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Sets With Type-2 Operations

Carol Walker Elbert WalkerDepartment of Mathematical Sciences Department of Mathematical Sciences

New Mexico State University New Mexico State UniversityLas Cruces, New Mexico 88003 Las Cruces, New Mexico 88003

Email: hardy@nmsu.edu Email: elbert@nmsu.edu

Abstract-In this paper, we examine the algebra formed by We denote by 1 and 0 the elements ofM defined bythe subsets of the unit interval with operations induced by theorder on the unit interval. This algebra can be obtained as a Il(x) = u ifx (4)subalgebra of the truth value algebra of type-2 fuzzy sets. I if x 1

I. INTRODUCTION (X) I i-f x O (5)0 ifX:z40Type-2 fuzzy sets were introduced by Zadeh [1], extending

the notion of ordinary fuzzy sets. There is now a rather 111. THE ALGEBRA (M, LI,71,* 0, 1)extensive literature on the subject, discussing both theoretical At this point, we have the algebra M = (M, LH, 1,*, 0, 1).and practical aspects. The paper [2] gives a mathematical This is the basic algebra for type-2 fuzzy set theory. Whatevertreatment of the algebra of fuzzy truth values for type-2 fuzzy equations this algebra satisfies will be automatically satisfiedsets and some of its subalgebras. This algebra has a number of by the set of all fuzzy type-2 subsets of U with the corre-interesting subalgebras, both from a theoretical and practical sponding pointwise operations.viewpoint. One subalgebra is the subset of all mappings of the The set M also has the pointwise operations V, A, ' on itunit interval into the two-element set {0,1}. Of course it is in coming from operations on [0,1], and is a De Morgan algebranatural one-to-one correspondence with the subsets of the unit under these operations. In particular, under these operations,interval, but the operations analogous to meet and join induced it is a lattice with order given by f < g if f = f A g, oron this subalgebra do not correspond to the usual ones of equivalently, if g = f V g. These operations are useful inintersection and union. This paper investigates this subalgebra, deriving properties of the algebra M. See [2].developing some of its basic properties. In so doing, it is Theorem 1. For f e M, let fL and fR be the elements ofconceptually and computationally convenient to view it as an M defined byalgebra of subsets of [0,1] with appropriate operations.

II. TYPE-2 FUZZY SETS fL (X) = VY<Xf(y) (6)

A type-2 fuzzy subset of a set U is a mapping from U fR(X) = VY>_f(y)into the set M = Map([0, 1], [0,1]) of all mappings from the Then for f and g C Munit interval into itself. Operations on the set of all such fuzzy f Hi = (f A gL) V (fL A g) (7)subsets of U come pointwise from operations on M. Let A, (fL L)V, and ' be the usual operations on [0,1] given by = (f Vg) A A g

x A y = minx,y} (1) fF g (f AgR) V (fR A g) (8)x V y = max{x,y}= (f V g) A (f A gR)

S/ = 1-xUsing these equations it is fairly straightforward to verify the

Let f and g be in M. The binary operations LH and n on following basic properties of M.M are defined by the equations Corollary 2: Let f, g, h C M. Then

(fL g) (x) V (f(y) A g(z)) (2) 1) fL f = f; f n f = fyVz=x 2) fLg=gLHf;fFg=gFf

(f n g) (x) = V (f(8) Ag(z)) 3) fnA

= f; fL L = LyAz=3x 4) fLi(g Lih) =jfLg) Lih; fHn(gHnh) =jfHng)Hnh

and the unary * operation by 5) f Li(fHng) =fHn(f Lig)

f*(z) V f(8)f(1') (3) 6) (f**f9 )n*(n)=*

1-4244-0363-4/06/$20.OO ©2006 IEEE

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IV. THE SUBALGEBRA OF SUBSETS OF M subalgebra is actually closed under several other operations.A subalgebra of an algebra is a subset of that algebra that is For example it is closed under unions (but not intersections,

