848 Jyung Ryun Seo and Chang Koo Lee that NFrm is complete, cocomplete and study permanence properties of important subcategories of NFrm. Furthermore

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1 Comm. Korean Math. Soc. 13 (1998), No. 4, pp. 847{854 CATEGORIES OF NEARNESS FRAMES Jyung Ryun Seo and Chang Koo Lee Abstract. We investigate categorical properties of the category NFrm of nearness frames and uniform homomorphisms. We introduce a concept of weakly strong nearness frames and study its permanence properties. 1. Introduction Benabou noticed that frames(=locales=complete Heyting algebras) can be used to study topological structure in the setting of order structure([5]) and Herrlich has introduced the concept of nearness which generalize topological structure, uniform structure, proximal structure and contiguity structure([6], [7]). Combining these two concepts, Banaschewski and Pultr have introduced the concept of nearness frames and then constructed completions of nearness frames([2], [4]) which give rise to coreection for strong nearness frames([3]). Hong and Kim constructed Cauchy completions of nearness frames and show that the category of strong Cauchy complete nearness frames is coreective in the category SNFrm of strong nearness frames and uniform homomorphisms([8]). The purpose of this paper is to study categorical properties of nearness frames. We show that the forgetful functor G of the category NFrm of nearness frames and uniform homomorphisms to the category Frm of frames and frame homomorphisms has the right adjoint. Moreover, we show Received March 2, Revised June 6, Mathematics Subject Classication: 06D99, 18A30, 18A40, 18B35, 54E17. Key words and phrases: Nearness frames, adjoint functors, uniform, strong, almost uniform frames, weakly strong nearness frames.

2 848 Jyung Ryun Seo and Chang Koo Lee that NFrm is complete, cocomplete and study permanence properties of important subcategories of NFrm. Furthermore, generalizing strong nearness frames, we introduce a concept of weakly strong nearness frames and study its categorical properties. We include some basic denitions for the completeness of the paper. Definition 1.1. A complete lattice L is said to be a frame if for any element x and any subset S of L, x ^ ; _ S = _ fx ^ y j y 2 Sg: In the following, top(bottom,resp) of a frame L will be denoted by e(0 resp) and for any a 2 L it has the pseudo complement a = W fx 2 L j a ^ x =0g: Furthermore, we write a b if a _ b = e and L is called a regular frame if for any x 2 L x = W fa 2 L j a xg: Definition 1.2. Amaph : L! M between frames is called a frame homomorphism if h preserves all nite meets and arbitrary joins. A subset C of a frame L is called a cover of L if W C = e and the set of covers of L will be denoted by CovL: For C D 2 CovL we say that C renes D if for any c 2 C there is d 2 D with c d: In this case, we write C D: Moreover, (CovL ) isa quasi-ordered set. For x 2 L and C 2 CovL C x = W fc 2 C j x^c 6= 0g and CC denotes the set fc c j c 2 Cg: For C D 2 CovL we write C D if CC D: Notation 1.3. For a subset N of CovL and x y 2 L we write x C N y or simply x C y if there is C 2N with C x y: The following is due to Banaschewski and Pultr ([4]). Definition 1.4. (1) Let L be a frame and N a subset of CovL: Then N is called a nearness on L if N is a lter in (CovL ) and for any x 2 L x = W fa 2 L j a C xg and (L N ) is called a nearness frame. (2) A map h : (L N )! (M M) between nearness frames is called a uniform homomorphism if h is a frame homomorphism and for any C 2N h(c) 2M:

