Intuitionistic Fuzzy Lattices and Intuitionistic Fuzzy Boolean Algebras

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1 Intuitionistic Fuzzy Lttices nd Intuitionistic Fuzzy oolen Algebrs.K. Tripthy #1, M.K. Stpthy *2 nd P.K.Choudhury ##3 # School of Computing Science nd Engineering VIT University Vellore , TN, Indi 1 tripthybk@vit.c.in * Deprtment of Mthemtics.G. College, Pdmpur Pdmpur, Odish, Indi 2 mst141106@gmil.com ## Trtrini College, Purusottmpur, Odish 3 Prmod_c32@yhoo.com Abstrct The concept of lttice plys significnt role in mthemtics nd other domins where order properties ply n importnt role. In fct, oolen lgebr, which is specil cse of Lttice, is of fundmentl importnce in Computer science. The notion of Fuzzy sets s n extension of Crisp sets is model to hndle uncertinty in dt. Following it, the notions of fuzzy lttices hve been introduced by mny reserchers [1, 11]. One of the pproches is the introduction of fuzzy prtilly ordered reltion on crisp set which stisfies the lttice property. Intuitionistic fuzzy sets [2] re generlistions of the notion of fuzzy sets. So fr the notion of Intuitionistic fuzzy lttices hs neither been introduced nor studied. In this pper, we introduce the notions of intuitionistic fuzzy lttices, intuitionistic fuzzy oolen lgebr nd study their properties. In this process, we provide n lternte definition of ntisymmetric property of intuitionistic fuzzy reltions nd compre it with the existing ones. Keyword- Intuitionistic fuzzy set, Intuitionistic fuzzy reltion, Intuitionistic fuzzy lttice, Intuitionistic fuzzy oolen lgebr I. INTODUCTION The notion of fuzzy sets [ 13] nd tht of fuzzy reltions [14 ] hve been introduced s extensions of crisp sets nd crisp reltions to model uncertinty in dt nd informtion. The concepts of intuitionistic fuzzy sets [2] nd intuitionistic fuzzy reltions [3, 5, 6, 7, 8, 9] re further extensions in this direction nd these notions generlise the notions of fuzzy sets nd fuzzy reltions respectively. The specil types of reltions like equivlence reltions nd prtilly ordered reltions hve importnt pplictions in mthemtics. The notions of lttices in generl nd tht of oolen lgebr in prticulr hve importnt role in Computer science nd rely on prtilly ordered reltions. The concept of fuzzy lttice hs been introduced in mny wys [1, 11]. Perhps the most nturl pproch mong ll these by defining prtilly ordered fuzzy reltion over crisp set is due to Tripthy et l [12]. Most importntly, besides the study of severl specil types of fuzzy lttices, they hve introduced the concept of fuzzy oolen lgebr. The study of vrious types of intuitionistic fuzzy reltions hs been done in literture [3, 5, 6, 7, 8, 9]. In this pper we introduce the notions of intuitionistic fuzzy lttice, intuitionistic fuzzy oolen lgebr nd study their properties nd lso properties of some specil type of such lttices. II. FUZZY LATTICES AND FUZZY OOLEAN ALGEAS A. Fuzzy Lttices We first introduce some preliminry definition, which shll be used to define fuzzy lttices. Definition 2.1.1: A fuzzy binry reltion on set X is fuzzy prtil ordering if nd only if it is fuzzy reflexive, fuzzy ntisymmetric nd fuzzy trnsitive under ny fuzzy trnsitivity. Here, we shll confine ourselves to the mx-min trnsitivity only. There re severl other definitions of fuzzy trnsitivity. Definition 2.1.2: Let be fuzzy prtil ordering defined on set X. Then, for ny element x X, we ssocite two fuzzy sets. The first one is the dominting clss of x( [10]) denoted by [ x] nd is defined s (2.1.1) ( y) ( x, y), [ x] = where y X. In other words, the dominting clss of x contins the members of X to the degree to which they dominte x. ISSN : Vol 5 No 3 Jun-Jul

