On the smoothness and the totally strong properties for nearness frames
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1 Int. Sc. Technol. J. Namba Vol 1, Issue 1, 2013 On the smoothness and the totally strong propertes for nearness frames Martn. M. Mugoch Department of Mathematcs, Unversty of Namba 340 Mandume Ndemufayo Avenue, Prvate Bag 13301, Poneerspark, Wndhoek, Namba Receved: 10th March, Accepted: 4th Aprl, Abstract In ths paper we brefly explore the propertes of smooth and totally strong nearness frames. It turns out that there s a relatonshp between them, n partcular, totally strong nearness frames are smooth. We also show that the category of smooth nearness frames s coproductve, and that, as s the case wth that of the totally strong ones, t s closed under completons. Keywords: nearness frame, nterpolatve, almost unform, strong, smooth, totally strong. Mathematcs Subject Classfcaton (2010): Prmary 06D22, Secondary 54E17, 54F50 ISTJN 2013; 1(1): Prelmnares Recall that a frame s a complete lattce L n whch the dstrbutve law a S = {a x x S} Correspondng author - E-mal: mmugoch@unam.na 35
2 holds for all a L and S L. We denote the top element and the bottom element of L by 1 L and 0 L respectvely, droppng the subscrpt L f t s clear from the context. For a general theory of frames we refer to [7], and for nearness frames we refer to [1], [2] and [3]. In ths ntroductory secton we collect a few facts that wll be relevant for our dscusson, and fx notaton. An element a of a frame L s sad to be rather below an element b, wrtten a b, f there s an element s such that a s = 0 and s b = 1. Further, a s sad to be completely below b, wrtten a b, f there exsts a doubly ndexed sequence (x nk ), known as a scale between a and b, wth n = 0,1,... and k = 0,1,...,2 n such that a = x n0, x nk x nk+1, x n2 n = b, and x nk = x n+12k, for all n = 0,1,... and k = 0,1,...,2 n. Call L regular f a = {x L x a} for each a L, and completely regular f a = {x L x a} for each a L. The pseudocomplement of an element a s the element a = {x L x a = 0}. We say a s complemented f a a = 1. A frame homomorphsm s a map between frames whch preserves fnte meets, ncludng the top element, and arbtrary jons, ncludng the bottom element. A homomorphsm s called dense f t maps only the bottom element to the bottom element. Assocated wth a homomorphsm h : L M s ts rght adjont h : M L gven by h (a) = {x L h(x) a}. A cover of a frame s a subset wth jon equal to the top. The set of all covers of L s denoted by CovL. For covers C and D of L, C s sad to refne D, wrtten C D, f for every c C there exsts d D such that c d. For any A CovL and x L, the element Ax of L s defned by Ax = {a A a x 0}. If A,B CovL, then AB s the cover gven by AB = {Ab b B}. A cover A star-refnes a cover B, wrtten A B, f AA B. For any N CovL, the relaton N (or smply ) on L s defned by x y f Cx y for some C N, 36
3 and N s sad to be admssble f a = {x L x a} for each a L. If a b, we say a s unformly below b. A nearness on L s an admssble flter N n (CovL, ). A nearness frame s a par (L,N) where N s a nearness on L. A frame has a nearness f and only f t s regular. Therefore all frames consdered here are assumed to be regular. We shall frequently abuse notaton and denote a nearness frame by ts underlyng frame. If we have not named a nearness n queston when talkng about a nearness frame L, we shall, at tmes, wrte NL for the nearness. Gven a nearness frame L, the covers n NL are called unform covers of L. If L s a nearness frame and C CovL, Č s the cover defned by Č = {x L x c for some c C}. A nearness frame L s sad to be: (1) strong f for every unform cover C, Č s also a unform cover. (2) fne f NL = CovL. Let L and M be nearness frames. A homomorphsm h: L M between the respectve underlyng frames s sad to be: (1) unform f h[c] NM for each C NL. (2) a surjecton, or quotent map, f t s onto and NM = {h[c] C NL}. In ths case we may refer to M as a quotent of L. (3) a strct surjecton f t s a dense surjecton and the unform covers h [C],C NM, generate NL. We shall need the followng results whch appear n [1]. Lemma 1.1. () If a b n L, then h(a) h(b) n M. () If h s a dense surjecton, then a b n L mples h h(a) b. () If h s a strct surjecton, then x y n M ff h (x) h (y) n L. 37