3 Categories of nearness frames 849 For any frame L there is a nearness on L if and only if L is a regular frame, because x C N y implies x y for a nearness frame (L N ) and for N = CovL x y implies x C N y (see [2] for the details). Definition 1.5. A nearness frame (L N )issaid to be : (1) uniform if for any C 2N there exists D 2N with D C: (2) strong if for any C 2 N C = fx 2 L j x C y for some y 2 Cg belongs to N. (3) almost uniform if it is strong and C interpolates, i.e., if x C y there is z 2 L such that x C z C y: (4) totally bounded if fc 2N j C is nite g generates N : For the terminology not introduced in the paper, we refer to [1] for the category theory and [9][11] for frames. 2. The category of nearness frames In what follows, the category of frames (regular frames, resp.) and frame homomorphisms will be denoted by Frm (RFrm, resp.) and NFrm denotes the category of nearness frames and uniform homomorphisms. One has immediately the forgetful functor U : NFrm! RFrm: We dene F : RFrm! NFrm as follows : for any L 2 RFrm F (L) =(L CovL) and f 2 RFrm F (f) =f: Then it is clear that F is a functor, for any frame homomorphism preserves covers. Proposition 2.1. The functor F : RFrm! NFrm is embedding and F is a right adjoint of U : NFrm! RFrm: Proof. The rst half is immediate from UF = 1 RFrm : For any L 2 RFrm let " L : UFL! L be the identity map of L: For any (M M) 2 NFrm and a frame homomorphism h : U(M M)! L the map h : (M M)! FL ; h(x) = h(x) is clearly a uniform homomorphism, because for any C 2 M h(c) 2 CovL: Furthermore, " L U(h) = h and such an h is unique. Thus F is the right adjoint of U: Corollary 2.2. The functor U : NFrm! RFrm preserves colimits and F : RFrm! NFrm preserves limits.

4 850 Jyung Ryun Seo and Chang Koo Lee Let G : NFrm! Frm denote the forgetful functor and E : RFrm! Frm the embedding functor, then G = EU: For anyframel it contains the largest regular subframe L and since the homomorphic image of a regular frame is again a regular frame, the inclusion map : L! L gives rise to the RFrm-coreection of L (see [2] for The details) and hence E has the right adjoint. Thus wehave the following by Proposition 2.1. Corollary 2.3. The functor G : NFrm! Frm has the right adjoint and therefore preserves colimits. Proof of the following lemma can be found in [10]. Lemma 2.4. Suppose that (h i : L i! M) is an extremal epi sink in Frm such that (L i N i ) is a nearness frame for each i 2 I. If M is a lter generated by B = f^j2j h j (A j ) j J is a nite subset of I A j 2N j g then M is the G-nal nearness on M with respect to (h i ) : Theorem 2.5. The category NFrm is cocomplete. Proof. Let D : I! NFrm be any diagram and let ; l i : GD(i)! L be the colimit of G D in Frm: Since (l i) is an extremal epi sink in Frm there is the G-nal nearness N on L with respect to (l i ) by the above lemma. We now show that ; l i : D(i)! (L N ) is the colimit for D: Take any natural sink ; f i : D(i)! (M M) for D then ; G(f i ) is a natural sink for G D and hence there is a unique frame homomorphism f : L! M such that for any i 2 I f l i = G(f i ): Since (l i ) is G-nal, there is a unique uniform homomorphism g : (L N )! (M M) with G(g) = f and g l i = f i : Since G is faithful, such a g :(L N )! (M M) isunique. This completes the proof. Theorem 2.6. The category NFrm is complete. Proof. It is enough to show that NFrm has equalizers and products. First we prove that NFrm has equalizers. Let (L N ) h 1 h2 (M M) be a pair of uniform homomorphisms and let j : E! L be the equalizer for h 1 h 2 in Frm: In fact, E is the subframe fx 2 L j h 1 (x) = h 2 (x)g of L: Let ; (E i N i ) be the family of nearness frames such that E i is a

