A Family of Finite De Morgan Algebras

Transcription

1 A Family of Finite De Morgan Algebras Carol L Walker Department of Mathematical Sciences New Mexico State University Las Cruces, NM 88003, USA hardy@nmsuedu Elbert A Walker Department of Mathematical Sciences New Mexico State University Las Cruces, NM 88003, USA elbert@nmsuedu Abstract The algebra of truth values for fuzzy sets of type- 2 consists of all mappings from the unit interval [0; 1] into itself, with operations certain convolutions of these mappings with respect to pointwise max and min This algebra has been studied rather extensively in the last few years, both from an applications point of view and a theoretical one Most of the theory goes through when [0; 1] is replaced by any two nite chains, in which case interesting nite algebras arise De Morgan algebras and Kleene algebras in particular and a basic question is just where these algebras t into the world of all such nite algebras We investigate one particularly interesting family of such De Morgan algebras Keywords nite De Morgan algebra; type-2 fuzzy sets I INTRODUCTION The basic mathematical object underlying the theory of fuzzy sets of type-2 is an algebra, called the algebra of truth values for fuzzy sets of type-2 This algebra has been studied quite extensively since its introduction by Zadeh [15] in 1975 A number of such papers are listed in the references at the end Its elements are all mappings [0; 1] [0;1] from the set [0; 1] to the set [0; 1] with operations certain convolutions of these functions The basic algebraic theory depends only on the fact that [0; 1] is a complete chain, so lends itself to various generalizations Our interest in this paper is in the special case where each of the two copies of [0; 1] is replaced by a nite chain That is, if m and n are nite chains, the set of elements of the algebra is m n, all mappings of n into m With operations we describe below, we have a nite algebra with basically the same algebraic properties as the algebra of truth values of fuzzy sets of type-2 One particular subalgebra consists of those elements that are convex and normal, as dened below This subalgebra contains subalgebras analogous to the truth value algebra of type 1 and of interval-valued fuzzy sets, which are respectively, Kleene and De Morgan It is a characteristic subalgebra and enjoys other special properties as well Due to space limitations, we conne our attention to this De Morgan subalgebra of convex, normal functions Thus, for any two nite chains, we have a nite De Morgan algebra, and we ask where these special algebras t into the world of all such nite algebras and what are their special properties Our principal result is the characterization of the De Morgan algebras as those whose poset of join irreducible elements has a particularly simple structure This leads to the determination of the automorphism groups of these algebras Our basic tool is an alternate representation of these algebras, making their operations much more intuitive In Section II we give a brief overview of some properties of the algebra of truth values for fuzzy sets of type-2 In Section III, we translate these properties to properties of nite type-2 algebras Then in Section IV, we develop three other representations of these algebras, leading eventually to an intuitive representation In Section V, we determine the size of the algebras in question, and in Section VI, we consider more general classes of De Morgan algebras, suggested by the previous work This leads to the determination in Section VI of the automorphism groups of the De Morgan algebras The setting for type-2 fuzzy sets, as proposed by Zadeh [15], generalized that of both type-1 and interval-valued fuzzy sets The references at the end of this paper contain many additional citations II TYPE-2 FUZZY SETS Following are the and, or and not operations dened by Zadeh [15] for type-2 fuzzy sets Denition 1: On [0; 1] [0;1], let (f t g) (x) = sup ff (y) ^ g (z) : y _ z = xg (f u g) (x) = sup ff (y) ^ g (z) : y ^ z = xg :f(x) = sup ff (y) : y = 1 xg = f(1 x) 1 if x = 1 1 if x = 0 1 (x) = 0 if x 6= 1, 0 (x) = 0 if x 6= 0 Denition 2: The algebra of truth values for type-2 fuzzy sets is M = ( [0; 1] [0;1] ; t; u; :;0; 1) Determining the properties of the algebra M is helped by introducing the following auxiliary operations Denition 3: For f 2 M, let f L and f R be the elements of M dened by f L (x) = sup ff(y) : y xg f R (x) = sup ff(y) : y xg The point of this denition is that the or and and operations t and u of M can be expressed in terms of the pointwise max and min of functions, as follows /09/$ IEEE