2 The second one is, the clss dominted by x ([10]), denoted by [ x] nd is defined s (2.1.2) ( y) ( y, x), [ x] = where y X. We hve, the clss dominted by x contins the elements of X to the degree to which they re dominted by x. Definition 2.1.3: An element x X is undominted if nd only if (x, y) = 0 for ll y X nd x y. X is sid to be undominting if nd only if (y, x) = 0 for ll y X nd x y. Definition 2.1.4: For crisp A of X on which fuzzy prtil ordering is defined, the fuzzy upper bound set of A is the fuzzy set denoted (, ) (2.1.3) U(, A) = x. U A nd is defined by ([10]) [ ] x A Definition 2.1.5: For crisp subset A of set X on which fuzzy prtil ordering is defined, the fuzzy lower bound set for A is the fuzzy set denoted by L(,A) nd is defined by (2.1.4) LA (, ) = [ x]. x A Definition 2.1.6: The lest upper bound of A with respect to the fuzzy prtil ordering reltion is unique element x in U(, A ) such tht (2.1.5) U (, A)( x) > 0 nd ( x, y) > 0, for ll elements y in the support of U(, A ). Note 2.1.1: If there re two such elements x nd y then by (2.1.5) we hve xy (, ) > 0 ndyx (, ) > 0. So, by the ntisymmetric property x = y. Definition 2.1.7: The gretest lower bound with respect to the fuzzy prtil ordering reltion is unique element x in L(, A) such tht (2.1.6) L(, A)( x) > 0 nd ( y, x) > 0, for ll elements in the support of L(,A). Definition 2.1.8: A crisp set X on which fuzzy prtil ordering is defined is sid to be fuzzy lttice if nd only if for ny two element set {x, y} in X, the lest upper bound (lub) nd the gretest lower bound (glb) exist in X. We denote the lub of {x, y} by x y nd the glb of {x, y} by x y. Mny properties of fuzzy lttices defined this mnner re estblished in [12].. Fuzzy oolen lgebr oolen lgebrs hve n importnt role in the ppliction res like computer since. It is specil kind of lttice. So, obviously one cn expect fuzzy oolen lgebr to be considered s specil cse of fuzzy lttice. We define it below ([12]). Definition 2.2.1: A complemented distributive fuzzy lttice is clled fuzzy oolen lgebr. Definition 2.2.2: Let = (,,,0,1,') be fuzzy oolen lgebr. For ny two elements nd b in, we define the opertion ring sum denoted by s (2.2.1) b = ( b ) ( b). Definition 2.2.3: In ny Fuzzy oolen lgebr = (,,,0,1,'), ring product is defined by (2.2.2) b = b,, b Definition 2.2.4: A complemented distributive fuzzy lttice with the binry opertion nd is fuzzy oolen ring with identity 1. Definition 2.2.5: Let ( L,, ) be fuzzy lttice with lower bound 0. An immedite successor of 0 is clled n tom. ISSN : Vol 5 No 3 Jun-Jul

3 Definition 2.2.6: Let ( L,, ) clled n ntitom. Definition 2.2.7: Let ( L,, ) (2.2.3) = 1 2 = 1 or = 2. L,, Definition 2.2.8: An element in fuzzy lttice ( ) (2.2.4) = 1 2 = 1 or = 2. be fuzzy lttice with n upper bound 1. An immedite predecessor of 1 is be fuzzy lttice. An element L is sid to be join irreducible if is sid to be meet irreducible if III. INTUITIONISTIC FUZZY LATTICES We generlise the notion of fuzzy lttices introduced by Tripthy nd Chudhury ([12]) to the setting of intuitionistic fuzzy lttices in this section. As is well known fuzzy sets re