,…m1}ý
i ≠ a_{1},a_{2},…,a_{r} with u_{i} = y_{i }≥ x_{i}

(3) 
For i ∈ {1,2,…,m1} ý i = a_{1},a_{2},…,a_{m1};
u_{i} = x_{i} ≥ 1_{i} = y_{i}

(4) 
From Eq. 3 and 4, u = x ∨ y.
JOINIRREDUCIBLE ELEMENTS
Theorem 3: Each joinirreducible element of (F_{P, m} ≤_{s}) is of the form ((1,2,2,…,2,a,…,a)), where a ∈ {2,4,6,…}.
Proof: Let ((1,2,2,…,2,a_{1},a_{2},…,a_{s})) be any joinirreducible element of F_{p,m}. Suppose that a_{1} < a_{2}.
So, ((1,2,2,…,2,a_{1},a_{2},…,a_{s})) is not joinirreducible, contradiction. Hence, a_{1} = a_{2}.
Similarly, it can be established that a_{2} = a_{3} = … a_{s}. Any element of F_{p,m} can be expressed is terms of joinirreducible elements of F _{p,m}, as shown in the following theorem.
Theorem 4: Let be an arbitrary element of F_{p,m}.
Then,
Where:
Proof: Obvious.
MULTIPLICATION AND THE RIGHTRESIDUAL
Definition: Multiplication in F_{P,m} defined by;
Theorem 5: (F_{P,m}, ≤_{s}) is a clsemigroup.
Proof:
• 
Clearly the unity e of (F_{P,m}, ≤_{s})
is ((1,2,2,…,2)). 
• 
(F_{P,m}, ≤_{s}) is associative. 
• 
(F_{P,m}, ≤_{s}) satisfies the unrestrictive
distributive laws. 
• 
(F_{P,m}, ≤_{s})is conditionally complete,
since if A is a bounded subset of F_{P,m}, then is the least upper
bound of A and is the greatest lower bound of A. 
Theorem 6: ∀ P, Q ∈ (F_{p,m},≤_{s}), (P ∧ Q)(P ∨ Q) = PQ
Proof: Let
P = ((1,a_{1},a_{2},…,a_{m1}))
Q = ((1,b_{1},b_{2},…,b_{m1})).
We assume that a_{i} < b_{i}, i ∈ {1,2,…,m1}.
Then the (i+1) th entry of (P ∧ Q)(P ∨ Q) = 1/2(a_{i})(b_{i})
= (i + 1)th entry of PQ (I)
Similarly, if a_{i} > b_{i} or a_{i} = b_{i}
(I) can still be shown to be true.
Theorem 7: Let
P = ((1,a_{1},a_{2},…,a_{m1}))
Q = ((1,a_{1},a_{2},…,a_{m1})) be elements of F_{p,m}.
Then if
and only if
• 
Q ≤_{s} P 
• 
b_{j} is a divisor of 2a_{j}; j = 1,2,…,m1 
• 

Proof: Let Suppose P = QR, where R = ((1,c_{1},c_{2},…,c_{m1}))
Then, ((1,a_{1},a_{2},…,a_{m1})) = ((1,b_{1},b_{2},…,b_{m1}))((1,c_{1},c_{2},…,c_{m1}))
• 
Clearly Q ≤ P. 
• 

• 

Conversely, suppose that the three conditions are true.
Where:
Then, clearly R ∈ F_{p,m} and P = QR.
Theorem 8: Let P, Q be as in the above theorem. If a_{i} = kb_{i}, k ∈ {2,3,…}, then QR = P, where R = ((1,c_{1},c_{2},…,c_{m1)}) and c_{i} = 2k.
Proof: If QR = P, then
Hence, c_{i} = 2k.
Theorem 9: Let P, Q, R be as in the above theorem. If a_{i} = b_{i}^{n} (n = 2,3,…) and QR = P, then c_{i} = 2b_{i}^{n1}.
Proof: If QR = P, then
Hence, c_{i} = 2b_{i}^{n1}.
THE RIGHT RESIDUAL
Theorem 10: Let
be elements of (F_{p,m}, ≤_{s}). Then (∧ P_{k}): Q exists if and only if
Proof: Suppose (∧ P_{k}:Q) exists. Then ∃ positive, even integers c_{1},c_{2},…,c_{m1} such that R = ((1,c_{1},c_{2},…,c_{m1}) and
Conversely, suppose Eq. 5 is true. Then,
Where:
Hence, (∧ P_{k}: Q) exists.
Theorem 11:
Proof: Since R = (∧ P_{i}): Q, QR ≤ P_{1}∧P_{2}∧…∧P_{u} and c_{1},c_{2},…,c_{m1} are the largest positive, even integers satisfying:
Also, since
are the largest positive even integers satisfying
Let
We will show that:
and that d_{1},d_{2},…,d_{m1} are the largest positive,
even integers that satisfy Eq. 6, 7,…,
m+5.
Equation (m+5) is obviously true.
Let D_{1},D_{2},…,D_{m1} be positive even integers satisfying d_{1} ≤ D_{1}, d_{2} ≤ D_{2},…,d_{m1}≤ D_{m1}.
We show by contradiction, that d_{1} = D_{1}, d_{2} = D_{2},…,d_{m1} = D_{m1}
Suppose that c_{1(m1)} < D_{m1}.
This means that: D_{m2} = d_{m2}. Continuing like this, we
can establish that d_{i} = D_{i}, ∀i ∈ {1,2,…,m1}.
Thus, the values we have chosen for d_{1},d_{2},…,d_{m1}
are the largest positive, even integers satisfying Eq. 6,
7,…,(m+5).
Theorem 12:
Proof:
THE DEFICIT
Definition: We define the deficit h •C k of h by k as the smallest x, if it exists, satisfying h ≤ xk.
Theorem 13:
Proof :
We claim that d_{1},d_{2},…,d_{m1} are the smallest positive even integers satisfying the same inequalities as r_{1},…r_{m1}, namely α1,α2,…,α(m1).
Suppose ∃ positive even integers
Theorem 14:
Proof:
CONCLUSIONS
The details characterizing the Algebraic Structures of the SKLattices are of immense usefulness in the context of the recentlygrowing interest in the area of the contemporarily modern Approach to Image Retrieval Based on Concept Lattices, as also in useful reference to Complexity reduction in Latticebased information retrieval, which requires that the reduction process must preserve the algebraic structure of a lattice (Kalinin and Spatzier, 2005; Karen and Vogel, 2005).