2 Theorem 4: The following hold for all f; g 2 M f t g = f ^ g L _ f L ^ g = (f _ g) ^ f L ^ g L f u g = f ^ g R _ f R ^ g = (f _ g) ^ f R ^ g R Introducing the operations L and R and using them in this general situation to express other operations in terms of pointwise ones, as in the theorem above, appears in many papers, for example in [7], [9] Using these auxiliary operations, it is fairly routine to verify the following properties of the algebra M The details of the proofs are in [9] Corollary 5: Let f, g, h 2 M The basic properties of M follow 1) f t f = f; f u f = f 2) f t g = g t f; f u g = g u f 3) 1 u f = f; 0 t f = f 4) f t (g t h) = (f t g) t h; f u (g u h) = (f u g) u h 5) f t (f u g) = f u (f t g) 6) ::f = f; : (f t g) = :f u :g; : (f u g) = :f t :g Notice that this list does not include the absorption laws or the distributive laws The algebra M is, in general, not a lattice However, M contains a host of subalgebras of interest, some of which are lattices In particular, it contains copies of the truth value algebras of type-1 fuzzy sets and of intervalvalued fuzzy sets We refer the reader to [9],[10],[11] for those and other details about M, and proceed directly to the subalgebras that motivated this paper Denition 6: An element f of M is normal if supff(x) : x 2 [0; 1]g = 1 Proposition 7: The normal functions form a subalgebra N of M: Denition 8: An element f of M is convex if for x y z, f(y) f(x) ^ f(z) Equivalently, f is convex if f = f L ^ f R Proposition 9: The convex functions form a subalgebra C of M: Theorem 10: The subalgebra D = C \ N is a De Morgan algebra, and is a maximal lattice in M Again, the proofs of the propositions above are in [9] The basic theory goes through when [0; 1] is replaced by any two nite chains In that case, D is a nite De Morgan algebra So any two nite chains give rise to a nite De Morgan algebra This family of nite algebras is the subject of this paper III FINITE TYPE-2 ALGEBRAS Here we introduce some notation and translate some pertinent results from the previous section to the nite case For a positive integer k, let k be the linearly ordered set with k elements As indicated, the basic theory above goes through if [0; 1] [0;1] is replaced by m n : We denote this algebra by M(m n ) and its elements by n-tuples (a 1 ; : : : ; a n ) where a i 2 f1; 2; :::; mg: In particular, the convex normal functions form a De Morgan algebra (a distributive lattice with negation satisfying the De Morgan laws) This De Morgan algebra is denoted D(m n ) For a n-tuple to be normal requires that it contains m as an entry, and for a normal n-tuple to be convex requires that it be increasing until the rst entry that is m, and be decreasing after that (Increasing means non-decreasing, and similarly for decreasing) The negation on n-tuples comes from the negation n! n given by i 0 = n i + 1 Thus :(a 1 ; a 2 ; : : : ; a n ) = (a 1 0; a 2 0; : : : ; a n 0) = (a n ; a n 1 ; : : : ; a 1 ) The lattice operations t and u are given by the formulas in Theorem 4 The tables below give the elements of some of these algebras for small m and n D(2 2 ) D(3 2 ) D(4 2 ) (1; 4) (1; 2) (1; 3) (2; 4) (2; 2) (2; 3) (3; 4) (2; 1) (3; 3) (4; 4) (3; 2) (4; 3) (3; 1) (4; 2) (4; 1) For n = 2, note that the algebras D(m n ) are chains of length 2m 1 Below are two examples for m = 2 (We omit the parentheses and commas from here on to preserve space) D D In Section V, we will develop formulas for the size of D(m n ): IV OTHER REPRESENTATIONS OF D(m n ) One problem in investigating the De Morgan algebra D(m n ) is that the partial order given by the lattice operations t and u is not the coordinate-wise partial order on the n- tuples To see this, just note that the identities of D(m n ) are