specil cses of intuitionistic fuzzy sets. Hence intuitionistic fuzzy sets hve better modeling power thn the fuzzy sets. Similrly, intuitionistic fuzzy lttices re more generl thn the fuzzy lttices. First we introduce some definitions which we define below: Definition 3.1: An intuitionistic fuzzy reltion is n intuitionistic fuzzy subset of X Y; tht is given by (3.1) = { < ( xy, ), μ( xy, ), ν ( xy, ) > / x X, y Y}, where μ, ν : X Y [0,1], stisfy the condition 0 μ ( xy, ) + ν ( xy, ) 1, for every ( x, y) X Y. Definition 3.2: We sy tht n intuitionistic fuzzy reltions is (3.2) reflexive, if for every x X, μ( xx, ) = 1 ndν ( xx, ) = 0. (3.3) irreflexive, if for some x X, μ( xx, ) 1 orν ( xx, ) 0. (3.4) ntireflexive, if for every x X, μ( xx, ) = 0 ndν ( xx, ) = 1. (3.5) symmetric, if for every ( x, y) X X μ ( x, y) = μ ( y, x) nd ν ( x, y) = ν ( y, x). In the opposite cse we sy is symmetric. (3.6) Perfect ntisymmetric if for every ( x, y) X X with x y nd μ( x, y) > 0 or ( μ( x, y) = 0 nd ν ( x, y) < 1) then μ( xy, ) = 0 ndν ( xy, ) = 1. (3.7) trnsitive if, where o is mx-min nd min-mx composition, tht is ( xz, ) mxmin { y ( xy, ), ( xy, )} ( x, z) min mx { ( x, y), ( y, z) } μ μ μ nd ν ν ν y. Note 3.1: The definition of perfect ntisymmetric given in (3.6) is equivlent to the following: n intuitionistic fuzzy reltion is perfect ntisymmetric if for every ( x, y) X X ( μ( x, y) > 0 nd μ( y, x) > 0) or ( ν ( x, y) < 1 (3.8) nd ν ( y, x) < 1) x = y. Observtion 3.1: First we see tht when X is fuzzy set, ν (x, y) = 1-μ (x, y) nd ν (y, x) = 1-μ (y, x). So tht the second hlf of condition (3.8) reduces to the first prt nd so we get the corresponding definition of ntisymmetry of fuzzy set. Observtion 3.2: Next, we note tht when first hlf of condition (3.8) is true the second hlf follows. ut, when the second hlf holds, we get tht μ ( x, y) 0 nd μ ( y, x) 0. So, the first hlf does not follow. To be ISSN : Vol 5 No 3 Jun-Jul

4 precise, the second hlf covers the cses μ ( xy, ) = 0 orμ ( yx, ) = 0. So, we cnnot do wy with ny of the prts in (3.8). Observtion 3.3: Finlly, we shll show tht (3.6) nd (3.8) re equivlent. (i) Proof of (3.8) (3.6) We hve (3.8) is equivlent to ( x y) ( μ( x, y) > 0 nd μ( y, x) > 0) (3.9) nd ( ν ( x, y) < 1 nd ν ( y, x) < 1). In (3.9) ' ' is the oolen negtion. In ddition if ( x, y ) 0 then of HS we get μ ( yx, ) = 0. μ > from the first expression From the second expression on the HS we get either both ν ( x, y) = 1 ndν ( y, x) = 1 or one of these is equl to 1 nd the other is not. ut in this cse s μ ( xy, ) > 0, we cnnot hve ν ( xy, ) = 1. So, we hve ν ( yx, ) = 1. Hence, in ny cse we get ν ( xy, ) = 1. On the other hnd suppose μ( xy, ) = 0 ndν ( xy, ) < 1. Then from the second expression on the HS of (3.9), we must hve ν ( yx, ) = 1. Hence, μ ( yx, ) = 0. (ii)proof of (3.6) (3.8) Suppose x y. Then we hve two cses. Cse-1: μ ( xy, ) > 0. In this cse we hve μ ( yx, ) = 0 by(3.6). Also ν ( yx, ) = 1. So tht HS of (3.9) is true. Cse-2: μ ( xy, ) = 0. Then either ν ( xy, ) = 1 orν ( xy, ) < 1. If ν (x, y) = 1 then μ (x, y) = 0. So, HS of (3.9) is true. On the other hnd if ν (x, y) < 1 then by (3.6) μ (y, x) = 0 nd ν (y, x) = 1.So, HS of (3.9) is true. Definition 3.3: An intuitionistic fuzzy reltion on X is sid to be ll intuitionistic fuzzy prtilly ordered reltion if is reflexive, perfect ntisymmetric nd trnsitive; tht is (3.2), (3.6) nd (3.7) hold. Definition 3.4: Let X be set with n intuitionistic fuzzy prtil ordered reltion defined over it. Then (X, ) is clled n intuitionistic fuzzy prtilly ordered set. Definition 3.5: When intuitionistic fuzzy prtil ordering in defined on set X, two intuitionistic fuzzy sets re ssocited with ech element x in X. The first is clled the dominting clss of x. We denote it by [ x] nd is defined by (3.10) y μ iff μ x, y > 0 or{ μ x, y = 0 ndν x, y < 1}. (3.11) [ x] ( ) ( ) ( ) The second is clled the clss dominted by x. we denote it by [ x] nd is defined by y μ iffμ yx, > 0 or{ μ yx, = 0 ndν yx, < 1}. [ x] ( ) ( ) ( ) Definition 3.6: An element x X is undominted if nd only if (3.12) μ ( xy) ndν ( xy), = 0, = 1 for ll y X nd y x An element x X is undominting if nd only if (3.13) μ ( ) ν ( ) y, x = 0 nd y, x = 1 for ll y X nd y x. Definition 3.7: For crisp subset A of set X on which n intuitionistic fuzzy prtil ordering is defined, the intuitionistic fuzzy upper bound for A in the intuitionistic fuzzy set denoted by (, ) U A nd is defined by ISSN : Vol 5 No 3 Jun-Jul

5 U A, x x A = (3.14) ( ) [ ] when denotes intersection of intuitionistic fuzzy sets. Definition 3.8: Let A be n intuitionistic fuzzy set. Then by the support set of A, We men ll elements x for μ x > 0 or μ x = 0 nd ν x < 1. which A( ) { A( ) A( ) } Definition 3.9: For crisp subset A of set X on which n intuitionistic fuzzy prtil ordering is defined, the intuitionistic fuzzy lower bound for A is the intuitionistic fuzzy set denoted by L(,A) nd is defined by L A =., x x A (3.15) ( ) [ ] Definition 3.10: The lest upper bound of A with respect to the intuitionistic fuzzy prtil ordering reltion is unique element x in support set of U(, A ) such tht (3.16) For ll other y in support set of U(, A ) μ( x y) > or{ μ( x y) = nd ν( x y) < }, 0, 0, 1. Definition 3.11: The gretest lower bound of A with respect to the intuitionistic fuzzy prtil ordering reltion is unique element x in the support set of L(, A) such tht μ y, x > 0 or μ y, x = 0 nd ν y, x < 1. (3.17) For ll other y in the support set of L(, A) ( ) { ( ) ( ) } Note 3.2: The uniqueness of x in definitions 3.10 nd 3.11 follows from the intuitionistic fuzzy ntisymmetric property of. We provide the proof for The cse of 3.11 is similr. Suppose tht there re two such elements x nd z. Then there re three cses. Cse (i) If both x nd z stisfy μl(, A) ( x) > 0 nd μl(, A) ( z) > 0 then μ( x, z) > 0 nd μ( z, x) > 0. So, by ntisymmetric property x = z. Cse (ii) If both stisfy μl(, A) ( x) = 0, ν L(, A) ( x) < 1 nd μl(, A) ( z) = 0, ν L(, A) ( z) < 1 then from ν ( x) < 1 nd ν ( z) < 1 we get x = z. L(, A) L(, A) Cse (iii) If μl(, A) ( x) > 0, μ( z, x) > 0( which impliesν ( z, x) < 1) nd μl(, A) ( z) = 0, ν L(, A) ( z) < 1 (which implies ν (x, z) < 1). So, from ν (, zx) < 1 ndν (, zx) < 1 we get by ntisymmetry tht x = z. Definition 3.12: A crisp set X on which intuitionistic fuzzy prtil ordering is defined is sid to be n intuitionistic fuzzy lttice if nd only if for ny two element set {x, y} in X, the lest upper bound (lub) nd gretest lower bound (glb) exist in X. We denote the lub of {x, y} by x y nd glb of {x, y} by x y. Exmple 3.1: Let X = {, b, c, d, e}. We define n intuitionistic fuzzy reltion over X, given by the mtrix: b c d e (1.0) ( 7, 2) ( 0,1) ( 0,1) ( 0,1) b ( 0, 8) ( 1,0) ( 0,1) ( 9, 1) ( 0, 8) c ( 5, 3) ( 7, 2) ( 1, 0) ( 1, 0) ( 8, 1) d ( 0, 8) ( 0,1) ( 0,1) ( 1,0) ( 0,1) e ( 0, 7) ( 1, 8) ( 0, 7) ( 9, 1) ( 1,0) Clerly the reltion is intuitionistic fuzzy reflexive nd intuitionistic fuzzy ntisymmetric from its definition. Also, it is mx-min, min-mx trnsitive (see (3.7)). The following tble describes the lub nd glb for different pin of elements of X. ISSN : Vol 5 No 3 Jun-Jul