closed under the operations of the algebra. The algebra M has since the intersection of two of these sets may be empty). Alsomany interesting subalgebras, some of which are investigated TLand SR are in this subalgebra whenever S is.in [2]. For S C [0,1], let S be the characteristic function of The following additional facts should be noted.S; that is, S(x) = 1 if x C S, and is 0 otherwise. We write -a . is not a lattice under the operations LH and H.for {a}, and these elements are called singletons. For a > b, . H does not distribute over LH.the elements [a, b] are called intervals. The set of singletons . H does not distribute over H.is a subalgebra of M, and a -+ -a is an isomorphism from thetruth value algebra of ordinary type-I fuzzy sets into M. We . The absortion laws S H (S Li T) =S S L (S H T) dowill denote the subalgebra of singletons by IL The mapping not hold.(a, b) -± [a, b] is an isomorphism from the truth value algebra . 1 LH S = 1; 0 H S = 0.of interval-valued fuzzy sets into M, whose image will be . The elements of $ are normal.denoted by 2 See [2] for details. The elements S are called The set of singletons in 5 is the subalgebra I, and is asubsets of M. The non-empty subsets form a subalgebra of Kleene algebra.M we denote by S. Observe that Cc 2] C $ C M.Almost all applications of type-2 fuzzy sets have actually . The set of closed intervals in $ is the subalgebra 2 and

been applications of interval-valued ones. One reason is that is a De Morgan algebra.certain computational difficulties are more tractable there. . The set of intervals (open, closed, half-open) in $ is a DeThere is a treatment of the fundamental properties of interval- Morgan subalgebra of S.valued fuzzy sets in [3]. In any case, $ is a subalgebra of M . The set of open intervals in $ is not a subalgebra sincecontaining the algebra of truth values of interval-valued fuzzy it does not contain 0 and 1.sets, so affords a more general situation. Conceivably, it couldbe of some practical use. An interval-valued fuzzy subset ofa set U is a mapping of U into the set {[a, b]: a < b} of V. THE ALGEBRA OF FINITE SUBSETSclosed intervals of [0, 1], or as elements of M, into the set of The subalgebra F of finite subsets may be a prime candidatecharacteristic functions of the form [a, b]. More generally, a for applications. A fuzzy subset of a set U is a mappingset-valued fuzzy subset of a set U would be a mapping of U U -± [0,1]. The image of an element u is interpreted as theinto the set of functions of the form S. These are not intervals, degree to which u belongs to that fuzzy subset, and such valuesbut have the value 0 or 1 at every point of their domain..

may be provided by "experts" or the reading of an instrument.If the functions S are identified with the corresponding For finite subsets of [0,1], a fuzzy subset of a set U is asubsets S of [0,1], then the following are clear for S,T C

m U -+ 5, and the image of each element of U is a[0,1]: mappigUXnd hmgefec lmn fU1finite subset of [0, 1]. That is, the degree to which an element u

S V T =S U T and S A T S n T "belongs" to this fuzzy subset is a finite subset of [0, 1] insteadSL = {X:X > some element of S} of a single element. For interval-valued subsets, an image is a

S' = {Sx: x < some element of S} closed interval of [0,1]. So for truth value algebras, we have

In the usual ordering of [0,1], SL is the upset of S and SRis the downset of S. Making this identification, the formulasfor LH and H become IF E[2]

SHT = (SuT)nSLnTL=(SnTL)U(SLnT)Sn-T = (SUT)nsRnTR= (SnTR) u (SRnT) and the indicated inclusions are as subalgebras. Thus for fuzzy

Thus the operations LH and H can be expressed in terms of sets themselves, those with values in IF are generalizations ofordinary union and intersection of sets. Also, 1 corresponds type-I fuzzy sets, as are those that are interval-valued. Theto the set {1}, 0 to the set {0}, and S corresponds to the arbitrary nonempty set-valued ones are generalizations of bothset {1- s s C S}, which we denote S*. The singletons those with values in IF, and interval-valued ones.{a} are denoted simply by a. In short, the mapping S -+ Sis an isomorphism from the algebra of non-empty subsets VI. AUTOMORPHISMS OF $of [0,1] with the operations indicated, to the algebra S. We We will show that the automorphisms of $ are all inducedwork with this algebra of non-empty subsets of [0,11 since by automorphisms of 1I. This is analogous to the situation forit is conceptually simpler than the subalgebra $ of M. We the truth value algebra of interval-valued fuzzy sets [3]. Sincestill denote this algebra of non-empty subsets of [0,11 by a set S =Uaes{a}, an automorphism bo of li does induce a$, and its operations by Li, H, *, 0 and 1. Of course, the mapping bo (5) =UaCS'o (a) of $ into itself. We will showequations 1-7 above hold since they hold in M itself. This that these mapping are precisely the automorphisms of the

