Modern Set

MODERN SET JUN T ANAKA 1. Introduction In this pap er, we in tend to generalize the classical set theory as m uch as p ossible. we will do this b y freeing sets from the r egular prop erties of classical sets; e.g., the law of exc luded middle, the la w of no n-contradiction, the distributiv e law, the commutativ e law,etc.... The fuzzy set theory succeeded in fre e ing sets from the law of excluded middle and the law of contradiction. Howev er, in order to extend our language, it is more o r les s unreasonable to keep the commutativ e la w and the distributiv e law. This moder n idea of sets keeps the conce pt of membership functions but their v a lue are not necessarily in [0,1]; nor do these mo der n s e ts form a lattice necessarily . Esp ecially noteworth y is that moder n sets are more general than ge ner alized fuzzy set, (P le ase refer to Nak a jima [6]). Here is the hierarch y of generality: Moder n sets ≥ L- fuzzy set ≥ Generalized F uzzy s et ≥ fuzzy set ≥ classical set. Thes e moder n sets b ecome the class ical sets under the restrictio n to O x , I x for a ll x in X, and no further conditions are required The w orld of natural languages can not b e ruled by a single c la ssical lo gic b ecause the rea l ph ysical world consists o f many asp ects, ea ch of which obey s a different logic. As is well-known, the distributive law do es not alwa ys hold in every logic system. F or example, in cla s sical and in tuitionistic log ic it holds, but in q uantum logic it fails. The Commutativ e law s e ems inv alid when we talk ab out anything with tempo r al order in our daily conv ersation since ev ery ev ent in time is not reversible. ”He is a student a nd a male”. I can also say , ”He is a male and a student” but mean the same thing. The tw o sig nifications commute of course. ”I hav e a health insura nce and a ca r insurance”. In this case, the tw o significations comm ute. But how a bo ut this case ” He wen t to a sup e r market and then a drug store ” . In this ca s e, the tw o significations do not commute anymore since there is temp or al or der. F urthermore, we have no expr ession for ”He wen t to a s up er market and then a dr ug store” in classical logic or fuzzy log ic. In o ther words, the pr op erties from classic al lo gic, which we take for granted, do not express our thoughts that well. 2. Preliminaries : Extension of La ttice In this section, we shall briefly review the well-known facts ab out lattice theor y (e.g. Birkhoff [1], Iwamura [4] ), prop ose an extension lattice, and in vestigate its prop erties. (L, ∧ , ∨ ) is ca lled a lattice, if it is closed under op erations ∧ and ∨ , and satisfies, for a n y elements x,y ,z in L: (L1) the co mm utative la w: x ∧ y = y ∧ x and x ∨ y = y ∨ x Date : May , 25, 2008. 2000 Mathematics Subject Classific ation. Primary: 03E72. Key wor ds and ph r ases. F uz zy set, L- fuzzy set, generalized fuzzy set. 1 2 JUN T ANAKA (L2) the a sso ciative la w: x ∧ ( y ∧ z ) = ( x ∧ y ) ∧ z and x ∨ ( y ∨ z ) = ( x ∨ y ) ∨ z (L3) the a bsorption law: x ∨ ( y ∧ x ) =x and x ∧ ( y ∨ x ) = x. Hereinafter, the la ttice (L, ∧ , ∨ ) will often b e wr itten L for simplicity . A mapping h from a la ttice L to another L ′ is called a lattice-homomor phism, if it satisfies h ( x ∧ y ) = h ( x ) ∧ h ( y ) and h ( x ∨ y ) = h ( x ) ∨ h ( y ) , ∀ x, y ∈ L. If h is a bijection, that is , if h is one-to- o ne and on to, it is called a lattice- isomorphism; and in this case, L ′ is said to be lattice-isomor phic to L. A lattice (L, ∧ , ∨ ) is called distributive if, for any x,y ,z in L, (L4) the dis tr ibutiv e law holds: x ∨ ( y ∧ z ) = ( x ∨ y ) ∧ ( y ∨ z ) and x ∧ ( y ∨ z ) = ( x ∧ y ) ∨ ( y ∧ z ) A lattice L is calle d co mplete if, a n y subset A of L, L c o ntain the supremum ∨ A and