We will prove that the class of Boolean algebras with quasi-semi-homomorphism is a category, denoted byBoQS

(1)

Vol. 19 (2018), No. 1, pp. 171–189 DOI: 10.18514/MMN.2018.1803

QUASI-SEMI-HOMOMORPHISMS AND GENERALIZED PROXIMITY RELATIONS BETWEEN BOOLEAN ALGEBRAS

SERGIO A. CELANI Received 13 October, 2015

Abstract. In this paper we shall define the notion of quasi-semi-homomorphisms between Boolean algebras, as a generalization of the quasi-modal operators introduced in [3], of the notion of meet-homomorphism studied in [12] and [11], and the notion of precontact or proximity relation defined in [8]. We will prove that the class of Boolean algebras with quasi-semi-homomorphism is a category, denoted byBoQS. We shall prove that this category is equivalent to the category StQBof Stone spaces where the morphisms are binary relations, called quasi-Boolean relations, satisfying additional conditions. This duality extends the duality for meet-homomorphism given by P. R. Halmos in [12] and the duality for quasi-modal operators proved in [3].

2000Mathematics Subject Classification: 06E25; 06E15; 03G05; 54E0

Keywords: proximity relations, precontact algebras, quasi-modal operator, quasi-semi-homomorphisms, Stone spaces, quasi-Boolean relations

1. INTRODUCTION

Recall that amodal algebrais a Boolean algebra Awith an operatorWA!A such that 1D1 , and .a^b/Da^b, for all a; b2A. It is well known that the variety of modal algebras is the algebraic semantic of normal modal logics [10,16]. Modal algebras are dual objects of descriptive general frames, also called modal spaces, i.e., Stone spaces with a relation verifying certain conditions (see [10], and [16]). P. R. Halmos define in [12] the notion of meet-homomorphism (or hemihomomorphism) between Boolean algebras. Recall that a meet-homomorphism between two Boolean algebrasAandB, is a functionhWA!Bsuch thath.1/D1, andh.a^b/Dh.a/^h.b/, for alla; b2A. IfADB, thenhis a modal operator [10,16]. LetXandY be the Stone spaces ofAandB, respectively. As it follows from [12] and [11], a meet-homomorphism hWA!B is dually characterized by means of a relation RY X such that R.y/ is a closed subset of X, for each y2Y, andhR.U /D fy2Y WR.y/Ugis a clopen subset ofY, for each clopenU X:

These relations are called Boolean relations in [12], or Boolean correspondences in [11] (see also [16]).

c 2018 Miskolc University Press

(2)

In [3], the notions of quasi-modal operator and quasi-modal algebra were introduced as a generalization of the notion of modal operator and modal algebra, respectively. A quasi-modal operator in a Boolean algebraA is a mapthat sends each elementa2Ato an idealaofA, and satisfies analogous conditions with the modal operatorof modal algebras. A quasi-modal algebra is a pairhA; iwhere Ais a Boolean algebra andis a quasi-modal operator. We note that a quasi-modal operator is not an operation, but has many similar properties to modal operators.

In this paper we shall introduce maps between a Boolean algebra Aand the set of all ideals of another Boolean algebraB satisfying analogous conditions with the meet-homomorphism between Boolean algebras [12]. We call these maps quasi- semi-homomorphisms. One of the main objectives of this paper is to study this class of maps, and their topological representation.

As we will explain below, the quasi-modal operators are closely connected with the proximity or precontact relations defined between Boolean algebras. We recall that a proximity relation defined on a setXis a binary relationıP.X /P.X /satisfying certains conditions (see Definition2). IfU; V 2P.X /, then the intuitive meaning of a proximity relationı is that UıV holds, whenU is close to V in some sense. A proximity or precontact space, also called a nearness space, is a pairhX; ıi, where X is a set and ıis a proximity relation. SinceP.X /is a Boolean algebra, we can introduced an abstract definition of proximity relation in the class of Boolean algebras (see [15] and [4]). In the literature, there exist many classes of Boolean algebras endowed with some type of proximity relations. As examples, we can mention the Boolean contact algebras defined in [9], or the Boolean connection algebras defined in [17]. For other versions of Boolean algebras endowed relations see [5], [8], [7], [19], and [18]. In [8] the notions ofproximity relationon a Boolean algebra and the proximity Boolean algebraswere defined as an abstract version of proximity spaces [15]. This class of structures is the most general class of Boolean algebras endowed with a proximity relation. We note that the notion of proximity Boolean algebras is equivalent to the notion of precontact algebras [8].

There exists a strong connection between proximity relations defined in a Boolean algebra and quasi-modal operator. Given a proximity relationıin a Boolean algebra A, we can prove that the set_ıbD fa2AW.a;:b/…ıgis an ideal ofA. So, we have a map_ı that send elements to ideals of the algebra A. As we shall see, this map is a quasi-modal operator. Conversely, if we have a quasi-modal operatordefined in a Boolean algebraA, then the relationaıbdefined bya…:b, is a proximity relation on A (for the details see Theorem1). Thus, we have that the notions of proximity relation and quasi-modal operator are interdefinable. Moreover, since the notion of quasi-semi-homomorphism is a generalization of the notion of quasi-modal operator, and this last is equivalent to the notion of proximity relation, we will get that it is possible to introduce a generalization of the notion of proximity relation.

(3)

The paper is organized as follows. In Section 2 we start recalling some basic defin- itions and results on Stone duality for Boolean algebras. In Section 3 we shall introduce the notion of quasi-semi-homomorphism and the notion of generalized proximity relation. Also, we shall prove that the notions of quasi-semi-homomorphism and generalized proximity relation are equivalent, and as consequence of this fact, we have that the notions of quasi-modal operator and proximity relation are equivalent. This fact has strong consequences, because it puts the proximity relations very close to the modal operators. We shall see that the class of Boolean algebras with the quasi-semi-homomorphism form a category denoted byBoQS. In Section 4, we shall introduce the notion of generalized quasi-Boolean relation between Stone spaces, and we shall prove some propierties. We shall prove that the class of Stone spaces with the generalized quasi-Boolean relations form a category, simbolized by StQB. In Section 5 we shall prove that the categories StQBandBoQS are dually equivalent. As an application of this duality we will prove a generalization of the result that assert that the Boolean homomorphisms are the minimal elements in the set of all join-homomorphisms between two Boolean algebras (see [11]). In this last section we prove that the minimal elements in the set of all quasi-Boolean relations defined between two Stone spaces is a Boolean relation.