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algebra S. Identifying the irreducible elements of $ is crucial 1) (O U 1) S =((O u 1) u S) n (O u i)R n SR =O U Sin showing this. Irreducible elements must go to irreducible 2) (O U 1) HS = ((O U 1) U S) n (O U I)L n SL = 1 U Selements under any automorphism. 3) ~o(0 U S)

=((0 U 1) H S) =o(0 U 1) H so(S) =o(0 U

Definition 3: An element S of $ is join irreducible if S 1) U po(S) n o(0 u 1)R n o(S)R = (O u 1 u p(S)) nS, LH S2 implies that S = S, or S = S2. An element S of $ o(S)R = 0 U so(S)is meet irreducible if S = S, I S2 implies that S = S, or 4) o(S U 1) ((O U 1) H S) = (o(O U 1) LH o (S)S S2. An element is irreducible if it is both join and meet (L (0 U 1) U f (S))L n o(O u )L 0 c(S)Lirreducible. (0UIU~o(s))0~O(S)L IU (S)

Proposition 4: The irreducibles in $ are the singletons and 5) a e SR implies that (a U 1) H S = ((a U 1) U S) nthe set {0, 1}. (aUl)RnSR = ((aUl) US) nSR= aUS

Proof Suppose x, y c S for some S c $, with x < y. 6) a c SLimplies that (O U a) n S = ((O Ua)U S) nIf x >0, then (O U a)L n SL = ((O U a) U S) n SL = a U S

((S u {0}) \ {x}) LH (S\ {}) = (S u {}) n [O, 1 n SL = S Theorem 6 o(aUS) o(a) U(S).shows that S is not irreducible. If y' < 1, then Proof: Suppose that a V S. Note that for any a and any(S\ {x}) H ((Su {}) \{}) = (Su {1}) n [0,1 nsR = s S, either a c SL or a c SR. Suppose that a c SR. Then,

using (4) and (5) above,again shows that S is not irreducible. Thus either S is asingleton or S = {0, 1}. (p(a U S) (p ((a U 1) n S)

If S H T = a for some a c [0,1], then =U c(a U 1)n((U (S)(SU T) n SL n TL = a (0p(a U 1)U0o (S))nO(0p(a U i))R n (0 (S))R

If a# b and b e S U T, then b V SL n TL implies b < a; = (1 U (a)U0 (S)) n(1 u0 (a))R n (0 (S))Rthat is, S U T C [0, a]. If b < a C S and c < a c T, then (1 U 0o(a)U0o (S)) n (0 (S))RbVc c SL nOTL implies that b V c = a. It follows that S = a (o (a)U f(So)) n (o(S))Ror T = a. Similarly, if SHT = a, then SUT = a. Thus a isirreducible. If S Hi T {0, 1}, then This last quantity is either bo (a) U bo (S) or bo (S), and the

(SU T) n SL nTL = {o, i} latter is impossible since a U S :4 S. Thus (o(a U S)((p (a) U bo (S)). Now suppose that.a c SL. Then similarly,

implies O C SL and O c TL, implying SL = TL = [O, 1] .

Thus S U T = {0, 1}. Since 0 c S and 0 c T, it follows that bo(a U S) = (p(O U a) HS)either S = {0, 1} or T = {0, 1}. The proof for H is similar. = (O U a) n o (S)Thus {0, 1} is irreducible. f (0 U a) H (5 ) L 0 (( S))= (fo(0 U a) U 9 (5)) 0 (Wo(0 u a)) n(())

Corollary 5: Every automorphism of$ induces an automor- (0 U (o (a) Up (s)) n (O U W (a))L n ((p (S))L

phism of the subalgebra of singletons and fixes the set {0, }. ( U (a) U (5)) 0 (O (5))LProof: We need only that for a c (0,1) and o an (0o(a)U:o(S)) n(O (S))L

automorphism of $, o(a) f {0, 1}. So suppose that o(a) ={0,1}. Let a < b < 1. Then a L b = b, and o (aL b) = = f(a) U o (S), (b)= ({0,1}U o(b)) n0(b)L =" (b) U 1 o(b). U