3 coordinate-wise incomparable For example, in D(3 3 ), those identities are 1 = (1; 1; 3) and 0 = (3; 1; 1) We now give another representation of the bounded lattice D(m n ) as n- tuples in which the partial order is coordinate-wise Denition 11: D 1 (m n ) is the algebra whose elements are decreasing n-tuples of elements from f1; 2; : : : ; 2m 1g which include m, and whose operations are given by pointwise max and min on these n-tuples, negation :(b 1 ; b 2 ; :::; b n ) = (2m b 1 ; 2m b 2 ; :::; 2m b n ); and the constants 1 = (2m 1; : : : ; 2m 1; m) and 0 = (m; 1; : : : ; 1) The proof of the following theorem is routine Theorem 12: D 1 (m n ) is a De Morgan algebra Theorem 13: The De Morgan algebras D(m n ) and D 1 (m n ) are isomorphic Proof: For a = (a 1 ; a 2 ; : : : ; a n ) 2 D(m n ), let i be the smallest index i for which a i = m For j < i, replace a j by 2m a j We get the n-tuple '(a) = (2m a 1; 2m a 2 ; : : : ; 2m a i 1 ; a i ; a i+1 ; : : : ; a n) It is easy to see that ' is a one-to-one mapping of D(m n ) onto D 1 (m n ) For an n-tuple a, we write a i for its i-th component For an n-tuple a, the balance point of a is the smallest index i such that a i = m Suppose the balance point of a and b are greater than j Then Thus (a t b) j = (a j _ b j ) ^ (_ ij a i ) ^ (_b ii ) = (a j _ b j ) ^ a j ^ b j = a j ^ b j ('(a t b)) j = 2m (a j ^ b j ) ('(a)) j _ ('(b)) j = (2m a j ) _ (2m b j ) = 2m (a j ^ b j ) Suppose the balance point of b is j and the balance point of a is j Then Thus (a t b) j = (a j _ b j ) ^ (_ ij a i ) ^ (_b ii ) = (a j _ b j ) ^ a j ^ m = a j ('(a t b)) j = 2m a j ('(a)) j _ ('(b)) j = (2m a j ) _ b j = 2m a j The last equality holds because a j and b j are m The other calculations to show that '(a t b) = '(a) _ '(b) are similar Likewise, that '(a u b) = '(a) ^ '(b) is routine The constants are obviously preserved, and a straightforward but tedious calculation shows the : is preserved 1 We make a slight simplication For each n-tuple in D 1 (m n ), remove the entry with the smallest index that is equal to m This yields all decreasing (n 1)-tuples from f1; 2; : : : ; 2m 1g; which we denote by D 2 (m n ) With pointwise operations of max and min and negation :(b 1 ; b 2 ; : : : ; b n 1 ) = (2m b n 1 ; 2m b n 2 ; : : : ; 2m 1 This representation was suggested by Professor John Harding b 2 ; 2m b 1 ), this makes D 2 (m n ) into a De Morgan algebra isomorphic to D 1 (m n ) Of course D 2 (m n ) is the set of all decreasing maps from n 1 into 2m 1 In any case, as De Morgan algebras we have D(m n ) D 1 (m n ) D 2 (m n ) V THE CARDINALITY OF D(m n ) To determine the size of D(m n ), it is easiest to use the representation D 2 (m n ), which is the set of decreasing n 1 tuples from f1; 2; : : : ; 2m 1g To simplify notation in the calculation, we prove the following proposition Proposition 14: The number of decreasing a-tuples from ((i 1)+a)! f1; 2; : : : ; ig is (i 1)!a! Proof: Induct on a For a = 1, the number is obviously i, and ((i 1)+a)!=(i 1)!a! = i Assume a > 1 and the formula holds for tuples of length a 1 The number of decreasing a-tuples from f1; 2; : : : ; ig ending with 1 is the number of decreasing a 1-tuples from f1; 2; : : : ; ig The number of decreasing a-tuples from f1; 2; : : : ; ig ending with 2 is the number of decreasing a 1-tuples from f2; 3; : : : ; ig, and so on Finally, the number of decreasing a-tuples from f1; 2; : : : ; ig ending with i is the number of decreasing a 1-tuples from fig The sum of these numbers, using ((i 1)+(a 1))! ((i 2)+(a 1))! induction on a is (i 1)!(a 1)! + (i 2)!(a 1)! + + ((i i)+(a 1))! (i i)!(a 1)! which, by a well-known combinatorial formula, or by an easy induction on i, is ((i 1)+a)! (i 1)!a! (2m 2+n 1)! (2m 2)!(n 1)! Theorem 15: jd(m n )j = Proof: Letting i = 2m 1 and a = n 1 in the previous proposition yields the desired result (2m 2+n 1)! Now (2m 2)!