2. PRELIMINARIES

We assume that the reader is familiar with basic concepts of Boolean algebras and topological duality (see [1] or [13]).

We recall that a subset of a topological space X is clopen if it is both closed and open, and thatX iszero-dimensionalif the set of clopen subsets ofX forms a basis for the topology. We shall denote byO.X /(C.X /) the set of all open subsets (closed subsets) of X. The closure of a subsetZ is denoted by cl.Z/. We shall denote by Clo.X /the set of all clopen subsets ofX. Clearly the notions of Hausdorff andT0coincide in the realm of zero-dimensional spaces. AStone spaceX iszero- dimensional, compact and Hausdorff topological space. We note that a Stone space istotally disconnected, i.e., given distinct pointsx; y2X, there isU 2Clo.X /ofX such thatx2U andy…U. IfX is a Stone space, then Clo.X /is a Boolean algebra under the set theoretical operations.

IfAD hA;_;^;:; 0; 1i is a Boolean algebra, by Ul.A/ we shall denote the set of all ultrafilters (or proper maximal filters) ofAwhile by Id.A/and Fi.A/we shall denote the families of all ideals and filters ofA, respectively.

LetX be a Stone space. The map"_XWX !Ul.Clo.X //given by

"X.x/D fU 2Clo.X /Wx2Ug

is a bijective and continuous function. LetAbe a Boolean algebra and let ˇAWA!P.Ul.A//

(4)

the Stone map defined byˇA.a/D fP 2Ul.A/Wa2Pg. Sometimes we will write ˇ instead of ˇA. With each Boolean algebra A we can associate a Stone space hUl.A/; Ai whose points are the elements of Ul.A/ and the topology A is de- termined by the clopen basis ˇ ŒAD fˇA.a/Wa2Ag. If misunderstanding is ex- luded, we write Ul.A/ instead of hUl.A/; _Ai. Thus, if X is a Stone space, then X ŠUl.Clo.X //, and ifAis a Boolean algebra, thenAŠClo.Ul.A//.

If Ais a Boolean algebra, then there exists a duality between ideals (filters) of Aand open (closed) sets of Ul.A/. More precisely, forI 2Id.A/andF 2Fi.A/.

The value of the function'AŒI D fP 2Ul.A/WI\P ¤¿gis an open of Ul.A/, and thus'Ais an one-to-one mapping between Id.A/and the set ofO.Ul.A//of all open subset of Ul.A/. The function Adefined by AŒF D fP 2Ul.A/WF Pg, is a one-to-one mapping between Fi.A/and the setC.Ul.A//of all closed subset of Ul.A/. We note that'AŒI DS

fˇ.a/Wa2Ig. If we denote byZand byYthe meet and the join in the set Id.A/, respectively, then'AŒI1YI2D'AŒI1['AŒI2, and 'AŒI1ZI2D'AŒI1\'AŒI2(see [13] and [16] for further information on Boolean duality).

Let Abe a Boolean algebra. The filter (ideal) generated by a subset Y A is denoted byF .Y /(I .Y /). IfY D fag, then we writeF .a/DŒa/(I.a/D.a). The set complement of a subsetY Awill be denoted byY^c orA Y.

3. QUASI-SEMI-HOMOMORPHISMS

In this section we introduce the main notion of this paper. We define the notion of quasi-semi-homomorphim as a generalization of the notion of quasi-modal operator [2,3] and the notion of semi-homomorphism between Boolean algebras [11,12].

Definition 1. LetAandBbe two Boolean algebras. Aquasi-semi-homomorphism is a function W A!Id.B/ such that it verifies the following conditions for all a; b2AW

Q1 .a^b/Da\b, Q2 1DB.

In the followingQS ŒA; Bstands for the set of all quasi-semi-homomorphism defined betweenAandB. If1; 22QS ŒA; Bwe define12 by 1.a/2.a/, for alla2A. This gives an order relation inQS ŒA; B. We note that whenADB, the elements ofQS ŒA; ADQS ŒAare called quasi-modal operators in [3]. A pair hA; i, where2QS ŒAis called aquasi-modal algebra.

If2QS ŒA; B, thenis monotonic, because ifab, thenaDa^b, and so aD.a^b/Da\b, i.e.,ab.

Example 1. Let A be a Boolean algebra. The map I_AW A!Id.A/ given by IA.a/D.a, for eacha2A, is clearly a quasi-semi-homomorphism.

Example 2. Let A and B two Boolean algebras. We recall first that a meet- hemimorphisms ormeet-homomorphism[11] [12], is a functionhWA!Bsuch that

(5)

h.1/D1, andh.a^b/Dh.a/^h.b/, for alla; b2A. The functionhcan be exten- ded to a map_hWA!Id.B/of the following form. Put_h.a/D.h.a/, for each a2A. It is clear thath verifies the equalitiesh.a^b/Dh.a/\h.b/and _h.1/DB. Thus,_his a quasi-semi-homomorphism. An element2QS ŒA; B

is called aprincipal quasi-semi-homomorphisms if a is principal ideal, for each a2A. In other words, for eacha2A, there exists b2B suh thataD.b. it is clear that ifis principal, then the maphWA!Bdefined byh.a/DbiffaD.b

is a meet-hemimorphisms. Thus, the class of principal quasi-semi-homomorphisms is equivalent to the class of meet-homomorphisms.

Recall that whenADBandWA!Ais a meet-homomorphism, the pairhA;i is called a modal algebra [16]. So, the class of modal algebras can be identified with the class of pairs hA; i whereA is a Boolean algebra andis a principal quasi- semi-homomorphism.

The following example is fundamental in the representation theory of quasi-semi- homomorphisms.

Example 3. Let X and Y be two set. Let R be a relation between X and Y. Define a functionNRWP.Y /!Id.P.X //;asNR.U /D.R.U /, whereR.U /D fx2X WR.x/Ug, withU 2P.Y /. Then it is easy to see that

NR2QS ŒP.Y /;P.X /.

LetA andB be two Boolean algebras. For each 2QS ŒA; B, we define the dual quasi-semi-homomorphism r WA!Fi.B/ by raD ::a, where :xD f:yWy2xg. We note thatc2 r.a_b/D ::.a_b/D :.:a^ :b/iff:c2 .:a^ :b/D:a\:biff:c2:aand:c2:biffc2 raandc2 rbiff c2 ra\ rb. Thus the mapr verifies the following conditions:

Q3 r.a_b/D ra\ rb, Q4 r0DB.

