Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.5, No.7, 2015 79 Projective and Injective Weak Distributive Lattice Rahul Dravid Department of Mathematics, Shri Vaishnav Institute of Management Indore Dr. M.R. Aloney Department of Mathematics TIT Bhopal Abstract: This Paper is Concerned with solving problems existence with quasi projective and quasi injective objects and retracts respectively over problems with projective and injective objects and retracts in the category whose objects are the complete quasi lattice and morphism are the complete quasi lattice homomorphism from the point of view .in this paper we mentioned here some necessary and sufficient conditions for the given lattice be quasi projective and quasi injective and retracts respect. 1. Basic Definitions and Theorems : There are used following symbols: Categories are denoted by 2I, B, C....... Objects of categories by A, B, C......morphism by letters f, g, h, If A, B are the objects of category 2Ithen H (A, B) denotes the set of all morphism from A to B. Identity morphism from A to A is given by id A . For f ∈ (, ) and g ∈ (, ) then there exist h such that h ∈ (, ) then composition of given monomorphism is denoted by goh or gh =f. If X, Y are sets then f: X→Y denotes mapping of the set X into Set Y, Put f(X) = {f(t) ∣ t∈ X } y∈ Y , f -1 (y) = { x∣ x∈X f(x) = y} For U⊆X, f ∣ U denotes the restriction of the mapping f on the set U. If f: X→Y and f(x 1 )≠ f(x 2 ) then it is injective for two elements x, y∈X, x 1 ≠x 2 and surjective if f(x) = y if f is both then it is bijective. In Partially ordered set that is a set which is reflexive, anti-symmetric and transitive if A is an ordered set ∅≠X⊆ A the least upper bound of subset X in the set A if it exist then denoted by sup A X and greatest lower bound of X is inf A X can also use as x⋁ y and x∧y instead of sup {x, y} and inf {x, y}, X-Y denotes difference of X and Y. For ordered set A, x and y ∈ A and x, y are incomparable that is neither x≤y <x, y> denotes closed interval {t∣ t∈ A, x≤ t≤ y} (x, y) denotes open interval {t∣ t∈ A, x<t<y} we can say y covers x. If y>x, <x, y> ={x, y} smallest and greatest element are 0 A and 1 A . If A, B are ordered sets then A+B denote cardinal sum and A⨁B denote direct sum or ordinal sum. AxB is their cardinal product [1]. If a∈A, b∈ B then [a, b] ∈ AXB and A⨂B is direct or ordinal product. Let 2I be a category an object A∈ 2I is called projective if for arbitrary B,C ∈ 2I for arbitrary epimorphism g ∈ H 2I (B,C) and arbitrary morphism f ∈ H 2I (A,C) we have h ∈ H 2I (A,B) so that gh=f. Let A∈2I is called projective retract if for every B ∈2I and for arbitrary epimorphism g∈ H 2I (A, B) we have h∈ H 2I (A, B) such that gh=idA. Let A∈2I is called injective object if for arbitrary B, C ∈2I arbitrary monomrphism g∈H 2I (C, B) and arbitrary monomrphism f∈H 2I (B, A) such that hg=f.