4 (v) If h s a strct surjecton, then for any a L and any x M, we have a h (x) n L ff h(a) x n M. The category of nearness frames and unform homomorphsms s denoted by NFrm. Its subcategory consstng of strong nearness frames wll here be denoted by StrNFrm. A nearness frame L s complete f any strct surjecton M L s an somorphsm, and a completon of L s any strct surjecton M L wth complete M. Fne nearness frames are complete. Any nearness frame L has a completon γ L : CL L, whch s unque up to somorphsm. In [4] a nearness frame s called quotent-fne f t s a quotent of a fne nearness frame. It s then shown that a nearness frame s quotent-fne f and only f ts completon s fne. In [5] a nearness frame L s called nterpolatve or has the nterpolaton property f, for every a, b L, a b mples a c b for some c L. An almost unform nearness frame s one whch s strong and nterpolatve. We denote the category of almost unform nearness frames by AuNFrm. We also recall from [9] how coproducts are constructed n NFrm. Gven a system {L } I of nearness frames, let L be the coproduct of the underlyng frames, and ι : L L the coproduct njectons. The frame L s generated by elements of the form a = ι (a ), where the a L are such that only fntely many of them are not equal to 1. The results n the followng lemma appear n [9]. Lemma 1.2. The elements a have the followng propertes: () a = 0 ff a = 0 for some I. 38
5 () 0 a b ff for all I, a b. () 0 a b n L ff for all I, 0 a b n L. Consder the covers C = { c c C } where each C s a unform cover of L, and only fntely many of the C are non-trval, that s, unequal to {1}. The collecton of all such C generates a nearness on L ; and the resultng nearness frame together wth the coproduct njectons ι j : L j L consttutes a coproduct n the category NFrm. 2 Smooth nearness frames Smooth nearness frames were ntroduced n [2] as an ad-hoc means to studyng completon n nearness frames. In ths secton we nvestgate some propertes of these nearness frames, culmnatng n showng that the smooth property s not changed under completons. Call a nearness frame (L, µ) smooth f for each unform cover C, the set C s = {x L x y for some y C} s also a unform cover. Wrte SmNFrm for the category of smooth nearness frames, whch, n consequence, s a full subcategory of NFrm Remark 2.1. Note here that snce x y mples x y, we have Č C s, so that, as observed n [2], Every strong nearness frame s smooth. However, a smooth nearness frame need not be strong. Thus, StrNFrm SmNFrm. It s clear that quotent-fne (and, hence, fne) nearness frames are smooth. 39
6 Let L be a nearness frame. If A s a unform cover, then the set A = {x L x A} s also a unform cover (snce A refnes A ). The followng characterzaton of smooth nearness frames follows naturally. Lemma 2.2. A nearness frame L s smooth ff for each A NL, there exsts B NL such that B refnes A. Proof. ( ) Suppose L s smooth. Let A be a unform cover. Then A s = {x L x a, some a A} s also a unform cover. By the hypothess, (A s ) s a unform cover (as observed above) of the desred knd refnng A. ( ) Conversely, assume that the condton holds. For A NL, let B refne A for some B NL. Then B A s, so that B A s. Consequently, A s NL, so that L s smooth. Example 2.3. It s worth notng here that the cover A, as ntroduced above, s not necessarly the same as the set A r = {x A x = x } of all regular elements n A. As an example, let L be a nearness frame where the underlyng frame s compact and non-boolean. (Trvally, every nearness on a Boolean frame s smooth). Let A be a unform cover of L consstng only of regular elements and let x L be a non-regular element such that x A. Put B = A {x}. Then B NL and B = A {x }, but B r = {y B y = y } = A. In order to show that SmNFrm s coproductve, we shall need the followng lemma. 40