Now we introduce a notion that generalizes the notion of proximity relation (also called precontact relation) defined in a Boolean algebra [8] [7] [14].

Definition 2. LetAandBbe two Boolean algebras. Ageneralized precontact or generalized proximity relationbetweenAandBis a relationıABsuch that

P1 Ifaıb, thena¤0andb¤0.

P2 aı.b_c/iffaıboraıc.

P3 .a_b/ıciffaıcorbıc.

WhenADB, a generalized precontact relationıis called aproximity or precontact relation, and the pairhA; ıiis called a proximity orprecontact algebra[6–8]. An important example of proximity relations are the proximity spaces. There are many other notions of proximity, and we suggest the reader consults the fundamental text by Naimpally and Warrack [15] for more examples, or the paper [18].

(6)

Theorem 1. LetAandBbe two Boolean algebras. There exists a bijective correspondence between quasi-semi-homomorphisms betweenAandBand the generalized proximity relations defined betweenAandB.

Proof. LetAandBbe two Boolean algebras. IfıABis a generalized proximity relation, then we prove that the subset ofA

_ıbD fa2AW.a;:b/…ıg;

is an ideal ofA. Let b2B. As .0;:b/…ı, we have that 02ıb. Let a1; a2 2 A. Suppose thata1a2 anda22_ıb. Then .a2;:b/…ı. Asa2Da1_a2, by condition P3 of Definition2we have that.a2;:b/D.a1_a2;:b/…ıiff.a1;:b/…ı and.a2;:b/…ı. Thus,.a1;:b/…ı, i.e., a12_ıb. Suppose thata12_ıb and a22_ıb. Then,.a1;:b/…ıand.a2;:b/…ı. Again, by condition P3 of Definition 2we have that.a1_a2;:b/…ı, i.e.,a1_a22_ıb. Thus,_ıb2Id.A/, for each b2B. Then the map_ıWB!Id.A/is well defined.

Let b1; b22B anda2 A. Then by condition P2 of Definition 2 we have the following equivalences:

a2ı.b1^b2/ iff .a;:.b1^b2//D.a;:b1_ :b2/…ı iff .a;:b1/…ıand.a;:b2/…ı

iff a2ıb1\ıb2:

By condition P1 of Definition2we have that.b;:1/D.b; 0/…ı, for allb2B. Thus, 12_ı.b/, for allb2B.

Conversely. LetWA!Id.B/be a quasi-semi-homomorphism. Define the relation

ıD f.a; b/2ABWa…:bg:

Let.a; b/2ı. IfaD0, then0…:b, which is a contradiction because:bis an ideal. IfbD0, thena…:0D1DA, which is a contradiction. Thus,a¤0 andb¤0.

Leta; b2A, andc2B. Taking into account that:c is and ideal ofB, we get the following equivalences:.a_b; c/2ıiffa_b…:ciffa…:c orb…:c iff.a; c/2ıor.b; c/2ı.

Leta2Aandb; c2B. Then.a; b_c/2ıiff

a…:.b_c/D.:b^ :c/D:b\:c

iff a…:b or a…:c iff .a; b/2ı or.a; c/2ı. Thus, ıis a generalized

proximity relation betweenAandB.

By Theorem1 we have that the notions of proximity relations and quasi-modal operators are interdefinable.

Definition 3. Let2QS ŒA; B. For eachC Aand for eachDBdefine (1) C DI.S

c2C

c/,

(7)

(2) rC DF .S

c2Crc/,

(3) ¹.D/D fa2AWa\D¤¿g, (4) r ¹.D/D fa2AW raDg.

(5) IfDDŒa/ ;we write ¹.a/instead of ¹.Œa//.

In the following lemma we summarize some properties well known in the theory of Boolean algebras with proximity relations (see [8,14,19]). For completeness we will give some proofs.

Lemma 1. Let2QS ŒA; B.

(1) Then ¹.F /DS

a2F ¹.a/2Fi.A/for eachF 2Fi.B/. Moreover, this union is directed.

(2) IfP 2Ul.B/, thenr ¹.P /^c2Id.A/.

(3) IDS

a2Iafor eachI2Id.A/. Moreover, this union is directed.

(4) .I1\I2/D.I1/\.I2/for allI1; I22Id.A/.

Proof. (1) LetF 2Fi.B/. It is easy to see ¹.F /2Fi.A/. Leta2A. Then a\F ¤¿ iff 9b2F .b2a/

iff 9b2F .Œb/\a¤¿/

iff 9b2F .a2 ¹.Œb//D ¹.b//

iff 9b2F .a2S

b2F ¹.b//:

In order to see that this union is directed suppose that a; b2F. Then it is easy to see that ¹.a/[ ¹.b/ ¹.a_b/, and asa_b2F, we get that this union is directed.

(2) We prove that r ¹.P /^c 2 Id.A/, when P 2 Ul.B/. Let a b and a 2 r ¹.P /^c. ThenraªP, and asrb ra, becauser is anti-monotonic, we have that b … r ¹.P /^c. Thus r ¹.P /^c is decreasing. Let a; b 2 r ¹.F /^c. Then raªP and rb ªP. Then there exist p1 2 ra P and p2 2 rb P. So, p1_p22 ra_ rb, and asP is prime,p1_p2…P. Then,p1_p22 r ¹.P /^c. It is clear that02 r ¹.P /^c, becauser0DB. Thus,r ¹.P /^c is an ideal ofA.

(3) LetI2Id.A/. We prove thatI.S

c2C

c/DS

a2Ia. It is clear thatS

a2Ia I.S

c2C

c/. Letc2I.S

c2C

c/. Then there existsai2I, and there existsxi 2ai, with1in, such thatcx1_: : :_xn:Sinceai.a1: : :_an/, for1in, thenx1_: : :_xn2.a1_: : :_an). Asa1_: : :_an2I, andc2.a1_: : :_an/, we get thatc2S

a2Ia. Thus the union is directed.

(4). Let I1; I2 2Id.A/. Asis monotonic, .I1\I2/.I1/\.I2/. Let c 2.I1/\.I2/. Then by item (3), there exist a2I1 andb2I2 such thatc 2 a\bD.a^b/. Asa^b2I₁\I₂, we get thatc2.I₁\I₂/.