the infimum ∧ A. If L is co mplete, then L itself includes the ma ximum and minim um elements, whic h a re often denoted by 1 and 0 , o r I and O , resp ectively . Definition 1. Co mplete He yti n g alge bra (cHa) A complete la ttice is called a complete Heyting alg ebra (cHa), if ∨ i ∈ I ( x i ∧ y ) = ( ∨ i ∈ I x i ) ∧ y holds for ∀ x i , y ∈ L ( i ∈ I ); where I is an index se t of arbitra ry cardinal num b er . A distributive lattice is called a Bo olean algebr a or a B o olean lattice, if for any element x in L, there exists a unique complement x C such that x ∨ x C = 1 and x ∧ x C = 0. It is well-known that for a se t E, the pow er set P(E ) = 2 E . The set of all s ubsets of E, is a Bo olean algebra. 3. Preliminaries : Generalized fuzzy sets In this sec tion we will c o nsider an alge braic s tructure of a family o f fuzzy sets. W e will show that the following three families ar e mutually equiv alent: a ring o f generalized fuzzy subsets, a n extension of (Boolea n) Lattice P(X), a nd a s et of L-fuzzy sets (introduced b y Gouen [3]). Definition 2. A family GF(X), whic h is closed under o per ations ∨ a nd ∧ , is calle d a ring of gene r alized fuzzy subsets of X, if it s atisfies: (1) GF(X) is a complete Heyting algebra with r esp ect to ∨ and ∧ , (2) GF(X) c ontains P (X) = 2 X as a sublattice of GF(X), (3) the ope r ations ∨ and ∧ coincide with set op e rations S and T , r esp ectively , in P(X), a nd (4) for a ny element A in P(X), A ∨ X= X and A ∧∅ = ∅ . Definition 3. Let L x be a lattice whic h is assigned to each x in X, a nd le t L denote { L x | x ∈ X } . An L-fuzzy s et A is characterize d by an L-v a lue d mem b ership function µ A which a sso ciates to each point x in X an element µ A ( x ) in L x . MODERN SET 3 Theorem 1 . If e ach L x is a cH a, then LF(x), the family of al l L-fuzzy sets, is a ring of gener alize d fuzzy subsets of X. We wil l show that LF(X) satisfies al l the c onditions in Definition 2. First, by the definition, LF(X) is close d under ∧ and ∨ . (1) LF(X) is a cHa, b e c ause e ach L x is a cHa. Condition (2) LF(X) k P(X) and (3) ∨ = ∪ and ∧ = ∩ in P(X) fol low fr om the fact that any element of P(X ) is define d with the element of LF(X ) whose memb ership function takes just two values, 1 and 0. ( 4) It fol lows fr om µ X ( x ) =1 and µ ∅ ( x ) = 0 that A ∪ X = X and A ∩ ∅ = ∅ , for any A in LF(X). 4. Modern Sets Definition 4. Bo olean Algebra In conformity with Birkhoff ’s b o ok [1], the fundamental oper ations of in tersection and union of elements will b e defined b y x ∩ y int ersec tio n x ∪ y union As is well known, it f ollows from the definition of Boolea n algebr a that there exis ts a unit element I and a null element O for which w e hav e the following: x ∩ I = x and x ∩ O = O x ∪ I = I and x ∪ O = x ∀ x ∈ X Note that ∩ a nd ∪ commute at this p o int . Definition 5. W eak B o olean algebra Let H b e an alg ebraic s pace with tw o distinct oper ators ∗ ∧ , ∗ ∨ from H to itself. H is called a W eak Bo olean Algebra if ∃ distinct O, I ∈ H such that O ∗ ∧ I = O and I ∗ ∧ O = O O and I commute O ∗ ∧ O = O and I ∗ ∧ I = I O ∗ ∨ I = I and I ∗ ∨ O = I O and I co mm ute O ∗ ∨ O = O and I ∗ ∨ I = I Please no te that ∗ ∧ ∗ ∨ are as so ciated with ∧ and ∨ , respe c tiv ely . O and I are asso ciated with the minim um element and the maximum element, respec tively . Definition 6. M o dern Sets Suppo se a w eak Bo o lean Algebra H x is assigned to each x in X and let H denote { H x | x ∈ X } . Each mo dern set A is characterized by a membership function µ A such that for each x ∈ X, µ A assigns a n element µ A ( x ) ∈ H x . W e define H ( X ) as the family of all modern s ets. When A is a set in the ordinary sense o f the term (in P(X)), its members hip function can take on only t wo v a lues O x