(n 1)! is the number of subsets of f1; 2; : : : ; 2m 2 + n 1g of size n 1 This is the same as the number of strictly decreasing n 1 tuples from f1; 2; : : : ; 2m 2 + n 1g This is yet another representation of the elements of D(m n ), but do the lattice operations correspond to pointwise max and min? Denition 16: D 3 (m n ) is the algebra whose elements are the n 1 tuples of strictly decreasing sequences from f1; 2; : : : ; 2m 2 + n 1g with operations pointwise max and min the obvious constants, and :(a 1 ; a 2 ; : : : ; a n 1 ) = (2m 2 + n a n 1 ; : : : ; 2m 2 + n a 1 ) Theorem 17: D 2 (m n ) D 3 (m n ) Proof: The mapping D 2 (m n )! D 3 (m n ) given by (a 1 ; a 2 ; : : : ; a n 1 ) 7! (a 1 + (n 2); a 2 + (n 3); : : : ; a n 2 + 1; a n 1 ) is rather obviously a lattice isomorphism The proof that : is preserved is straightforward though a bit tedious We now have four representations: 1) D(m n ), the normal convex functions of n-tuples from f1; 2; : : : ; mg, 2) D 1 (m n ), the decreasing n-tuples of elements from f1; 2; : : : ; 2m 1g, that have m as an entry, 3) D 2 (m n ), the decreasing (n 1)-tuples of elements from f1; 2; : : : ; 2m 1g,

4 4) D 3 (m n ), the strictly decreasing (n 1)-tuples of elements from f1; 2; : : : ; (2m 2) + (n 1)g In the last three representations, the lattice operations are pointwise, and the negations are as indicated earlier We will not use representation 4, but just note that it came about from the combinatorial result in Theorem 15 To illustrate, below we show each representation for m = 2, n = 3 D D D D There is difculty in depicting such lattices as those above for larger m and n because of the following Proposition 18: The De Morgan algebras D 1 (m n ) are not planar if m 4 and n 3 Proof: A nite distributive lattice is planar if and only if no element has 3 covers ([1] page 90, problem 45) For example, in D 1 (4 3 ) the 3 tuple (3; 2; 1) has covers (4; 2; 1),(3; 3; 1), and (3; 2; 2) VI THE DE MORGAN ALGEBRAS H(m n ) In the algebra D 2 (m n ), the tuples can be of any positive integer length, but entries must come from a set with an odd number of elements, namely f1; 2; : : : ; 2m 1g This suggests considering a more general class, namely the one in the following denition Denition 19: For positive integers m and n, let H(m n ) be the algebra of all decreasing n-tuples from f1; 2; : : : ; mg, with pointwise operations _ and ^ of max and min, with negation :(a 1 ; a 2 ; : : : ; a n ) = (m + 1 a n ; m + 1 a n 1 ; : : : ; m + 1 a 2 ; m + 1 a 1 ) and the obvious constants It is easy to see that H(m n ) is a De Morgan algebra Actually, H(m n ) is the set of all anti-homomorphisms from the poset n to the poset m It should also be clear that the De Morgan algebras D 2 (m n ) and H((2m 1) n 1 ) are the same Thus the De Morgan algebras D 2 (m n ), and of course our original algebras D(m n ); are special cases of the algebras H(m n ) So, we now investigate the larger family H(m n ) of nite De Morgan algebras The diagrams below illustrate some features of these algebras Note that H 3 2 has a non-trivial automorphism, and H 4 2 and H 3 3 are isomorphic and have no non-trivial automorphisms So these algebras may or may not have non-trivial automorphisms, and H(m n ) may be isomorphic to H(p q ) with m 6= p and n 6= q: H j 32 j 11 H H From Proposition 14, we get Theorem 20: jh(m n )j = ((m 1)+n)! (m 1)!n! Note that jh(m n )j = H((n + 1) m 1 ) We will see the signicance of this in Theorem 24 The De Morgan algebra H(m n ) is in particular a nite distributive lattice The set of join irreducible elements of a lattice is a poset under the induced order The point here is that nite distributive lattices and nite posets are equivalent categories, with maps in each case being order preserving ones A nite distributive lattice corresponds to its poset of non-zero join irreducible elements under the induced order, and a nite poset to the lattice of its downsets with order given by set inclusion The details may be found in [1] We determine now the poset of join irreducible elements of H(m n ): Denition : An element a in a lattice is join irreducible if a = b _ c implies that a = b or a = c Denition 22: For 1 i n 1, an n-tuple in H(m n ) has a jump at i if its i + 1 entry is strictly less than its i th entry It has a jump at n if the n th entry is at least 2 For example, the 5-tuple (5; 5; 5; 3; 1) has jumps at 3 and 4, the 6-tuple (8; 7; 2; 2; 2; 2) has jumps at 1 and 2 and 6, and the 6-tuple (5; 5; 5; 5; 1; 1) has a jump at 4 The only n-tuple