Let A; B and C be Boolean algebras. Let 12QS ŒA; B and 2 2QS ŒB; C .

We define the composition of 2 with 1. Recall that for each subset D of B,

(8)

we can consider an ideal2.D/DW

f2bWb2Dg. Then, as for eacha2A, we consider the ideal2Œ1.a/2Id.C /. Then, define the composition of2with1, in symbols2ı1, as

.2ı1/.a/D2Œ1.a/ ;

for eacha2A. We need to prove that2ı12QS ŒA; C . In the following result we use the quasi-semi-homomorphism defined in Example1.

Lemma 2. LetA,BandC be Boolean algebras. Let12QS ŒA; Band22 QS ŒB; C . Then:

(1) 2ı12QS ŒA; C .

(2) 1ıIAD1andIBı2D2. Proof. (1) By (4) of Lemma1we get that

.2ı1/.a^b/ D 2Œ1.a^b/ D 2Œ1.a/\1.b/

D 2Œ1a\2Œ1b D .2ı1/.a/\.2ı1/.b/:

Moreover,.2ı1/.1/D2Œ11D2ŒBDA. Thus,2ı1is a quasi-semi- homomorphism.

(2) Leta2A. Then.1ıIA/.a/D1ŒIAaD1Œ.aD1a. The proof of the

identityIBı2D2is similar.

Thus we can conclude that we have a category, denoted by BoQS, whose objects are Boolean algebras and whose morphism are quasi-semi-homomorphisms. In the next section we will prove that the categoryBoQSis dually equivalent to a category whose objects are Stone spaces, and whose morphism are a particular class of binary relations between Stone spaces.

In the following result we will characterize the isomorphisms (or iso-arrow) in the categoryBoQS. This result will be needed later.

Lemma 3. Let A andB be Boolean algebras and2QS ŒA; B. Then the following conditions are equivalent:

(1) is an iso-arrow in the categoryBoQS.

(2) There exists an one to one and onto function hWA!B such thataD .h.a/, for eacha2A.

Proof. .1/).2/Sinceis an iso-arrow in the categoryBoQS, there exists˘ 2 QS ŒB; Asuch thatı˘ DI_B and˘ıDI_A, whereI_A andI_B are the quasi- semi-homomorphisms defined in Example1. Leta2A. Then.˘ı/.a/DIA.a/D .a. As .˘ı/.a/D˘ ŒaDS

f˘ bWb2ag, there exists b2a such that

˘ b D.a. We prove thatb is unique. Suppose that there are b1; b22B such that

˘ b1D˘ b2. Ası˘DIB, we get

.b1D.ı˘ /.b1/D Œ˘ b1D Œ˘ b2D.ı˘ /.b2/D.b2 :

(9)

So,b1Db2. Then for eacha2Athere exists a uniqueb2B such that˘ bD.a.

So, we can consider the functionhWA!Bdefined by:

h.a/Dbiff˘ bD.a ; for eacha2A. We note that

˘.h.a//D.a : (3.1)

Similarly we can prove that there exists a functionkWB!Asuch that k.b/DaiffaD.b ;

for eachb2B. Also, we note that

.k.b//D.b : (3.2)

We prove thatkıhDId_AandhıkDId_B. Leta2A. Then as˘ıDI_Awe get that:

..kıh/.a/D˘ Œ.k.h.a/D˘ Œ.kıh/.a/^.3:2/D ˘ .h.a/

D˘ h.a/^.3:1/D .a

.2/).1/Assume that there exists an one to one and onto functionhWA!Bsuch thataD.h.a/, for eacha2A. So, there exists an one to one and onto function gWB!Asuch that.gıh/.a/Dafor alla2A, and.hıg/.b/Db for allb2B.

Consider the quasi-semi-homomorphism˘ WB!Id.A/defined by˘.b/D.g.b/.

Then we prove thatı˘ DIB and˘ıDIA. We prove that.ı˘ /.b/D.b.

Letb; d 2B such thatd 2.ı˘ /.b/D Œ˘ bDS

fcWc2˘ bD.g.b/g. So, there existsc2Aandd2Bsuch thatcg.b/andd 2cD.h.c/. So,dh.c/, and thusd h.c/h.g.b//Db, i.e., c 2.b. So, .ı˘ /.b/.b. The other inclusion it is left to the reader. Thus, ı˘ DIB. Similarly we can prove that

˘ıDIA. Therefore,is an iso-arrow in the categoryBoQS.

4. GENERALIZED QUASI-BOOLEAN RELATIONS

LetX andY be two topological spaces. LetRXY be a relation. We shall say thatRisupper-semi-continuous(u.s.c) ifR.O/D fx2XWR.x/Ogis an open subset ofXfor every open subsetOofY. We note thatR.O/is open for each open OofY iffr^R.C /D fx2X WR.x/\C ¤¿gis an closed ofXfor each closedC of Y. We shall say thatRispoint-compact (point-closed)ifR.x/is a compact (closed) subset ofY, for eachx2X. Clearly, ifY is a compact space, a relationRXY is point-compact iff it is point-closed.

Lemma 4. Let X and Y be two topological space. Suppose that Y is zero- dimensional space. LetRbe a point-compact relation. Then the following conditions are equivalent:

(1) Ris upper-semi-continuous,

(10)

(2) R.U /2O.X /;for eachU 2Clo.Y /.

Proof. IfRis upper-semi-continuous, thenR.U /2Clo.X /;for eachU 2Clo.Y /, because Clo.Y /O.Y /.

Conversely. Assume thatR.U /2O.X /;for eachU 2Clo.Y /. LetO2O.Y /.

As Y is zero-dimensional, Clo.Y / is a basis, soO DS

fUi2Clo.Y /WUi Og. SinceR is monotonic,

[fR.Ui/WUiOg R.[

fUi2Clo.Y /WUiOg/DR.O/:

We prove the other inclusion. Letx2R.S

fUi 2Clo.Y /WUi Og/, i.e.,R.x/

SfUi 2Clo.Y /WUi Og/. AsR.x/is a compact subset ofY, there exists a finite family fUi OW1ing such that R.x/U1[: : :[UnDU O, i.e., x 2 R.U /. Thus, S

fR.Ui/WUi Og DR.O/. ConsequentlyR.O/is an open subset ofX, because_R.Ui/2Clo.X /, for eachU 2Clo.Y /.