5 Categories of nearness frames 851 subframe of E and the inclusion frame homomorphism j i : E i! (L N ) is uniform for i 2 I: Let E o be the subframe of L generated by S fe i j i 2 Ig which is clearly a subframe of E: Let h i : E i! E o be the corestriction of j i to E o then (h i : E i! E o ) is an extremal epi sink in Frm therefore there is the G-nal nearness N o on E o with respect to ; h i : (E i N i )! E o : Let m : E o! L be the inclusion homomorphism. Since m h i = j i and j i is uniform for all i 2 I m :(E o N o )! (L N ) is also a uniform homomorphism. Clearly h 1 m = h 2 m: Take any uniform homomorphism h :(H H)! (L N ) with h 1 h = h 2 h: Then there is a unique frame homomorphism k : H! E with j k = h: Let k o : H! k(h) be the corestriction of k to k(h) which is an onto frame homomorphism. Thus there is the G-nal nearness H 0 on k(h) with respect to k o :(H H)! k(h): Let n : k(h)! L be the inclusion homomorphism, then n k o = h implies that n : ; k(h) H 0! (L N ) is uniform, because k o : (H H)! (k(h) H 0 ) is G-nal. Thus there is 2 I such that n = j i.e., ; k(h) H 0 = (E N ): Thus k(h) E o : Let h : H! E o be the corestriction of k to E o then h = (H H)! ko (E N ) j! (E o N o ) is a uniform homomorphism and m h = h: Since m is 1 ; 1 such an h is unique. Thus m = equ(h 1 h 2 ) in NFrm: Regarding products in NFrm let ; (L i N i ) be a family of nearness frames. Let L = Q L i be the product of (L i ) in Frm: For each 2 I and C 2N, dene C ~ = f(x i ) 2 L jx 2 C x i =0for i 6= g: Take C = (C i ) 2 Q N i and let C ~ = C S ~ i then W C ~ = el : Let B = f C ~ j C = (Ci ) 2 Q N ig. Choose C ~ and D ~ in B with C = (C i ) and D =(D i ). For each i 2 I E i = C i ^ D i 2 N i and E =(C i ^ D i ) 2 Q N i. Since E ~ C ~ ^ D ~ ( 2 I), E ~ C ~ ^ D: ~ Then E ~ 2 B for E 2 Q N i : Hence B is a lterbase on ; CovL : Let N be the lter on (CovL ) generated by B. We claim that for x = (x i ) y = (y i ) 2 L x C N y if x i C y i for all i 2 I: Indeed, let C i 2N i with C ixi y i and C =(C i ) : Take any (c i ) 2 C ~ with (c i ) ^ (x i ) 6=0 then there is 2 I with c ^ x 6= 0 hence c i =0 for all i 6= : Thus c y which implies (c i ) (y i ) : Take any x = (x i ) 2 L then x = W fy 2 L j y C x g for each 2 I: For any a 2 L we dene

6 852 Jyung Ryun Seo and Chang Koo Lee 0 i 6= ~a = i = a in L and ~x = W f~y j y C x g then x = W 2I ~x W fy j y C xgx: Thus (L N )isa nearness frame. For each 2 I the -th projection Pr : L! L is a frame homomorphism. For any C ~ 2 B Pr ( C) ~ = S Pr ( Ci ~ ) = f0g[c 2 N ( 2 I): Hence each Pr is uniform. Take any source ; f i : (M M)! (L i N i ) in NFrm let f = u f i : M! L be the frame homomorphism with Pr i f = f i (i 2 I): It remains to show that f is uniform. Take any D 2 M, then f i (D) 2 N i for all i 2 I: Put C i = f i (D) and let C = (C i ) 2 Q N i then C ~ 2 N. Take any x = (xi ) 2 C ~ then there is 2 I with x 2 C ~ hence x 2 C and x i = 0 for i 6= : Let d 2 D with f (d) = x then x f(d): Thus C ~ f(d) so that f(d) ; 2 N : Such an f is unique, for (Pr i ) is a mono-source. In all, Pri :(L N )! (L i N i ) is a product in NFrm: Proposition 2.7. The following subcategories of NFrm is closed under the formation of products in NFrm: (1) The category UFrm of uniform frames, (2) The category SNFrm of strong nearness frames, (3) The category AUFrm of almost uniform frames. Proof. Let (L i N i ) be a family in NFrm and (L N ) the product of the family. Using the notions in the proof of Theorem 2.6, B = f ~ CjC = (C i ) 2 Q N ig generates N. 1) Suppose that each (L i N i ) is a uniform frame for all i 2 I. Take any ~ C 2B,i.e., ~ C = S ~ C i, where C =(C i ) 2 Q N i. Then there is D i 2N i (i 2 I) with D i C i and hence D =(D i ) 2 Q N i and ~D 2 B: Take any (d i ) 2 ~ D then there is 2 I with d 2 D and d i =0for all i 6= so that there is c 2 C with D d a i = c if i = 0 if i 6=, c : Put then (a i ) 2 C ~ and D(di ~ ) (a i ), because for (x i ) 2 D, ~ (x i ) ^ (d i ) 6= 0 implies x ^ d 6= 0 and hence x i = 0 for i 6=. Thus D ~ C, ~ so that (L N )isa uniform frame.