In the following paragraphs, bo will be an automorphism of Suppose now that a C S. If S = a, o (a) c o (S). If Sthe algebra (S, L, H,* , 0, 1), where S is the set of non-empty has at least two elements, then S = a U S\a, and o (S)subsets of [0, 1], and o (a U S\a) = o(a) U o (S\a), whence o (a) c o (S). c

S H T (S n T) n SL n TL Corollary 7: a c S if and only if o (a) c o(S).S n T (S n T) n SR n TR Proof: If a c S, then o(a U S) =o (a) U (S) = o (S),

S* - 1l 8 : 8 c St so O (a) C o (S). If o (a) c o (S), then S = -1(o (S))_ s- tot) l(> (a) U o (S)) = (fo (a)) U (Wo (S)))

1 = {1 } Corollary 8: co (S) = UaCSO (a).In general, we will write a for the singleton {a}. We need Crlay9 fadol f~()C~()some elementary facts, which we list below. Always, S will It is not known whether or not an automorphism of Mbe a non-empty subset of [0,11. induces an automorphism of $. Such an automorphism does

induce an automorphism of IF [4].

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VII. HOMOMORPHISMS This is immediate from A H A = A = A L A and theThere are some interesting homomorphisms between the commutative and associative laws for H and LH. These two

algebras M, $, IF, and [2]. Viewing all these as type-2, all are partial orders are not the same, and neither implies the other.subalgebras ofM [4]. First we show that $ is a homomorphic For example, A L 1, but it is not true that A -C 1.image of M. Proposition 15: The following hold for A, Be S.

Theorem 10: For an element fC M, let C(f) be its support. 1) Under the partial order c, any two sets A and B haveThen f - C(f5) is a homomorphism of M onto S. a greatest lower bound. That greatest lower bound is

Proof: Observe that u(f V g) = o (f) V oj(g); u(f A A n B.g) = o(f) A 5 (g); o (fL) = (f)L; o7 (fR) = (f)R; 2) Under the partial order i, any two sets A and B have

7 (f*) = 7 (f)*; C(1) 1, and v(0) = 0. The rest is easy a least upper bound. That least upper bound is A LH B.using 7 and 8. * Proposition 16: The pointwise criteria for c and C are

Theorem 11: The function 5y$ -k $: S 4 sL n SR is an these:idempotent endomorphism. The image is the set of all intervals 1) A c B if and only if AR n B C A C BR.(including closed, open, and half-open/half-closed intervals). 2) A C B if and only if A n BL C B C AL.

Proof: Let S, T C S. Proposition 17: The following hold for A, B e S.

y(SLHT) (SUHT)L n (SHUT)R 1) A c 1 and 0- A.=(SLUTL)qn(SRULTR) 2) ALB ifand only ifB* A*.[(SL U TL) n (sL n TL)] IX. PROPERTIES OF T-NORMS ON $

n [(SR U TR) n (SRL nTRL)] If o is any binary operation on [0,1], then its convolution

[(SL U TL) n (SL nTL)] n (SR U TR) (f * g) (x) V (f(y) A g(z)) (9)=(SL nTL) n (Sp' U TR) yo

is a binary operation on M. There is a discussion of properties

(S) H (T) (SL n SR) H (TL nTR) of such a type-2 t-norm in [2]. In [5], there is a thorough(T)SkL) SJ) Li

T LnTdevelopment of such type-2 t-norms and related operations for=[(sL n R) U (TL 0 TR)] the subalgebra of M consisting of normal, convex, and uppern (SL n SR) L n (TL n TR) L semicontinuous functions.[(SL n SR) U (TL n TR)] n (SL n TL) For subsets A and B, (9) translates to

(SL nTL) n (SRUTR) AoB=U{{aob}:acA, bCB} (o0)

In particular, if o is a t-norm on [0,1], then * is a type-2(a(s))* = (SL n SR)* t-norm on S. Because $ is not a lattice we will not obtain

