ALGEBRAIC PROPERTIES OF THE LATTICE R nx

Transcription

1 CHAPTER 6 ALGEBRAIC PROPERTIES OF THE LATTICE R nx 6.0 Introduction This chapter continues to explore further algebraic and topological properties of the complete lattice Rnx described in Chapter 5. We also investigate certain algebraic structures that can be formed on the lattice Rnx under various fuzzy set theoretic operations. These investigations are valuable for a deeper understanding of the theory of ordered possibility distributions and also for obtaining further useful results regarding the theory. Throughout this chapter, Rnx denotes the complete distributive lattice of ordered possibility distributions with bounds OR =(I, 0,..., 0) and IR=(l, 1,..., 1) 6.1 Further algebraic properties of R "x In this section, we investigate further algebraic properties of Rnx w.r.t. different topologies on it. Throughout this section, Px denotes the metric on R nx defined in Chapter Proposition R is a topological lattice (See: Def. (1.1.16)) in its metric topology induced by nx the metric Px. As any metric lattice is a topological lattice in its metric topology (See: Result (1.1.17)), the result follows.

2 6.1.2 Proposition Let d, be a metric on the lattice Rnx defined by d,(r, t) = y$ lri - $1 ;r, t E Rnx Then d, is a complete metric on Rnx. The result can be easily verified Proposition The metrics Px and 4, defined on Rnx are equivalent in the sense that they give rise to same family of open sets. Let SS(r), 6 > 0 be an open sphere in the metric space (Rnx, P, ) If $(r) is an open sphere in the metric space (Rnx, d,), then = [ri - til (log,i - log2(i - 1)) < a log,n

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4 Thus for every S: (r). 2 > 0 there exists Sg(r) with 5 = a(l0g 2n - 1% 2(n- 1)) such that Ss(r) c ~:(r) Thus Px and d, are equivalent rnetrics on Rnx and hence they induce same topology on Rnx Proposition The topology on Rnxinduced by d, is the order topology (See: Def. (1.1.15)) on Rnx. It is enough to prove that a net { a r) in Rnx converges to r E Rnx in the order topology iff { a r) converges to r w.r.t. the metric d,. Let { a r) be a net in Rnx which converges to r in Rnx w.r.t the order topology on R nx ' a belongs to any directed index set. a r + r in the order topology e lirninf ar =lirnsup ar =r e lirninf ari=lirnsupari=ri V~EN,

5 G liminf)ar,-r,\=limsuplari-rii=o b'i~n, G lim inf max 1 a ri -ri I= lim sup max 1 a r, -ri 1 = 0 i EN,, i EN" G lim inf dx(a r, r) = lim sup dx(a r, r) = 0 G a r + r in the metric topology induced by dx. Hence the result Corollary The metric topology induced by PX on Rnx is equivalent to the order topology on R "x' An immediate consequence of Prop. (6.1.3) and Prop. (6.1.4) Corollary Rnx is a topological lattice w.r.t. the order topology on Rnx. As the order topology on Rnx is equivalent to the metric topology on Rnx induced by the metric Px, the result follows fiom Prop.(6.1.1)

6 6.1.7 Proposition Rnx satisfies the infinite distributive laws. Since any complete distributive lattice which is a topological lattice under order convergence satisfies the infinite distributive laws(see: Result (1.1.18)), the result follows Remark The above result can be obtained directly. For any set { a r) and any t in Rnx, the infinite distributive laws t A (V ar) = V(t A ar) a a tv (A ar) = A (tvar) a a hold in Rnx as they hold in the case of real numbers in [0, 11 w.r.t. arbitrary maximum and arbitrary minimum Proposition R is both Brouwerian and dually Brouwerian. "X As any complete lattice which satisfies the infinite distributive laws is both Brouwerian and dually Brouwerian (See: Def. (1.1.19) and Result (1.1.20)), the result follows.

7 Remark R is both Brouwerian and dually Brouwerian.. For any given elements, "X r = (r,,..., rn), t = (t,,..., tn) in Rnx, the greatest element t: r of the set. {x Rnx : r /\ x 5 t) is given by t:r= a = (a,,..., a,) E R wherefori EN", "X 1 if i=l a,, if ti > ri The least element r:t of the set. {x Rnx : r v x 2 t) is given by. r: t = p = (PI,..., P,) R where for i E "X N,,, 1 ifi=l if ti 5 ri where P, + = 0 ti if ti > ri Definition Let r, t E Rnx. Then for any A. [O, I], the hction A r + (1- )t defined by (A r + (1 - A )t) (x) = A r(x) + (1 - A )t (x) ; x E "X is called a convex linear combination of r and t.

8 Proposition If r, t E Rnx are not comparable w.r.t. the partial order relation ' 5 ' on the lattice Rnx, then for no A, p E (0, I), A+ p, hr + (1-h)t and pr +(l-p)t are comparable. that Let r, t E R be not comparable. If possible let there exist A, p E (0, 1)such "x hr+(l-h)t hr+(l-h)t 5 pr+(l-p)t with h # p 5 pr+(l-p)t 3 (h-p)ri 5 (h-p)ti v i Nn 3 ri 5 ti tf i d n, (-:h# P) which is a contradiction to our assumption. Hence if two elements in Rnx are not comparable, then their convex linear combinations are also not comparable Note The following results hold true in R "x as they hold in any complete lattice. 1. Every complete automorphism on the lattice Rnx is a homeomorphism in the order topology.