6 Element pir lub glb {, b} b {, c} c {, d} d {, e} e {b, c} b c {b, d} d b {b, e} b e {c, d} d c {c, e} e c {d, e} d e So, the set X with intuitionistic fuzzy prtil ordering defined over it s bove, is n intuitionistic fuzzy lttice. Theorem 3.1: In n intuitionistic fuzzy lttice (L, ) for ny two elements, b L, ( b) or ( b) nd ( b) μ, > 0 μ, = 0 ν, < 1 b = b = b Proof We hve b= μ ( ) ( ) L > 0, b, μ b, > 0 or μ b, = 0 ndν ( b, ) < 1. Conversely, This together with ( b ) ( ) { ( ) } ( ) ( ) [ b] μ, b > 0 or μ, b = 0 nd ν(, b) < 1 L(,{, b}). μ, > 0 implies tht is the glb of {, b} or b =. This completes the proof. The following results which hve been estblished in cse of fuzzy lttices cn be esily extended to the setting of intuitionistic fuzzy lttices s we hve shown in proving the bove theorem. Theorem 3.2: Let (L, ) be n intuitionistic fuzzy lttice. Then the idempotent, commuttive, ssocitive nd bsorption properties for the opertions nd hold. Theorem 3.3: For ll, b, c L, where (L, ) is n IF-lttice, μ b, > 0 or μ b, = 0 ndν b, < 1 ( ) { ( ) ( ) } { μ( c, b c) 0 nd μ( c, b c) 0 } > > or { μ( c, b c) 0, μ( c, b c) 0, ν( c, b c) 1, ν( c, b c) 1} = = < <,. Theorem 3.4: For ll,b,c L, where (L,) is n IF-lttice, (i) { μa( b, ) > 0 nd μ ( c, ) >.0} or { μ ( b, ) = 0, ν ( b, ) < 1, μ c, = 0, ν c, < 1 ( ) ( ) } { μ (, b c) > 0 nd μ (, b c) > 0} or { μ (, b c) = 0, μ (, b c) = 0, ν (, b c) < 1, ν (, b c) < 1} (ii) { μa( b, ) > 0 nd μ ( c, ). < 0} or { μ ( b, ) = 0, ν ( b, ) < 1, μ ( c, ) = 0, ν ( c, ) < 1} μ ( b c, ) > 0, μ ( b c, ) > 0 or { } ISSN : Vol 5 No 3 Jun-Jul

7 μ μ Theorem 3.5: Theorem 3.6: ( b c ) ν ( b c ) ( b c ) ν ( b c ), = 0,, < 1,, = 0,, < 1 For, b, c L, where (L, ) is n IF-lttice the distributive inequlities hold. For ll, b, c L, where L is n IF-lttice, μ ( ) ( c) μ ( b c) ( b) c { μ( c, ) = 0, ν( c, ) < 1} { μ ( ( ) ( ) ) = ν ( ( b c), ( b) c) < 1}, > 0, > 0 nd b c, b c, Definition 3.13: An IF-lttice (L, ) is sid to be complete if every nonempty subset of L hs lub nd glb. Definition 3.14: An IF-lttice (L, ) is sid to be bounded if two elements, 0, 1 L such tht ( x) or ( x) nd ( x) ( x ) or ( x ) nd ( x ) μ 0, > 0 μ 0, = 0 ν 0, < 1 μ,1 > 0 μ,1 = 0 ν,1 < 1 for ll x L. Definition 3.15: IF-lttice (L, ) is sid to be distributive if nd only if for ll, b, c L, ( b c) = ( b) ( c) nd ( b c) = ( b) ( c) Definition 3.16: An IF-lttice (L, ) is sid to be modulr if ( b c ) = ( b) c whenever μ ( c, ). > 0 or μ ( c, ) = 0 ndν ( c, ) < 1 for ll, b, c L. Definition 3.17: Let (L, ) be bounded IF-lttice nd we denote the lower nd upper bounds of L by 0 nd 1 respectively. An element L is sid to be complement of L if nd only if = 0 nd = 1. The following theorems cn be proved s in the corresponding crisp cses: Theorem 3.7: Every distributive IF-lttice is modulr. Theorem 3.8: If (L, ) is complemented distributive IF-lttice then the two De Morgn s Lws b b b = b hold for ll, b L. ( ) = nd ( ) Exmple 3.1: We define n intuitionistic fuzzy pentgonl lttice s follows: Let L { 0,,,, 1} nd =. The intuitionistic fuzzy prtil ordering reltion is defined in terms of the mtrix given below. Here the entries denote the vlues of the membership nd non-membership function s n ordered pir ISSN : Vol 5 No 3 Jun-Jul