and I x with µ A ( x ) = O x or I x according to whether x do es or do es not belong to A. The o p er ations ∗ ∨ x and ∗ ∧ x coincide with the set op eratio ns S and T , resp ectively , in P(X). A mo dern s et is empty if and only if its mem b ership function is identically O x for all x in X. Two mo dern sets A a nd B are equal if and only if µ A ( x ) = µ B ( x ) for all x in X. If H x is partially or dered by ≤ for ea ch x, then we can define containmen t as follows: A is contained in B if and only if µ A ( x ) ≤ µ B ( x ) for a ll x in X. 4 JUN T ANAKA Union . The union of tw o mo dern sets A a nd B with res pective membership functions µ A ( x ) and µ B ( x ) is a mo dern set, written as C = A ∨ B , who se member- ship function is related to those of A and B by µ C ( x ) = µ A ( x ) ∗ ∨ x µ B ( x ) , x ∈ X Note that the order do es matter if H x is not commutativ e. In tersection The intersection of tw o mo der n sets A and B with resp ective mem b ership functions µ A ( x ) and µ B ( x ) is a mo dern set, written as C = A ∧ B, whose membership function is related to tho s e of A a nd B by µ C ( x ) = µ A ( x ) ∗ ∧ x µ B ( x ) , x ∈ X The notion of complement was no t given in Definition 5 since we ca n trivially define the complemen t on a Mo der n Set for each x in X such as · C : H x → H x where ( { O x } ) C = I x and ( { I x } ) C = O x even while ( { A x } ) C can be an ything for all A x ∈ H x , A x 6 = O x , I x such that (( { A x } ) C ) C = { A x } Theorem 2. F or any mo dern sets A, B , C whose memb ership function take on values O x or I x , (L1)-(L4) hold. Pr o of. Since the pro of is mo re or less clear, herein w e will brie fly indicate the pro of for the commutativ e a nd distributiv e cases. Let’s c heck the comm utative law. Call C = A ∨ B. W e only hav e to chec k the four p oss ible c a ses: O x ∗ ∧ I x = O x and I x ∗ ∧ O x = O x O x ∗ ∧ O x = O x and I x ∗ ∧ I x = I x for all x ∈ X Thu s, µ C ( x ) = µ A ( x ) ∗ ∨ x µ B ( x ) = µ B ( x ) ∗ ∨ x µ A ( x ). Therefore, C = A ∨ B = B ∨ A. W e c a n show the comm utative la w for intersection similarly . W e will show the t wo most impo r tant cases of the distributive law briefly . Call C = A ∨ ( B ∧ C ). O x ∗ ∨ ( I x ∗ ∧ O x ) = O x ∗ ∨ O x = O x = I x ∗ ∧ O x = ( I x ∗ ∨ O x ) ∗ ∧ ( O x ∗ ∨ O x ) O x ∗ ∨ ( I x ∗ ∧ I x ) = O x ∗ ∨ I x = I x = I x ∗ ∧ I x = ( O x ∗ ∨ I x ) ∗ ∧ ( O x ∗ ∨ I x ) for a ll x ∈ X Please chec k the rest o f cases. Tha t should not be to o difficult.  Theorem 3. H x is c ommutative for al l x in X iff H = { H x | x ∈ X } is c ommu t ative. Pr o of. It is obvious.  Theorem 4. H x is distributive for al l x in X iff H = { H x | x ∈ X } is distributive. Pr o of. It is obvious.  Theorem 5. Similar e quivalent st atemen t (as ab ove) hold for the absorption law, the law of ex clude d midd le, the law of non-c ontr adiction, the asso ciative law, etc... MODERN SET 5 Remark 1. As you se e fr om the pr evious the or ems, if ( H x , ∗ ∨ x , ∗ ∧ x , · C ) is define d to b e in the sense of fuz zy sets such as ∗ ∨ x = max, ∗ ∧ x = min, ( · ) C = 1- ( · ) in the interval [0,1] , then H ( X ) b e c omes the family of fuzzy set s. If H x satisfy the definitio n of L att ic e for e ach x in X , then H ( X ) b e c omes a family of L-fuzzy set. If H x additiona l ly satisfy the d efinition of a c omplete Heyting algebr a, it is a ring of gener alize d fuzzy subsets. Pr o of. The pro of is clear.  