Remark 1. By the previous Lemma, whenX andY are Stone spaces, andR XY, we have that the following conditions are equivalent:

(1) Ris a point-compact relation and_R.O/2O.X /;for eachU 2O.Y /.

(2) Ris a point-closed relation andR.U /2O.X /;for eachU 2Clo.Y /.

Definition 4. LetXandY be two Stone spaces. We shall say that a binary relation RXY is aquasi-Boolean relationif

(1) Ris a point-closed relation,

(2) _R.U /2O.X /, for eachU 2Clo.Y /:

IfR.U /2Clo.X /, for eachU 2Clo.Y /, thenRis called aBoolean relation[12], also called a Boolean correspondence in [11]. It is clear that every Boolean relation is a quasi-Boolean relation.

Remark2. LetX be a Stone space. A pairhX; Ri, whereR is a quasi-Boolean relation defined inX is called aquasi-modal space. The quasi-modal spaces are the dual objects of the quasi-modal algebras (see [3] and [2]). IfRis a Boolean relation, then the pairhX; Ri is called amodal space or descriptive general frame [10,16].

The modal spaces are the dual of the modal algebras, i.e., pairshA;i, whereAis a Boolean algebra andis a modal operator.

Given a Stone spaceX, the map"X WX !Ul.Clo.X //defined by

"X.x/D fU 2Clo.X /Wx2Ug

, is a bijective and continuous function. Thus, for eachP 2Ul.Clo.X //there exists a uniquex2X such that"X.x/DP.

LetX andY be two Stone spaces. LetRXY be a relation. For eachx2X we can consider the set

_R¹."X.x//D fU 2Clo.X2/Wx2R.U /g D fU 2Clo.X2/WR.x/Ug:

(11)

We define the relationRRUl.Clo.X //Ul.Clo.Y //;as follows:

."X.x/; "Y.y//2RRiff_R¹."X.x//"Y.y//:

In the following Lemma we shall give an equivalent condition to the condition (1) of Definition4.

Lemma 5. LetX1 andX2 be two Stone spaces. LetRX1X2 be a relation.

Suppose that R.U / is an open subset of X1, for each U 2Clo.X2/. Then the following conditions are equivalent

(1) R.x/is a closed subset ofX2, for eachx2X1, (2) .x; y/2Riff."1.x/; "2.y//2R_R.

Proof. (1) implies (2). Letx; y2X1. It is clear that if.x; y/2Rthen

."1.x/; "2.y//2R_R. Suppose that y…R.x/. AsR.x/is a closed subset ofX2, there exists U 2Clo.X2/ such thaty…U andR.x/U. So, x2R.U /. Then U 2_R¹."1.x//andU …"2.y/, i.e.,."1.x/; "2.y//…R_R.

(2) implies (1). We prove that cl.R.x//DR.x/. Suppose that there exists y 2 cl.R.x//buty…R.x/. Then."1.x/; "2.y//…R_R, i.e., there existsU 2Clo.X2/ such thatU 2_R¹."1.x//andU …"2.y/. Thenx2R.U /andy…U, i.e.,R.x/

U andy…U. So,y…cl.R.x//, which is a contradiction. Thus, cl.R.x//R.x/,

and consequentlyR.x/is a closed subset ofX2.

LetXandY be two Stone spaces. By Lemma5we have that a relationRX1X2

is a quasi-Boolean relation iffRsatisfies the following conditions:

(1) .x; y/2Riff."1.x/; "2.y//2RR, (2) R.U /2O.X /, for eachU 2Clo.Y /:

We denote byQB ŒX; Y the set of all quasi-Boolean relations between two Stone spacesX andY.

Lemma 6. LetX andY be Stone spaces. Let R2QB ŒX; Y . ThenR ŒC is a closed subset ofY for each closed subsetC ofX.

Proof. LetC be a closed subset ofX. We note thatR ŒC DS

fR.x/Wx2Cg. It suffices to prove that for anyy…RŒC there existsU 2Clo.Y /such thatR ŒC U andy…U. Takey …RŒC . Theny…R.x/for eachx2C. AsRis point-closed, for each x 2C there exists Ux 2Clo.Y / such that R.x/Ux andy …Ux. So, x2_R.Ux/, for eachx2C. Thus,CS

f_R.Ux/Wx2Xg, and asC is compact, there existsx1; : : : xn2C such that

C R.Ux1/[: : :[R.Uxn/R.Ux1[: : :[Uxn/DR.U /;

i.e., R ŒC U. Therefore there existsU 2Clo.Y / such thaty …U andR ŒC

U.

Lemma 7. IfR2QB ŒX; Y , thenNR2QS ŒClo.Y /;Clo.X /.

(12)

Proof. If R 2QB ŒX; Y , then by Example 3 it is clear that NR W Clo.Y /! Id.Clo.X //is generalized quasi-semi-homomorphism.

Thus,NR2QS ŒClo.Y /;Clo.X /.

LetX; Y andZ be Stone spaces. Let R2QB ŒX; Y , and S2QB ŒY; Z. The compositionofRwithSis the relation

RıSD f.x; ´/2XZW 9y2Y Œ.x; y/2Rand.y; ´/2H g: We note that.RıS /.x/DS ŒR.x/DS

fS.y/Wy2R.x/g.

Lemma 8. The composition of quasi-Boolean relations is a quasi-Boolean relation.

Proof. LetX; Y andZ be Stone spaces. LetR2QB ŒX; Y andS2QB ŒY; Z.

We prove thatRıSXZis point-closed. Letx2X. AsR.x/is a closed subset ofY, by Lemma6, we get thatS ŒR.x/D.SıR/.x/is a closed subset ofZ.

We prove that.RıS/.U /DRıS.U /, for eachU 2Clo.Z/. Letx2.Rı _S/.U /D_R._S.U /), i.e., R.x/_S.U /. Let ´2.RıS /.x/DS ŒR.x/D SfS.y/Wy2R.x/g. Then there exists y2R.x/such that´2S.y/. AsR.x/

S.U /,y2S.U /, i.e.,S.y/U. So,´2U. Thus,.RıS/.U /RıS.U /.

Letx2RıS.U /. Then.RıS /.x/DS ŒR.x/DS

fS.y/Wy2R.x/g U, i.e., S.y/U, for ally2R.x/. So,y2S.U /for ally2R.x/, i.e.,R.x/S.U /.