=s* n sR* all of the usual properties of a t-norm. However, convolutions=S*R n S*L of t-norms do have some interesting and useful properties, as

follows.7(Q~*) Proposition 18: Let * be the convolution of a t-norm. Then

That ay preserves H follows from the fact that it preserves LH for A, B, C C $,and I, and it is trivial that ay preserves 0 and 1. Clearly, ay is 1) * is commutative and associative.an idempotent function. * 2) Ao1 = A

Corollary 12: The restriction of ay to IF is an idempotent 3) A * (B U C) = (A * B) U (A * C)homomorphism from IF onto [2]. 4) If B C C, then (A * B) C (A * C)

VIII. Two ORDER RELATIONS The dual operation defined byIn a lattice with operations V and A, a partial order is given A 0 B = (A* * B*)* (11)

by a < b if a A b = a, or equivalently if a V b = b. This gives a is a t-conorm. For subsets A and B, (11) translates tolattice order, that is, a partial order in which any two elementshave a least upper bound and a greatest lower bound. Even A 0) B U{{1- ((1- a) o (1 - b))}: a C A, b C B}though the algebra $ is not a lattice under the operations LHand H, these operations have the requisite properties to definepartial orders. Proofs of the following propositions are in [2]. Proposition 19. Let 0 be the dual of a type-2 t-norm. Then

Definition 13. ACgifAHB=A,ACB ifALiB =B. foA B,CeC,1) 0 is commutative and associative.

Proposition 14. The relations c and C~are reflexive, anti- 2) A @ 0 A.symmetric, and transitive, thus are partial orders. 3) A @ (B U C)(4 (A o B) UJ(A o C).

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4) If B C C, then (A@B) C (A@C).Proposition 20: If the t-norn o is continuous1, then

1) A [O,1] AP'2) A [0 1] AL

3) (AeB)R =PAR BR4) (AeB)L AAL BL5) (A©B)RAPAR©BR6) (A@B)L AAL BLThe convex sets in $ are the intervals. This includes the

singletons and open and half-open/half-closed intervals as wellas the closed intervals. These play a special role here.

Proposition 21: The distributive laws

A (B [n C) (A.B) C (A.C)A (B HC) (A mB) H (A C)A (BnC) (A B) C (A C)A (B H C) (A B) H (A C)

hold for all B, C C $ if and only if A is an interval.On the unit interval, t-norms are increasing in each variable.

Type-2 t-norms behave in a similar way with respect to theorders c and -q in the special case considered below.

Corollary 22: If A is an interval and B c C, then

A.BFIA.C and A@BLIA@CIf A is an interval and B -q C, then

A.B - A.C and A@B - A@CIf A and B are finite sets, then A * B is also finite. Thus

the convolution of a t-norm induces a type-2 t-norm on IF.

X. CONCLUSIONSThe algebra of subsets of the unit interval with type-2

operations has many interesting properties, in spite of thefact that it is not a lattice. The computational requirementsfor working in this algebra are computing the union andintersection of sets, which means computing the maximumand minimum of functions [0,1] -+ {0, 1}, and computingthe upset and downset of a set, which requires computing thebounds on a subset of the unit interval. The algebra of finitesubsets of the unit interval is a potentially interesting algebraof truth values for applications.

REFERENCES[1] L. Zadeh, "The concept of a linguistic variable and its application to

approximate reasoning," Inform Sci., vol. 8, 1975, pp. 199-249.[2] C. Walker and E. Walker, "The algebra of fuzzy truth values," Fuzzy Sets

and Systems, vol. 149, 2005, pp. 309-347.[3] M. Gehrke, C. Walker, and E. Walker, "Some comments on interval-

valued fuzzy sets," Int. J of Intelligent Sys., vol. 11, 1996, pp. 751-759.[4] C. Walker. and E. Walker, "Automorphisms of the algebra of fuzzy truth

values," unpublished.[5] M. F. Kawaguchi and M. Miyakoshi, "Extended t-norms as logical

connectives of fuzzy truth values," Multi. Vat. Logic, vol. 8, no. 1, 2002,pp. 53-69.

'In[2] properties 1 and 2 were claimed to hold without the assumption ofcontinuity. But this assumption is clearly necessary.