9 2. Every principal ideal of the lattice Rnx is a convex sublattice and is closed in the order topology. 6.2 Further algebraic structures on the lattice R "X In this section, we shall discuss what kinds of algebraic structures, the lattice R form under some well known fuzzy set theoretic operations. "X Definition The fuzzy set theoretic operations called algebraic product (-), algebraic sum (i), bounded product (O), bounded sum (@), drastic product (A) and drastic sum (v) can be defined on RnX as follows. For r, t E R "x' (1) algebraic product r.t is defined as (r.t) (x) = r(x) t(x) v x E "X (2) algebraic sum r i t is defined as (rit) (x) = r(x) + t(x) - r(x) t(x) v X' E "X (3) bounded product r 0 t is defined as (r~t)(x)= ov(r(x)+t(x)-1) vx E "X (4) bounded sum r CB t is defined as (r $ t) (x) = I A (r(x) + t(x)) t/ x E "X

10 (5) drastic product r A t is defined as (6) drastic sum r \;/ t is defined as Proposition Let * E{., 4, 0, 63, A, V ) Then for any r, s, t E R, the following results hold. "x (i) r * t E R (closure property) "x (ii) r * t = t * r (commutativity) (iii) r * (t * s) = (r * t) * s (associativity) (iv) r * (t v s) = (r * t) v (r * s) r*(t A s)=(r* t) ~ (r*s) (distributivity) Commutativity, associativity and distributivity hold in R "x as they hold true in fuzzy sets with membership values in [0, I]. (See: [60,61] )

11 The closure property can easily be verified in all cases except for (4) Proof of closure property for (4) Let r, t E RnX. (r / t) (x) = r(x) + t(x) - r(x) t(x) = 1 - (I-r(x))(l-t(x)) E [0, 11 y x E "X and (r i t) (x,) = 1 Also, if i < j, (r -it) (xi) = 1 - (1-r(xi)) (1-t(xi)) :. (r i t) E Rnx (1 -r(x,)) (1 - t(x>) = (r / t) (x,) (-: r (xj 2 r (x,) and t(x> 2 t(x,) Proposition R forms a lattice ordered semigroup with unity I, and zero OR under A, v and "X bounded product 0 where '0' is the semigroup operation. Dually RnX forms a lattice ordered semi group with unity 0, and zero I, under v, A and bounded sum $ where ' $' is the semigroup operation. RnX also forms a unitary (= 1, ) commutative semi ring with zero (= 0, ) under

12 v as addition and bounded product a as multiplication. Duality holds for A and bounded sum 8 with unity (= 0, ) and zero (= I, ). The universal bounds OR and I, of Rnx satisfy the identities r 8 I, = I, = I, 8 r Then the results follow from the Prop.(6.2.2) and Definitions (1.1.22) and Proposition Rnx forms a lattice ordered semigroup with unity 1, and zero OR under A, v and algebraic product 6 3 where is the semigroup operation. Dually Rnx forms a lattice ordered semi group with unity 0, and zero I, under v A and algebraic sum i where is the semigroup operation.

13 Rnx also forms a unitary (= I, ) commutative semi ring with zero (= OR ) under v as addition and algebraic product. as multiplication. Duality holds for A and algebraic sum i with unity (= 0, ) and zero (= I, ). In addition to the identity laws w.r.t. A and v, the universal bounds 0, and I, of Rnx also satisfy the identities r. 0, = OR= 0, -r - r.1, - r = I; r r +OR= r = 0,ir r +IR = I, = IR+r Then the results follow from the Prop. (6.2.2) and Definitions (1.1.22) and (1.1.23) Proposition Rnx forms a lattice ordered semigroup with unity I, and zero OR under A, v and drastic product A, where A is the semigroup operation. Dually Rnx forms a lattice ordered semi group with unity 0, and zero I, under v, A and drastic sum V. Rnx also forms a unitary (= 1, ) commutative semi ring with zero (= OR ) under v as addition and drastic product A as multiplication. The duality holds for A and drastic sum \;I with unity (= 0, ) and zero (= I, ).

14 In addition to the identity laws w.r.t. A and v, the universal bounds 0, and I, of Rnx also satisfy the identities ra OR = OR= OR Ar ra I,= r = IRA r rvor= r = ORvr r v IR = IR = IRv r Then the results follow from the Prop (6.2.2) and Definitions (1.l.22) and (1.1.23). The following proposition gives the identities satisfied by the non-specificity measure U, on Rnx (See: Def. (5.2.2)) w.r.t. the fuzzy set theoretic operations Proposition The non-specificity measure U, on Rnx satisfies the following identities. (i) U, (r. t) + U, (r + t) = U, (r) + U, (t) (ii) U, (r t) + U, (r $ t) = U, (r) + U, (t) for all r, t E Rnx.

15 The results follow immediately from the Definitions(6.2.1) and (5.2.2) Remark The following result need NOT hold for all r, t E Rnx ux (r A t) + Ux (r \;I t) = ux (r) + Ux (t) Example: Let r = (1,.8,.3) t= (1,.6,.4) be elements in R3X rat =(l,o,o), r\;it =(I, 1, 1) Then it is obvious that Remark The set Rnx does not satisfy idempotency, absorption and distributivity w.r.t. algebraic product (.) and algebraic sum (4). Hence Rnx does not constitute algebraic structures such as lattice or semiring w.r.t. these operations. The same is true of (a, $) and (A, V).