Thenx2R.S.U //D.RıS/.U /. Thus,RıS.U /.RıS/.U /.

Letf WX !Y be a function between two Stone Spaces. Consider the relation fXY defined by

fD f.x; y/2XY Wf .x/Dyg:

Lemma 9. LetX andY be two Stone spaces. Iff WX !Y is a function such thatf ¹.U /is an open subset ofX for eachU 2Clo.Y /, thenf2QB ŒX; Y .

Proof. It is clear that_f.U /D fxWf.x/Ug D˚

xWxf ¹.U / Df ¹.U /.

Thus,_f.U /is an open subset ofX for eachU 2Clo.Y /. Also, asY is a Stone Space, we havef.x/is a closed subset ofY, for eachx2X. Thus,f2QB ŒX; Y .

Using the previous lemma we obtain the following result.

Corollary 1. LetX be a Stone space. Consider the"X WX !Ul.Clo.X //. Then the relation"_X XUl.Clo.X //given by.x; P /2"_X iff"_X.x/DP is a generalized quasi-Boolean relation.

By Lemma8 we conclude that the Stone spaces with generalized quasi-Boolean relations is a category, denoted byStQBwhere the identity morphism is the identity map IdX, whereX is a Stone space. The careful reader may have realized that the notation of composition of relations reverses the order of the actual composition in

(13)

the category. We have decided to preserve this usual notation, instead of giving a new one, in order to make the paper more readable.

In the following result we characterized the isomorphisms (or iso-arrow) in the categoryStQB.

Lemma 10. LetX andY be Stone spaces andR2QB ŒX; Y . Then the following conditions are equivalent:

(1) Ris an iso-arrow in the categoryStQB.

(2) There exists an one-to-one and onto functionf WX!Y such thatRDf, satisfying the conditionf ¹.U /is an open set for eachU 2Clo.Y /. i.e.,f is a continuous function between the Stone spacesX andY.

Proof. .1/).2/ LetS 2QB ŒY; X such that RıS DId_X and RıS DId_Y, whereId_YandId_Xare quasi-Boolean relations corresponding to the functionsId_X andIdY, respectively (see Lemma9). Then for everyx2X,.RıS /.x/DS.R.x//D Id_X.x/D fxg. Using the fact thatSıId_XDS, andx2S.R.x//, then there exists y2R.x/such thatS.y/D fxg. We prove thaty is unique. Suppose that there are y1; y22Y such thatS.y1/DS.y2/. AsSıRDId_Y, we get

fy1g D.SıR/.y1/DR.S.y1//DR.S.y2//D.SıR/.y2/D fy2g: Thus,y1Dy2. So we conclude that for eachx2X there exists a uniquey2Y such thatS.y/D fxg. Let us denote byf WX !Y the function defined by

f .x/Dy iffS.y/D fxg;

for each x 2X. Similarly we can prove that there exists a function g WY !X such that R.g.y//D fyg, for each y 2Y. As Id_YıS DS, we get that .Id_Yı S /.y/DS.Id_Y.y//DS.fyg/DS.y/, for eachy2Y. Then for everyx2X we have thatfg.f .x//g D.RıS /.g.f .x///DS.R.g.f .x///DS.ff .x/g/D fxg. So, g.f .x//Dx, for eachx2X, i.e.,gıf is the identity function onX.

Changing the roles off andg, we obtain thatf ıgis the identity function onY. We conclude thatf is a one to one map fromX ontoY andgis its inverse. Observe thatR.x/DR.g.f .x///D ff .x/g. ThenRDf. Similarly, we have thatSDg.

Consider nowU 2Clo.Y /. SinceRis a quasi-Boolean relation, we have that R.U /D fx2XWR.x/Ug D˚

x2X Wf.x/U D fx2XW ff .x/g UgD fx2XWf .x/2Ug Df ¹.U /: D

So,f ¹.U /is an open subset ofX.

The direction.2/).1/follows straightforward from Lemma9and the definition

of iso-arrow.

(14)

5. CATEGORICAL DUALITY

In this section we prove that there exists a category, whose object are Boolean algebras, and whose morphism are quasi-Boolean relations. In order to complete the duality we need to see how to define quasi-Boolean relation from each quasi-semi- homomorphism between two Boolean algebras.

Let2QS ŒA; B. We define a relationRUl.B/Ul.A/by .P; Q/2R , 8a2AWa\P ¤¿thena2Q

, ¹.P /Q:

We note that whenADB, the relationRis the relation used in [3] in the representation of quasi-modal algebras.

We give now a equivalent characterization for the relation R. We recall that, given2QS ŒA; B, the generalized proximity relationıBAis defined as .b; a/2ıiffb…:a.

Lemma 11. Let A andB be two Boolean algebras. Let 2QS ŒA; B. Let .P; Q/2Ul.B/Ul.A/. Then the following conditions are equivalent:

(1) .Q; P /2R, (2) QP ı

Proof. Let.P; Q/2Ul.B/Ul.A/. Assume that.Q; P /2R. Let.q; p/2Q P. If.q; p/…ı, thenq2:p\Q, i.e.,:p2 ¹.Q/P. So,:p^pD02P, which is a contradiction. Thus,QP ı.

Assume thatQP ı. Leta\Q¤¿. Then there existq2aandq2Q.

Suppose thata…P. Then:a2P. So,.q;:a/2QP ı, i.e.,q…::aDa

, which is a contradiction. Thus,.Q; P /2R:

Remark3. WhenADB, the relation given in (2) is the definition used in [14] for the topological representation of some extensions of proximity Boolean algebras.

Lemma 12. Let2QS ŒA; B. LetP 2Ul.B/andI2Id.A/. Then I\P D¿, 9Q2Ul.A/

¹.P /QandI\QD¿ :

Proof. LetP2Ul.B/andI2Id.A/. We note thatI\PD¿iffI\ ¹.P /D

¿. Indeed. Suppose thatI\P D¿and suppose that there existsa2I\ ¹.P /.

Then a\P ¤¿, i.e., there exist p 2P and p 2a. As a2I, we get that p2I\P, which is a contradiction. ThusI\ ¹.P /D¿. The other direction is similar and left to the reader.