7 Categories of nearness frames 853 2) Suppose that each (L i N i ) is strong for all i 2 I: For any C ~ 2 B(C =(C i ) 2 Q N i ), Ci 2N i (i 2 I): Let D =( Ci ) 2 Q N i and take any(d i ) 2 D, ~ then there is (ci ) 2 C ~ with (di ) / (c i ) because 0 / 0 in any frame. Thus D ~ C ~ and therefore C ~ 2N: 3) Suppose that each (L i N i ) is almost uniform. By 2), it remains to show that / in (L N ) interpolates, which is immediate from the fact that (x i ) / (y i ) in (L N )ifor all i 2 I, x i /y i : Generalizing strong nearness frames, weintroduce a concept of weakly strong nearness frames. Definition 2.8. A nearness frame (L N ) is said to be weakly strong if for any C 2N C = fy 2 Ljy x for some x 2 Cg belongs to N. Since x/y in a nearness frame (L N ) implies x y, for any C 2N, C C C hence every strong nearness frame is weakly strong. For a regular frame L, let FCov(L) denote the nearness on L generated by thesetofnitecovers of L then for any x y 2 L, x y i x/y in (L F Cov(L)). Thus (L F Cov(L)) is weakly strong i it is strong and hence a uniform frame, because a totally bounded frame is uniform i it is strong. Let WSNFrm denote the subcategory of NFrm determined by weakly strong nearness frames. Theorem 2.9. The subcategory WSNFrm is closed under the G- nal extremal epi sinks in NFrm and therefore closed under colimits in WSNFrm. Proof. Let (h i : (L i N i ) ;! (L N )) be a G-nal extremal epi sink such that each (L i N i )(i 2 I) is weakly strong. By Lemma 2.4, B = f j2j h j (A j )jj is a nite subset of I and A j 2 N j g generates N : Take any C = j2j h j (A j ) 2 B then A j 2 N j for each j 2 J: Since each h j (j 2 J) is a frame homomorphism, h j (A j ) h j (A j ) hence j2j h j (A j ) j2j h j (A j ) C, because is a sublattice of L L: Thus C 2 N, so that (L N ) is weakly strong. The second half follows from the proof of Theorem 2.5. Corollary The category WSNFrm is cocomplete.

8 854 Jyung Ryun Seo and Chang Koo Lee Proposition The subcategory WSNFrm is productive in NFrm. in WSNFrm, take any basic Proof. For any family ; (L i N i ) cover C ~ in the product (L N ) of the family, i.e., C ~ = [ Ci ~, where (C i ) 2 Q N i : Since each (L i N i )(i 2 I) isweakly strong, C i 2N i (i 2 I) hence ~ D = [ ~ Ci 2N: Since (x i ) (y i ) i x i y i for all i 2 I, ~ D ~ C and therefore ~C 2N. Thus (L N )isweakly strong. References [1] J. Adamek, H. Herrlich, and G. E. Strecker, Abstract and Concrete Categories, John Wiely and Sons Inc., New York, [2] B. Banaschewski, Completion in point-free topology, Math. Monographs of the Univ. of Cape Town (to appear). [3] B. Banaschewski, S. S. Hong and A. Pultr, The Completions of nearness frames, Quaest. Math. (to appear). [4] B. Banaschewski and A. Pultr, Cauchy points of uniform and nearness frames, Quaest. Math. 19 (1996), 101{127. [5] J. Benabou, Treillis locaux et paratopologies, seminaire Ehresmann (Topologie et Geometrie Dierentielle), lre annee, expose 2, [6] H. Herrlich, A concept of nearness, General Topol. Appl. 4 (1974), 191{212. [7], Topological structures I, Math. Centre Tracts 52 (1974), 59{122. [8] S. S. Hong and Y. K. Kim, Cauchy Completions of nearness frames, Appl. Categorical Structures 3 (1995), 371{377. [9] P. T. Johnstone, Stone Spaces, Cambridge Univ. Press, Cambridge, [10] Y. K. Kim, A Study on Cauchy Completions, Ph. D. Thesis, Sogang University, [11] S. Vickers, Topology via Logic, Cambridge Tracts in Theor. Comp. Sci., vol. 5, Cambridge Univ. Press, Cambridge, Jyung Ryun Seo Department of Mathematics Hanyang University Seoul , Korea Chang Koo Lee Hanyang Women's College Seoul , Korea