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2. Duality in Mathematical Structure by Tadashi OHKUMA Departmentof Mathematics,Tokyo Institute of Technology,Tokyo (Comm 6 [Vol. 34, 2. Duality in Mathematical Structure By Tadashi OHKUMA Departmentof Mathematics,Tokyo Institute of Technology,Tokyo (Comm. by Z. SUETUNA,M.J,A., Jan. 13, 1958) 1. In many representation theorems of different branches of ab- stract mathematics, for example, representation of ordered sets by cuts, that of lattices by sets of ideals, the conjugate spaces of Banach spaces, or the duality of groups, there seem to appear some similar concep- tions. In order to deal with those representation theorems simultane- ously, we made an attempt to set up a concept of a universal mathe- matical structure, in which the main role is played by some selected applications of a system to a system, which we call homomorphisms, but they may be continuous mappings in topological spaces, order- preserving mappings in ordered sets or linear operators in linear spaces. The definitions of our structure and some results of them will be stated in this note, but we omit the detail of proofs which, as well as the applications to individual specialized systems, will be published elsewhere. 2. Let be a family of sets. A set in is called a system. If X and Z are systems and Z C X, then Z is called a subsystem of X. We assume that, to each pair of systems X and Y, a family Hom (X, Y) of applications which map X into Y is distinguished. An application co in Hom (X, Y) is called a homomorphism of X in Y. Further we assume that those homomorphisms and the family satisfy the following axioms which fall into five groups (A)-(E). In these statements of axioms, the letters X, Y and Z denote systems. The axioms of the group (A) are concerned with the conditions for an application of a system to be a homomorphism. (A1) If ZC X, and IZ is the identical mapping on Z, then I~ EHom (Z, X). (A2) I f q e Hom (X, Z) and EHom (Z, Y), then ]icpe Hom (X, Y). (A3) I f cpe Hom (X, Y) and ~o(X)C Z, then cpe Hom (X, Z). The axioms of the group (B) are concerned with the conditions for a set to be a system. (Bl) I f ~pE Hom (X, Y), then tp(X) E~. (B2) If cpe Hom (X, Y) and Z,q (X) c, then q -'(Z) E (B3) There exists a one element system {e} such that for any system X the application which maps each element x of X onto e is 1) ~c(X)=f c(x);x e X }, c1(Z)=x; ~o(x)EZ}. No. 1] Duality in Mathematical Structure 7 a homomorphism of X in {e}.2' If p EHorn (X, Y) and cp(X) = Y, then p is called a homomorphism of X on Y, and Y is called homomorphic to X. A homomorphism cp of X in Y is called an isomorphism, if it is one-to-one and its inverse is also a homomorphism of yp(X) on X. If there exists an isomorphism of X on Y, then they are called isomorphic. Before stating the remaining axioms we shall introduce the defini- tion of the direct product.3' DEFINITION 1. Let X~; AE A, where A is a set of indices, be a family of systems. A system W, denoted by II Xa, is called the direct product of Xi's, if W satisfies the following conditions: i) W is a set-product of X2's, that is, there exists a one-to-one correspondence between an element w of W and a sequence {x}~of elements x~ EXa. This correspondence will be denoted by w='r{xa} and x~= pe(w), where w corresponds to {xA}. ii p~ EHom ( W, XA) for each d EA. iii) I f Z is a system and co e Hom (Z, Xa) for each AE A, then the mapping (p which is defined by q(z) = it{q (z)} for z e Z is a homo- a morphism of Z in W. Especially if the set A is finite and A = {1, 2, .. n}, then we may use the following notation: X1 X X~X ... X Xn = H X~. If each X~ is a same set X, then X' is defined as II Xa. Now we shall give a further axiom (C): (C) For any family {X;~A EA} of systems, where A is a set of indices, there exists a system which is the direct product H It is easily seen that if both U and W are direct products of Xx's, then U and W are isomorphic. Next, let X and Y be any systems. We put Yx = Y for each x EX and make the direct product YX. Then YX consists of all ap- plications from X in Y. Hence Hom (X, Y) is considered as a subset of YX. Under this interpretation we give the fourth axiom (D). (D) Hom (X, Y) is a subsystem of YX.5> The family may be a very vast one. For example it may be a family of whole topological spaces. Here, to avoid some pathology 2) It seems to be convenient in many investigations to add the followingaxiom: (B4) If X~, Au A, are systems and n X~ ~, then n X2, is a system. But this axiom is not necessary in this paper. 