Assume thatI\ ¹.P /D¿. Consider the family F D˚

H 2Fi.A/WI\H D¿and ¹.P /H :

As ¹.P /is a filter of Aand ¹.P /2F;thenF ¤¿:By Zorn’s Lemma, we can take a maximal Q2F. It remains to show that Q is an ultrafilter ofA. Let

(15)

a2A. Take the filtersFa DF .Q[ fag/andF_:aDF .Q[ f:ag/. Ifa;:a…Q, thenFa; F_:a…F:So,Fa\I¤¿andF_:a\I ¤¿. Then there existsq1; q22Q such thatq1^a2I andq2^ :a2I. TakeqDq1^q2. AsI is an ideal ofA, we have that.q^a/_.q^:a/Dq^.a_:a/Dq^1Dq2I;which is a contradiction.

Thus,Q2Ul.A/. So, ¹.P /QandQ\I D¿. The other direction it is easy

and left to the reader.

Theorem 2. Let2QS ŒA; B. Leta2AandP 2Ul.B/ :Then (1) a2 ¹.P /, 8Q2Ul.A/W ¹.P /Qthena2Q, (2) a2 r ¹.P /, 9Q2Ul.A/WQ r ¹.P /anda2Q.

Proof. We prove (1). The proof of (2) follows by duality. Assume that a … ¹.P /, i.e., a\P D¿. By Lemma 12 we get that there exists Q2 Ul.A/

such that ¹.P /Qanda…Q. The other direction is immediate.

Recall that ifI is an ideal of a Boolean algebraB, then 'BŒI D fP 2Ul.B/WI\P ¤¿g is an open subset of the Stone space ofB.

Theorem 3. LetAandBtwo Boolean algebras. Let2QS ŒA; B. Then (1) 'BŒaDR.ˇA.a//;for alla2A:

(2) R2QBŒUl.B/;Ul.A/.

Proof. (1) Let a2A. Let P 2R.ˇA.a//. Then R.P /ˇA.a/. If P … 'BŒa, thena\P D¿. So, there exists Q2R.P /such thata…Q. Then, R.P /ªˇ_A.a/, which is a contradiction. Thus,P 2'_BŒa. The other inclusion is easy and left to the reader. Thus,R.ˇA.a//is an open subset.

(2) By Theorem2we deduce thatR.P /DT

fˇA.a/Wa\P ¤¿g. Therefore, R.P /is a closed subset for eachP 2Ul.A/, i.e.,Ris point-closed.

We recall that if12QS ŒA; Band22QS ŒB; C , then2ı12QS ŒA; C .

Thus,R2ı1Ul.C /Ul.A/.

Lemma 13. LetA; BandC be Boolean algebras. Let12QS ŒA; Band22 QS ŒB; C . ThenR₂ı₁DR₂ıR₁.

Proof. Let .P; Q/ 2Ul.C /Ul.B/ such that .P; Q/2 R₂_ı₁. Then .2ı 1/ ¹.P /Q, i.e., for alla2A such that .2ı1/.a/\P ¤¿, then a2Q.

We note that asQ^cDA Qis an ideal, we have that1.Q^c/is an ideal. We prove that

₂¹.P /\1.Q^c/D¿:

Otherwise there existsa…Qandb2Bsuch that2b\P ¤¿andb21a. So, there existsc 22b\P. Then, c 2.2ı1/.a/\P, i.e.,a2.2ı1/ ¹.P /.

Thus, a2 Q, which is a contradiction. Thus, there exists D 2Ul.B/ such that

(16)

₂¹.P /D and₁¹.D/Q , i.e,.P; D/2R2and.D; Q/2R1:Therefore, .P; Q/2R₂ıR₁.

To prove the other inclusion, let .P; Q/2R2ıR1 . Then there exists D 2 Ul.B/such that₂¹.P /Dand₁¹.D/Q. Leta2Asuch that.2ı1/.a/\ P D2Œ1.a/\P ¤¿. Then there existsb2B and there existsc2C such that b21aandc 22b\P. So, b2₂¹.P /. So,b 21a\D, and consequently a2Q. Thus,.2ı1/ ¹.P /Q, i.e.,.P; Q/2R₂ı₁.

Define a contravariant functor˛WBoQS!StQBby

˛.A/D hUl.A/; _Ai if Ais a Boolean algebra

˛./DR if 2QS ŒA; B : Define a contravariant functorWStQB!BoQSas

.X /DClo.X / if X is a Boolean space .R/DR if R2QB ŒX; Y .

Since for each Boolean algebraAthe mapˇAWA!Clo.Ul.A//is an isomorphism inBoQS, we get that Theorem3means that the composite functorı˛is naturally equivalent to the identity functor, the natural equivalence being given by the iso- morphismsˇA. On the other hand, since for each Stone space X, the map "X is a homeomorphism fromX ontoX.Clo.X //, it follows that the relation"_X defined by

.x; P /2"_X iff"X.x/DP

is a quasi-Boolean relation, and by Lemma5we have that"_X is an isomorphism in StQB. It is easy to see"_X is a natural equivalence from the composite functorı˛ to the identity functor from in StQB, i.e., R_Rı"_X DRı"_Y for R2QS ŒX; Y : Similarly, it is easy to see thatˇAis a natural equivalence between the identity functor inBoQSand˛ı. Thus, we have the following result.

Theorem 4. The contravariant functors and ˛ and the natural equivalences

" andˇ define a dual equivalence between the category of Boolean algebras with quasi-semi-homomorphisms and the category of Stone spaces with quasi-Boolean relations.

As an application of the above duality we prove a generalization of the result that asserts that the Boolean homomorphisms are the minimal elements in the set of all join-homomorphisms between two Boolean algebras (see [11]). Now we prove that the minimal elements in the set of all quasi-Boolean relations defined between two Stone spaces is a Boolean relation.

LetAandBbe two Boolean algebras. LetX andY be the Stone spaces ofAand B, respectively. LetQS ŒX; Y the set of all quasi-Boolean relations defined between X andY endowed with the order given by the inclusion between relations. Let1

(17)

and22QS ŒA; Band letR1andR22QS ŒX; Y the associated quasi-Boolean relations. It is clear that12if and only ifR₁R₂:

Theorem 5. Let X and Y be two Stone spaces. An element of QS ŒX; Y it is minimal if and only if is a Boolean relation .