5 with no jumps is (1; 1; : : : ; 1), the zero of the lattice H(m n ) Theorem 23: The non-zero join irreducibles of H(m n ) are those n tuples with exactly one jump Proof: Let i < j and (a 1 ; a 2 ; : : : ; a n ) have a jump at i and at j Then (a 1 ; a 2 ; : : : ; a i ; a i+1 ; : : : ; a j ; a j+1 ; : : : ; a n ) = (a 1 ; a 2 ; :::; a i 1; a i+1 ; :::; a j ; a j+1 ; :::; a n ) _(a 1 ; a 2 ; :::; a i ; a i+1 ; :::; a j 1; a j+1 ; :::; a n ) Note that j may be n Therefore a join irreducible can have at most one jump It is not hard to see that if a decreasing n tuple has exactly one jump, it cannot be the join of two such pointwise smaller n tuples Since the only element with no jumps is the n-tuple (1; 1; : : : ; 1), the non-zero join irreducible elements of H(m n ) are of the form (a; a; : : : ; a; 1; 1; : : : ; 1), with a > 1 and at least one a in the tuple Thus with each non-zero join irreducible, there is associated a pair of integers, the integer a and the index of the last a For example, we have the following associations (5; 5; 1; 1; 1)! (5; 2) (5; 5; 5; 5; 5)! (5; 5) This association gets a map from the non-zero join irreducibles of H(m n ) to the poset (m 1)n: (Here, we are associating the poset f2; 3; : : : ; mg with the poset m 1) This is rather obviously a one-to-one mapping of the non-zero join irreducibles of H(m n ) onto the poset (m 1)n, and preserves component-wise order Thus we have the following Theorem 24: The poset JIH(m n ) of non-zero join irreducibles of H(m n ) is isomorphic to the poset (m 1) n: (Note that the poset (m 1) n is actually a bounded distributive lattice) The diagrams below illustrate this theorem JIH JIH JIH JIH JIH Because of the categorical equivalence of nite distributive lattices and nite posets, with a nite distributive lattice corresponding to its poset of non-zero join irreducible elements, the lattice H(m n ) is isomorphic to the lattice H(p q ) if and only if the posets (m 1) n and (p 1) q are isomorphic Further, it is clear that the poset (m 1) n has only the trivial automorphism unless m 1 = n; in which case it has exactly two automorphisms Thus the lattice H(m n ) has only the trivial automorphism unless m 1 = n; in which case it has exactly two automorphisms, its poset of non-zero join irreducibles being the poset n n: Thus we get the following corollaries Corollary 25: The lattices H(m n ) and H(p q ) are isomorphic if and only if m = p and n = q, or m 1 = q and p 1 = n Corollary 26: The automorphism group Aut(H(m n )) of the lattice H(m n ) has only one element unless m 1 = n, in which case it has exactly two elements Corollary 27: The automorphism group Aut(D(m n )) of the lattice D(m n ) has only one element unless 2m 1 = n 1, in which case it has exactly two elements If m 1 6= n; then the lattice H(m n ) has only the trivial automorphism, hence so does H(m n ) as a De Morgan algebra If 2m 1 6= n 1; the analogous statement holds for D(m n ): However, if m 1 = n, H(m n ) as a lattice has two automorphisms, and the question is whether or not the non-trivial lattice automorphism is also a De Morgan automorphism Theorem 28: Lattice automorphisms of H(m n ) are De Morgan automorphisms of H(m n ): Proof: If m 1 6= n; then H(m n ) has only the trivial lattice automorphism, hence only the trivial De Morgan automorphism If m 1 = n; H(m n ) has one non-trivial automorphism Denote this automorphism by ; and let be