7 Lemma 2.4. Let L be the coproduct of a famly {L } I of frames. Then for each element a L, ( a ) = (a ). Proof. Let ι : L L be the th coproduct njecton. We frst show that ( ) ( a ) = ι (a ). By defnton, for each ndex k, and for any x L k, ι k (x) = b, where b k = x and b = 1 for k. Now f c s any element of L such that ( a ) ( c ) = 0, then (a c ) = 0, so that a k c k = 0 for some ndex k. Ths mples c k a k. Consequently c ι k (a k) ι (a ). Snce the elements x generate the frame L, t follows that f any element of L does not meet a, then t s below ι (a ). Therefore ( a ) ι (a ). But, by applyng the nfnte dstrbutve law, ( ) ( a ) ι (a ) = 0. Therefore ι (a ) s the largest element of L dsjont from a. Thus, ( ) holds. Second, we apply ( ) and the fact that for each ndex k, (ι k (x)) = ι k (x ), to obtan ( ) ( a ) = ι (a ) = (ι (a )) = establshng the desred result. ι (a ) = (a ), 41
8 Proposton 2.5. SmNFrm s coproductve n NFrm. Proof. Let {L } I be a famly of smooth nearness frames. We show that ther coproduct L s also smooth. To see that, let A be a unform cover n the coproduct, and let A be a refnement of A (where each A NL ). Let A 1,...,A m covers A s. be the nontrval covers among the We construct a unform cover of the form B (n the coproduct nearness) refnng A as follows: For each { 1,..., m }, let B Then, makng use of the above lemma, NL refne A. We let B B = ( B ) = B refnes A whch refnes A. Hence the desred result follows. = {1} for the other s. In [2] t s shown that any dense surjecton h : L M wth L smooth s n fact a strct surjecton, and consequently any weak completon h : L M, where L s smooth, becomes a completon. Here we show the followng result. Proposton 2.6. If h : L M s a dense surjecton, then L s smooth ff M s smooth. Proof. ( ) Suppose L s smooth. Let C NM. To show that M s smooth, we need D NM such that D C. Snce h [C] NL, there exsts B NL such that B h [C], snce L s smooth. Snce h preserves pseudocomplements, beng a dense onto map, we have h[b ] = h[b]. Thus, h[b] s a unform cover of M such that h[b] refnes C, and therefore M s smooth. ( ) Suppose M s smooth. Let A be a unform cover of L. Snce h s a strct surjecton, there s a unform cover B of M such that h [B] A. Snce M s smooth, by the hypothess, B s a unform cover of M. Therefore h [B ] s a unform cover of L snce h s a strct surjecton. Snce h s a dense onto homomorphsm, h commutes wth pseudocomplementaton; so that h (b ) = h (b) 42
9 for each b B, and hence h [B ] = h [B]. But now h [B] refnes A ; therefore A s also a unform cover, and hence L s smooth. The followng corollary s evdent from the above result, snce the completon map s a strct surjecton. Corollary 2.7. A nearness frame s smooth ff ts completon s smooth. 3 Totally strong nearness frames By mposng a stronger refnement orderng on unform covers, n partcular, one whch uses scales n the manner n whch the completely below relaton s defned, we ntroduce, n ths secton, a type of nearness frames called the totally strong ones and establsh that ther category, namely TStrNFrm, s closed under completons, and that the nclusons AuNFrm TStrNFrm StrNFrm hold. Defnton 3.1. Let L be a nearness frame, and A, B NL. Wrte A s B f there s an nterpolatng sequence of unform covers (C nk ) between A and B, where C 00 = A, C 01 = B, C nk = C n+12k, and C nk C nk+1 for all n = 0,1,... and k = 0,1,...,2 n. In ths case we say A scale refnes B. We call a nearness frame totally strong f every unform cover s scale refned by a unform cover. Clearly, f A s B, then A B. Consequently, every totally strong nearness frame s strong. We wrte TStrNFrm for the category of totally strong nearness frames. Thus, TStrNFrm StrNFrm, and, snce StrNFrm SmNFrm, every totally strong nearness frame s smooth. In order to show that every almost unform nearness frame s totally strong, we need the followng result whch shows that nterpolaton n the underlyng frame L s transferred to ts nearness NL. Lemma 3.2. Suppose L s an nterpolatve nearness frame, and suppose A, B NL wth A B. Then there exsts C NL such that A C B. 43