Proof. LetRXY be a minimal element QS ŒX; Y . We prove that R is a Boolean relation. AsRis point-closed, we have to see thatR.U /is a closed subset ofX, for eachU 2Clo.Y /. Letx2cl._R.U //. Suppose thatx…_R.U / . Then R.x/ªU. So there existsy2R.x/such thaty…U. Define the relationRU as:

RU.´/DR.´/\U^c;

for each´2X. It is clear thatRU.´/is a closed subset for each´2X. ThusRU is point-closed. Moreover, forV 2Clo.Y /we have that

RU.V /D f´2X WRU.´/Vg D f´2X WR.´/\U^cVg D f´2X WR.´/U[Vg D f´2XW´2R.U[V /g DR.U [V /:

SinceRis a quasi-Boolean relation,R.U[V /is an open subset ofX. ThenRU is a quasi-Boolean relation. It is clear thatRU R. ThusRis not minimal element in QS ŒX; Y , which is a contradiction. Therefore, cl.R.U //DR.U /, i.e.,R.U / is a closed subset ofX. Consequently,Ris a Boolean relation.

FINAL REMARKS

In this paper we have proved a generalization of the Halmos’s duality [12] [11], and the duality given in [3] for quasi-modal algebras.

There are several possibilities to extend the results given in this work. One pos- sibility is to consider local Boolean algebras with a special class of morphisms. We recall that a local Boolean algebra is a pair of the formhA; Ii, whereAis a Boolean algebra andI is an ideal ofA, such thatŒI /DA. A local homomorphism between two local algebrashA; IiandhB; Jiis a Boolean homomorphismhWA!Bsatis- fying the following condition:

(LH): For eachb2J there existsa2I such thatbh.a/, i.e.,J .h ŒI .

A meaningful extension of the Stone duality is given by Geogi Dimov in [6]. In this paper it is shown that the category of local Boolean algebras with local homomorphism is dually equivalent to the category of Boolean spaces (= zero-dimensional locally compact Hausdorff spaces) with continuous maps. In a future work we shall study local Boolean algebras with meet-homomorphisms satisfying the condition (LH), and the representation theory by means of Stone spaces with a relation satisfaying certain conditions.

(18)

ACKNOWLEDGEMENTS

The author is grateful to the referee for their valuable comments. The author also acknowledges the support of the grant PIP 11220150100412CO of CONICET (Ar- gentina), and the partial support by the SYSMICS project (EU H2020-MSCA-RISE- 2015 Project 689176)

REFERENCES

[1] R. Balbes and P. Dwinger, “Distributive lattices.” Columbia, Missouri: University of Missouri Press. XIII, 294 p. $ 25.00 (1974)., 1974.

[2] S. Celani, “Subdirectly irreducible quasi-modal algebras.”Acta Math. Univ. Comen., New Ser., vol. 74, no. 2, pp. 219–228, 2005.

[3] S. Celani, “Quasi-modal algebras.”Math. Bohem., vol. 126, no. 4, pp. 721–736, 2001.

[4] G. Dimov and D. Vakarelov, “Contact algebras and region-based theory of space: a proximity approach. I.”Fundam. Inform., vol. 74, no. 2-3, pp. 209–249, 2006.

[5] G. Dimov and D. Vakarelov, “Contact algebras and region-based theory of space: a proximity approach. I.”Fundam. Inform., vol. 74, no. 2-3, pp. 209–249, 2006.

[6] G. D. Dimov, “Some generalizations of the Stone Duality Theorem.”Publ. Math., vol. 80, no.

3-4, pp. 255–293, 2012, doi:10.5486/PMD.2012.4814.

[7] I. D¨untsch and E. Orłowska, “Discrete dualities for some algebras with relations.”J. Log. Algebr.

Methods Program., vol. 83, no. 2, pp. 169–179, 2014, doi:10.1016/j.jlap.2014.02.006.

[8] I. D¨untsch and D. Vakarelov, “Region-based theory of discrete spaces: A proximity approach.”

Ann. Math. Artif. Intell., vol. 49, no. 1-4, pp. 5–14, 2007, doi:10.1007/s10472-007-9064-3.

[9] I. D¨untsch and M. Winter, “Algebraization and representation of mereotopological structures,”

Journal on Relational Methods in Computer Science, vol. 24, pp. 161 – 180, 2004.

[10] R. Goldblatt,Mathematics of modality. Stanford, CA: Stanford University, Center for the Study of Language and Information, 1993.

[11] S. Graf, “A selection theorem for Boolean correspondences.”J. Reine Angew. Math., vol. 295, pp.

169–186, 1977, doi:10.1515/crll.1977.295.169.

[12] P. R. Halmos, “Algebraic logic.” 1962.

[13] S. Koppelberg, Topological duality, in Handbook of Boolean algebras. North-Holland, Amsterdan-New York-Oxford-Tokyo, 1989, vol. 1, pp. 95–126.

[14] S. Koppelberg, I. D¨untsch, and M. Winter, “Remarks on contact relations on Boolean algebras.”

Algebra Univers., vol. 68, no. 3-4, pp. 353–366, 2012, doi:10.1007/s00012-012-0211-2.

[15] S. A. Naimpally and B. D. Warrack,Proximity spaces. Reprint of the 1970 original. Cambridge:

Cambridge University Press, 2008.

[16] G. Sambin and V. Vaccaro, “Topology and duality in modal logic.”Ann. Pure Appl. Logic, vol. 37, no. 3, pp. 249–296, 1988, doi:10.1016/0168-0072(88)90021-8.

[17] J. Stell, “Boolean connection algebras: A new approach to the Region-Connection Calculus.”

Artif. Intell., vol. 122, no. 1-2, pp. 111–136, 2000, doi:10.1016/S0004-3702(00)00045-X.

[18] D. Vakarelov,Region-Based Theory of Space: Algebras of Regions, Representation Theory, and Logics. New York, NY: Springer New York, 2007, pp. 267–348, doi:10.1007/978-0-387-69245- 6˙6.

[19] D. Vakarelov, G. Dimov, I. D¨untsch, and B. Bennett, “A proximity approach to some region- based theories of space.” J. Appl. Non-Class. Log., vol. 12, no. 3-4, pp. 527–559, 2002, doi:

10.3166/jancl.12.527-559.

(19)

Author’s address

Sergio A. Celani

Departamento de Matem´atica, Facultad de Ciencias Exactas, UNCPBA, 7000 Tandil, Argentina E-mail address:scelani@exa.unicen.edu.ar