3) This definitionand Theorem1 are stated by V. S. Krishnan: Closureoperations on c-structure, Indagationes Math., 15, 317-329(1953). 4) For example, the family of all totally ordered sets does not satisfy this axiom. In Banach spaces, the definition of direct products must be modified. 5) The set of all representations of a group by n-square matrices does not make itself a group, and our theorems can not be applied to Tannaka's duality theorem. a 8 T. OHKUMA [Vol. 34, of the set-theory, we assume ( E ) For any subfamily y of such that each system in has a power less than some cardinal number, a family 8 of systems can be selected in such a way that this family 3 makes itself a set with some power, and any system in ti is isomorphic to one of the systems in s3. 3. Let q and fr be homomorphisms of a system X. We say that q and / are equivalent if there exists an isomorphism of cp(X ) on i(X) with k=8cp. Let Horn (X) be the family of all homomorphisms of X. If we identify equivalent homomorphisms in Horn (X), then by the axiom (E), Horn (X) makes itself a set with some power. If we introduce an order < between homomorphisms in Horn (X) in such a way that p<r implies that there exists a 8 EHorn ('(X), cp(X )) with p = 8, then Horn (X) becomes a partially ordered set. But by the definition of the direct product and the axioms (C) and (B3) we have THEOREM 1. Horn (X) is a complete lattice by the order _<. Let P be a subset of Horn (X), and let V P denote the least upper bound of homomorphisms in P, then we can easily see LEMMA 1. Put _ V P. We have 'J'(x0) _ itt(x1), x0, x1 EX, if and only i f cp(xo)_ p(x1) for any ~pE D. Hereafter we fix a system L, and Horn (X, L) is denoted by X*. The first main theorem which is deduced from the axioms (A) (B) (C) (D) and (E) is the following THEOREM 2. (1) Let X be a system. I f we put x(cp)=~(x) for x EX and cPE X*, then x e X * * =Horn (X*, L). (2) If we put T(X)=,xthen T EHom (X, X**). (3) T is equivalent to V Horn (X, L). (4) The necessary and sufficient condition that we have IX= V Horn (X, L) s' is that for a sufficiently large set A of indices, X is isomorphically embedded in L1. Similarly, by putting ~b( ) _ e(p) for cpe X* and e e X* *, we have a homomorphism ~ EX * * *, and putting T° (cp)_ cp, we have a homo- morphism T° EHorn (X*, X***). But by the definition of Horn (X, L) =X, *X * is a subsystem of Lx , and hence by Theorem 2 (3) and (4), T° is an isomorphism. But further we have THEOREM 3. (1) Let be a homomorphism in X * * * =Hom (X**, L). Put q X)=), for x EX, (=T(X),xsee Theorem 2. (1)), then we have q:1E X* =Horn (X, L). (2) Put x (?/) _ q , then we have x e Horn (X***, X*). (3) The contraction of x on T°(X*) is the inverse mapping of the isomorphism T°. 6) Ix is the identical mapping on X, see axiom (AI). No. 1] Duality in MathematicalStructure 9 4. By the axiom (A2), for any 8 EHorn (X, Z) and cpe Horn (Z, Y), we have a homomorphism cp8 EHorn (X, Y). Here we have THEOREM 4. (1) Put *p(8)=cpO, then we have *~pE Horn (Horn (X, Z), Horn (X, Y)). ( 2 ) Put e*(co)=qp9, then we have 8* EHorn (Horn (Z, Y), Horn (X, Y)). ( 3 ) The mapping ~P **cp is a homomorphism of Horn (Z, Y) in Horn (Horn (X, Z), Horn (X, Y)). ( 4 ) The mapping 0 8* is a homomorphism of Horn (X, Z) in Horn (Horn (Z, Y), Horn (X, Y)). ( 5 ) If 8 is a homomorphism of X on Z, then B* is an iso- morphism of Hon (Z, Y) in Horn (X, Y).
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