10 Proof. Let A, B NL be such that A B. Then for each a A, there exsts b a B such that a b a. Snce L s nterpolatve, there exsts c a L such that a c a b a. Form the set C = {c a L a A}. Then C s a unform cover, snce A refnes t. Furthermore A C B by the way C s constructed. Proposton 3.3. If L s almost unform, then t s totally strong. Proof. Let B NL. Snce L s strong, there exsts A NL such that A B. By Lemma 3.2, nterpolates n NL, snce L s nterpolatve. Therefore A s B, snce, by the axom of countable dependent choce, a scale of unform covers wtnessng ths can be constructed n the same manner as done n [8, Lemma 1.5]. As an observaton from the above two results, t should be evdent that f L s a strong nearness frame wth the property that whenever A B n NL, there exsts C NL such that A C B, then L s totally strong. In our next set of results we am to show that the category TStrNFrm s closed under completons. Our proof wll be facltated by notng the followng: f h : L M s a unform frame homomorphsm and A scale refnes B n L, then h[a] scale refnes h[b] n M, for f (C nk ) s a scale of unform covers of L wtnessng A s B, then clearly (h[c nk ]) s a scale of unform covers of M wtnessng h[a] s h[b]. On the other hand, f h s a strct surjecton and U scale refnes V n M, then h [U] scale refnes h [V ] n L, for f (W nk ) s a scale of unform covers of M wtnessng U s V, then, by the strctness of h, (h [W nk ]) s a scale of unform covers of L wtnessng h [U] s h [V ]. Lemma 3.4. Let h : L M be a strct surjecton. Then L s totally strong ff M s totally strong. Proof. ( ) Suppose L s totally strong and let U be a unform cover of M. Then, by strctness, h [U] s a unform cover of L. Snce L s totally strong, there s a unform cover A of L that 44
11 scale refnes h [U]. By what we have observed above, h[a] s a unform cover of M scale refnng h[h [U]] = U. Therefore M s totally strong. ( ) Conversely, suppose M s totally strong and let A be a unform cover of L. By strctness, there s a unform cover U of M such that h [U] A. Snce M s totally strong, there s a unform cover V of M whch scale refnes U. Then h [V ] s a unform cover of L scale refnng h [U], and hence scale refnng A. Therefore L s totally strong. Snce completon maps are strct surjectons, we deduce the followng result. Corollary 3.5. A nearness frame s totally strong f and only f ts completon s totally strong. Acknowledgement: Parts of ths paper come from the PhD thess [6] wrtten under the supervson of Prof T. Dube of the Unversty of South Afrca. References [1] Banaschewsk B., Unform completon n pontfree topology, Topologcal and Algebrac Structures n Fuzzy Sets, Trends n Logc 20 (2003), [2] Banaschewsk B., Hong S.S. and Pultr A., On the completon of nearness frames, Quaest. Math. 28 (1998), [3] Banaschewsk B. and Pultr A., Cauchy ponts of unform and nearness frames, Quaest. Math. 19 (1996), [4] Dube T. and Mugoch M.M., Thoughts on quotent-fne nearness frames, Appl. Categor. Struct. 19 (2011), [5] Dube T. and Mugoch M.M., A note on almost unform nearness frames. Quaestones Mathematcae, 34 2 (2011), [6] Mugoch M.M., Contrbutons to the theory of nearness n pontfree topology, PhD thess, Unv. South Afrca (2009). 45
12 [7] Johnstone P.T., Stone Spaces, Cambrdge Unv. Press, Cambrdge, [8] Pultr A., Pontless unformtes I: Complete regularty. Comment. Math. Unv. Carolnae 25(1) (1984), [9] Pultr A. and Tozz A., Completon and coproducts of nearness frames, Symposum on Categorcal Topology (Rondebosch, 1994), , Unv. Cape Town, Rondebosch,
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