6 any negation of H(m n ): Now is an automorphism, so is the identity automorphism 1; or is :If = ; then since is or order two, = 1; whence = ; an impossibility So = 1; and multiplying on the right by and then by yields = : That is, commutes with ; so is a De Morgan automorphism We now examine the lattice isomorphisms from H(m n ) to H( (n + 1) m 1 ) If m 1 6= n; then each has only the trivial automorphism, so there is only one lattice isomorphism from H(m n ) to H( (n + 1) m 1 ) Of course it is the one given by the isomorphism of the posets (m 1) n and n (m 1) of non-zero join irreducibles of H(m n ) and H( (n + 1) m 1 ); respectively Since H(m n ) is De Morgan, its negation and the lattice isomorphism induce a negation on H( (n + 1) m 1 ), and is a De Morgan isomorphism But H( (n + 1) m 1 ) has only one negation since the product of two different negations would be a non-trivial automorphism of H( (n + 1) m 1 ): Therefore, is a De Morgan isomorphism We have Theorem 29: If m 1 6= n; then there is exactly one lattice isomorphism H(m n )! H( (n + 1) m 1 ); and it is a De Morgan isomorphism If m 1 = n; then H(m n ) = H( (n + 1) m 1 ); and by Corollary 26 and Theorem 28 H(m n ) has exactly one non-trivial lattice automorphism, and it is a De Morgan automorphism Since the poset of non-zero join irreducibles of H(m n ) is the lattice (m 1) n, this lattice is in turn determined by its poset of non-zero join irreducibles That poset is simply the disjoint chains m 2 and n 1 This again shows, for example that the automorphism group of H(m n ) has exactly one element unless m 2 = n 1, in which case it has exactly two automorphisms It is not true that jh(m n )j determines H(m n ) That is, it can happen that jh(m n )j = jh(p q )j without the two being isomorphic lattices For example H(8 3 ) = H(15 2 ) = 120, yet the criteria of Corollary 25 are not met VII SOME TABLES OF CARDINALITIES There are many, many combinatorial identities between the various entities above For example H(m 7 ) = H((m 1) 7 ) + H(m 6 ) This is not surprising since jh(m n )j is a binomial coefcient Below are some tables of sizes of various of these algebras These computations come directly from Theorem 20 m H(m 2 ) H(m 3 ) H(m 4 ) m H(m 5 ) H(m 6 ) H(m 7 ) VIII COMMENTS Finite truth value algebras may be of some interest in fuzzy theory, in particular the algebra M (m n ) We have limited our discussion to the particular subalgebra D (m n ) of M (m n ) because of its very special properties The algebra D (m n ) contains special Kleene subalgebras (De Morgan algebras satisfying the Kleene inequality x^:x y _:y), but we have not characterized them in the same sense that we have the De Morgan algebras D (m n ) It may be of some interest to characterize the nite analogues of the truth value algebras of type-1 and interval-valued fuzzy sets, which are, respectively, Kleene and De Morgan REFERENCES [1] Grätzer, G, General Lattice Theory, Birkhauser, Boston, 1998 [2] John, R, Type-2 fuzzy sets: An appraisal of theory and applications Int J Uncertainty, Fuzziness Knowledge-Based Systems 6 (6) (1998) [3] Karnik, N and Mendel, J, Operations on type-2 fuzzy sets, Fuzzy Sets and Systems, 122 (2001) [4] Mendel, J and John, R, Type-2 Fuzzy Sets Made Simple, IEEE Transactions on Fuzzy Systems, vol 10, No 2, , April 2002 [5] Mizumoto, M and Tanaka, K, Some Properties of Fuzzy Sets of Type-2, Information and Control 31, (1976) [6] Mizumoto, M and Tanaka, K, Fuzzy sets of type-2 under algebraic product and algebraic sum, Fuzzy Sets and Systems 5 (1981) [7] Emoto, M and Mukaidono, M, Necessary and Sufcient Conditions for fuzzy Truth Values to Form a De Morgan Algebra, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 7(4) (1999) [8] Nieminen, J, On the Algebraic Structure of Fuzzy Sets of Type 2, Kybernetika, 13, (1977) [9] Walker, C and Walker, E, The Algebra of Fuzzy Truth Values, Fuzzy Sets and Systems, 149 (2005) [10] Walker, C, and Walker, E, Automorphisms of Algebras of Fuzzy Truth Values, International Journal of Uncertainty, Fuzziness and Knowledgebased Systems, 14(6) (2006) [11] Walker, C, and Walker, E, Automorphisms of Algebras of Fuzzy Truth Values II, International Journal of Uncertainty, Fuzziness and Knowledge-based Systems, 16(5) (2008) [12] Walker, C and Walker, E, Sets With Type-2 Operations, International Journal of Approximate Reasoning (2008), doi:101016/jijar [13] Walker, C and Walker, E, Points With Type-2 Operations, Proceedings of the IFSA Conference, Cancun, June 2007, [14] Walker, C and Walker, E, Type-2 Intervals of Constant Height, Proceedings of the NAFIPS Conference, San Diego, June 2007, [15] Zadeh, L, The concept of a linguistic variable and its application to approximate reasoning, Inform Sci 8 (1975)