8 IFP 0 0 (1,0) >0 or (0,<1) >0 or (0,<1) >0 or (0, <1) 1 >0 or (0, <1) (0,1) (1,0) (0,1) - >0 or (0, <1) (0,1) - (1,0) - >0 or (0, <1) (0,1) - - (1,0) >0 or (0, <1) 1 (0,1) (0,1) (0,1) (0,1) (1,0) Here >0 mens ny rel number in (0, 1) cn be ssigned for the membership functionl vlue nd there is no restriction on the non-membership functionl vlue. Also, (0, <1) mens the vlue of the membership function is zero nd tht of non-membership function in strictly less thn 1. Finlly, mens tht no vlue exists for these slots; tht is the vlues re undefined. Exmple 3.2: We define the intuitionistic fuzzy dimond lttice s follows: Let L = { 0, b1, b2, b3, 1}. The intuitionistic fuzzy prtil ordering reltion is defined in term of the mtrix given below: IFD 0 b 1 b 2 b (1,0) >0 or >0 or >0 or >0 or (0, <1) (0,<1) (0,<1) (0, <1) b 1 (0,1) (1,0) - - >0 or (0, <1) b 2 (0,1) - (1,0) - >0 or (0, <1) b 3 (0,1) - - (1,0) >0 or (0, <1) 1 (0,1) (0,1) (0,1) (0,1) (1,0) The interprettions of the entries in the tble hve some menings s in cse of Exmple 3.1. In both those cses we get n infinite number of such lttices. Definition 3.18: An intuitionistic fuzzy chin is n IF-prtilly ordered set (L, ) in which for two ( ) ( ) ( ) ( b) or ( b) nd ( b) elements,b L, either μ, b > 0 or μ, b = 0 ndν, b < 1 or μ, > 0 μ, = 0 ν, < 1 We stte the following two properties: Theorem 3.9: Every intuitionistic fuzzy chin is distributive IF-lttice. Theorem 3.10: In complemented distributed IF-lttice (L, ) b, L ( b, ) 0 or ( b, ) 0, ( b, ) 1 ( b, ) 0 or { ( b, ) 0, ( b, ) 1} μ > μ = ν < b = 0 b = 1 μ > μ = ν <. IV. INTUITIONISTIC FUZZY OOLEAN ALGEA In this section we define specil type of IF-lttice which is clled IF-oolen lgebr nd estblish some of its properties. Definition 4.1: A complemented distributive IF-lttice is clled n IF-oolen lgebr. ISSN : Vol 5 No 3 Jun-Jul

9 Every complement IF-lttice is necessrily bounded. So, n IF-oolen lgebr is necessrily bounded. We denote it by (,,,0,1,) =, where is distributive bounded IF-lttice with bounds 0 nd 1 nd every element hs n unique complement denoted by. Definition 4.2: Let = (,,,0,1,') be IF-oolen lgebr. For ny two elements nd b in, we define the opertion ring sum denoted by s (4.1) b ( b ) ( b) is well defined opertion on. =. The following properties of IF-oolen lgebr cn be proved s in cse of fuzzy oolen lgebr. We only stte these results. Theorem 4.1: Let be n IF-oolen lgebr. Then b = b b b (4.2) ( ) ( ),, (4.3) b = b,, b (4.4) is ssocitive (4.5) 0 = 0 =, (4.6) 1= 1 =, (4.7) = 0, (4.8) ( b c) = ( b) ( c),, b, c Definition 4.3: In ny IF-oolen lgebr = (,,, 0,1, ') ring product (1) is defined by (4.9) b = b,, b Definition 4.4: A complemented distributive IF-lttice with the binry opertions nd oolen ring with identity 1. Theorem 4.2: In IF-oolen lgebr, (4.10) b = 0 = b,, b is IF- (4.11) ( 1 ) = 1 nd (4.12) ( 1 ) = 0, for ll Definition 4.5: Let ( L,, ) be n IF-lttice with lower bound 0. An immedite successor of 0 is clled n tom. {, > 0, > 0} { = < = < } = or b=, where is the IF Thus, 0 is n tom if μ( b) nd μ ( b ) or μ (, b) 0, ν (, b) 1, μ ( b, ) 0, μ ( b, ) 1 b 0 prtilly ordered reltion on L. Definition 4.6: Let ( L,, ) be on IF-lttice with upper bound 1. An immedite predecessor of 1 is clled on ntitom. Thus 1 is n ntitom if μ ( b) nd μ ( b ) Definition 4.7: Let (,, ), > 1,1 > 0 b = or b = 1. L be n IF-lttice. An element L is sid to be join irreducible if (4.13) = 1 2 = 1 or = 2 is sid to meet irreducible if ISSN : Vol 5 No 3 Jun-Jul