Example 6. We wil l pr esent an ex ample of a non- c ommut ative mo dern set. L et H x b e a sp ac e of line ar b ounde d op er ators on a Hilb ert sp ac e H for e ach x in X . Then we take the zer o O x and the identity I x in H x . We define the c omp osition ◦ = ∗ ∧ x and the addi tion + = ∗ ∨ x . In or der to cr e ate a we ak Bo ole an Algebr a, we must define an e quivalenc e class ∼ as A ∼ B iff A = B or ther e exist n, m ∈ N − { 0 } such t hat A = nI x and B = mI x . O x ( I x ) = O x and I x ( O x ) = O x O x and I x c ommu te O x ( O x ) = O x and I x ( I x ) = I x O x + I x = I x and I x + O x = I x O x and I x c ommu te O x + O x = O x and I x + I x = 2 I x = I x Now we have a we ak Bo ole an Al gebr a H x wher e the c omp osition op er ation is typ- ic al ly not c ommu tative. Th us H(X) is a family of mo dern sets of non-c ommutative typ e if one of H x is not c ommutative under the c omp osition. Ne e d less to say, if al l of H x is c ommutative un der the c omp osition, then H(X) satisfies the c ommut ative law. Example 7. L et (M, n × n , + , · ) b e an n × n matr ix sp ac e close d under addi tion + and matrix multiplic ation · . Then c al l the matrix multiplic ation identity I and the addition identity matrix O. Now we take · = ∗ ∧ x and + = ∗ ∨ x . By c onsidering the same e qu ivalenc e class as in Ex ample 6, we c an cr e ate a mo dern set fr om t he n × n matrix sp ac e. Theorem 8 . Gelfand The or em If U is a c ommutative a C ∗ -algebr a, t hen U is ∗ -isomorph ism to C ( X ) , some c omp act Hausdorff sp ac e X . Remark 2. F or a given c ommutative C ∗ -algebr a, we have a r epr esentation of it as a sp ac e of c ontinuous functions on some c omp act Haus dorff by the or em 8. Thus we c an always cr e ate a c ommu tative Mo dern set fr om any c ommutative C ∗ -algebr a under the same c onstr u ction as in Example 6. W e give the following definition and theorem as more examples of Mo dern Sets. Definition 7 . By a repr esentation of a C ∗ -algebra U on a Hilb ert space H , w e mean a ∗ homomorphism ϕ from U in to B ( H ). If in addition, ϕ is one - to-one, it is called a faithful representation. Theorem 9 . The Gelfand-Neumark The or em Each C ∗ -algebr a has a faithful r epr esentation on some Hilb ert sp ac e. Example 10. F or a given C ∗ -algebr a, we have a r epr esent ation of it as b ounde d line ar op er ators on a Hilb ert sp ac e by the or em 9. Thus we c an always cr e ate a Mo dern set fr om any C ∗ -algebr a under the same c onst r u ction as in Example 6. 6 JUN T ANAKA 5. Conclusion and obser v a tion As we men tioned in the Intro duction, systematic expr ession of o ur thought r e- quires r o om for at least the non-commutativ e pr op erty . I strongly b elieve that this new lo gic system w ill op en up a new blanch of Artificial Intelligence. P rop erty-like verbs such as ”be” , ” hav e”, and ”o wn” se em v a lid in classica l logic. How ever, most of the other v erbs are required to be non- commutativ e with res pec t to ob jects a nd time. This mo der n set do es not need to be co mm utative, in so me sens e , this is closer to the s ystem of our thought. W e need further inv estigation to improve the systematic expres s ion o f our thought in order to create a real Artificial I ntelligence. I dream the day will come, when we mak e a real AI. References [1] G. Birkhoff, Lattice Theory , 3r d ed. AMS collo quim Publication, Pr o vidence, RI, 1967. [2] J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl., 18, 145-174, 19 67 [3] M. Heidegger, b eing and tim e, H ar perOne; Revised ed ition, 1962 (English) [4] T. Iwam ura, Sokuron (Kyoritsu Shuppan, T okyo, 1966). [5] R. V. Kadison, J.R . Ri ngrose, F undamen tals of the Theory of Opreator Algebra, AMS, 1997. [6] N. Nak amu ra, generalized fuzzy sets, F uzzy set s and Syst ems 32 (1989), 30 7-314. [7] G. T ake uti, Senk ei-Daisu to ryoushi-rikigaku, 131-162, Shok abo, T okyo , 1981 (Japanese) [8] L.A. Zadeh, F uzzy sets, Infor mation and Con trol 8 (19 65), 33 8-353. University of California, Riverside, USA E-mail addr ess : juntanaka@ math.ucr.edu, yonigeninnin @gmail.com, junextension@h otmail.com