10 (4.14) = 1 2 = 1 or = 2. Theorem 4.3: Let (,,, 0,1, ') be IF-oolen lgebr. Then the following hold: (4.15) is join irreducible = 1 2 μ ( 2, 1) > 0 or ( ) { μ( 2, 1) = 0, ν ( 2, 1) < 1} μ ( ) ν ( ν ν ) μ 1, > 2 0 or { 1, 2 0, 1, 2 1} or nd (4.16) is meet irreducible = 1 2 = <, μ ( 2, 1) > 0 or ( ) 1 2 { μ ( 1, 2) =, ν ( 1, 1) < 1} or μ ( ) = ν ( ) < μ, > 0 or { 2, 1 0, 2, 1 1} Theorem 4.4: In ny IF-lttice with lower bound 0, every tom is join irreducible. Theorem 4.5: Let (,,,0,1,1) be IF-oolen lgebr. Then 0 is n tom if nd only if it is join irreducible. V. CONCLUSIONS The notion of Intuitionistic fuzzy lttice introduced in this pper is n extension of the corresponding definition of fuzzy lttice introduced in [12]. The dvntge in this definition is tht we consider prtilly ordered intuitionistic fuzzy reltion is defined over set, which provides nturl prtil ordering insted of the erlier cses where n intuitionistic fuzzy set is tken nd norml prtilly ordered reltion is defined. Mny concepts relted to this notion re defined nd properties re estblished. Some of these properties hve been proved nd the others cn be proved in wy similr to the fuzzy cse. Two specil IF-lttices hve been defined. Another specil cse nd perhps the most importnt one tht of IF-oolen lgebr is defined nd its properties re estblished. Since the intuitionistic fuzzy lttices re more relistic thn the fuzzy lttices the results estblished in this pper will cover wider re of pplictions. The notion of intuitionistic fuzzy oolen lgebr will led to the study of intuitionistic fuzzy oolen expressions nd possibly the notion of intuitionistic fuzzy gtes cn be developed. EFEENCES [1] N. Ajml: Fuzzy Lttices, Informtion Sciences, 79, (1994),pp [2] K.T.Atnossov: Intuitionistic Fuzzy Sets, Fuzzy sets nd systems, 20, (1986), pp [3] K.T. Atnossov: Intuitionistic fuzzy reltions, First scientific session of the Mthemticl Foundtion Artificil Intelligence, Sofi, IM- MFAIS, (1989), pp.1-3. [4] G.irkoff.: Lttice theory, 3 rd edition, Americn Mthemticl Society, Colloquium publiction, 25, (1984) [5] T.T. uhescu: Some observtions on intuitionistic fuzzy reltions, Itimest Seminr on Functionl Equtions, (1990), pp [6] P. urillo nd H.ustince: Intuitionistic fuzzy reltions (Prt-I), Mthwre Soft Computing, 2 (1995), pp.5-38 [7] P.urillo nd H.ustince: Intuitionistic fuzzy reltions (Prt-II), Mthwre Soft Computing 2 (1995), pp [8] P.urillo nd H.ustince: Structures on initutionistic fuzzy reltions, Fuzzy sets nd systems, 78 (1996), pp [9] H.ustince: Construction of intuitionistic fuzzy reltions with predetermined properties, Fuzzy sets nd systems, 109 (2000), pp [10] G.J.Klir nd.yuon: Fuzzy sets nd fuzzy logic, Theory nd Applictions, prentice Hll of Indi, (1997). [11] S.Nnd: Fuzzy Lttices, ulletin of the Clcutt Mthemticl Society, 81, (1989). [12].K.Tripthy nd P.K.Choudhury: Fuzzy Lttices nd Fuzzy oolen Algebr, The Journl of Fuzzy Mthemtics, Vol.16, no.1, (2008), pp [13] L.A.Zdeh: Fuzzy sets, Informtion nd Control, 8, (1965), pp [14] L.A.Zdeh: Similrity reltions nd fuzzy orderings, Informtion Sciences, vol.3, no.2, (1971), pp ISSN : Vol 5 No 3 Jun-Jul