Varieties of two-dimensional cylindric algebras II

Transkript

1 Algebra uivers. 51 (004) /04/ DOI /s c Birkhäuser Verlag, Basel, 004 Algebra Uiversalis Varieties of two-dimesioal cylidric algebras II Nick Bezhaishvili Abstract. I [] we ivestigated the lattice Λ(Df ) of all subvarieties of the variety Df of two-dimesioal diagoal free cylidric algebras. I the preset paper we ivestigate the lattice Λ(CA ) of all subvarieties of the variety CA of two-dimesioal cylidric algebras. We prove that the cardiality of Λ(CA ) is that of the cotiuum, give a criterio for a subvariety of CA to be locally fiite, ad describe the oly pre locally fiite subvariety of CA. We also characterize fiitely geerated subvarieties of CA by describig all fiftee pre fiitely geerated subvarieties of CA. Fially, we give a rough picture of Λ(CA ), ad ivestigate algebraic properties preserved ad reflected by the reduct fuctors F : CA Df ad Φ: Λ(CA ) Λ(Df ). 1. Itroductio This paper is a sequel to [] ad i it we ivestigate the lattice Λ(CA )of all subvarieties of the variety CA of two-dimesioal cylidric algebras. The variety CA is widely studied i the literature. Oe of the mai refereces is the fudametal work by Heki, Mok, ad Tarski [8]. Amog may other thigs it is well kow that Ulike the variety Df of two-dimesioal diagoal free cylidric algebras, ot every member of CA is represetable; The represetable members of CA form a proper subvariety of CA, usually deoted by RCA ; Both CA ad RCA are fiitely axiomatizable ad their equatioal theories are decidable; Both CA ad RCA are fiitely approximable, that is, geerated by their fiite members. However, either of them is locally fiite. To these results we add a criterio for a variety of two-dimesioal cylidric algebras to be locally fiite, a characterizatio of fiitely geerated ad pre fiitely geerated varieties of two-dimesioal cylidric algebras, ad a rough descriptio of the lattice Λ(CA ). Preseted by B. Jósso. Received November 9, 00; accepted i fial form February 6, Mathematics Subject Classificatio: 03G15, 08B15. Key words ad phrases: Two-dimesioal cylidric algebras, represetable algebras, locally fiite varieties, pre fiitely geerated varieties, reduct fuctors. 177

2 178 N. Bezhaishvili Algebra uivers. The paper is orgaized as follows. Sectio has a prelimiary purpose ad it cotais all the iformatio about Df ad CA eeded i subsequet sectios. I Sectio 3 we characterize represetable two-dimesioal cylidric algebras. I Sectio 4 we show that there exists a cotiuum of subvarieties of RCA,adthat there exists a cotiuum of varieties i betwee RCA ad CA. I Sectio 5 we describe the oly pre locally fiite subvariety of CA, ad characterize locally fiite varieties of two-dimesioal cylidric algebras. I Sectio 6 we characterize fiitely geerated subvarieties of CA by describig all fiftee pre fiitely geerated subvarieties of CA. Fially, i Sectio 7 we give a rough picture of the lattice structure of Λ(CA ), defie the reduct fuctors F: CA Df ad Φ: Λ(CA ) Λ(Df ), ad ivestigate algebraic properties preserved ad reflected by F ad Φ. Ackowledgemets. Special thaks go to my brother Guram. I would also like to thak Leo Esakia, Revaz Grigolia, ad Yde Veema for helpful discussios, as well as the referee for valuable suggestios icludig may poiters to [8].. Prelimiaries.1. Df. I this subsectio we review the results about two-dimesioal diagoal free cylidric algebras which will be used subsequetly. Defiitio.1 ([7, p.40]). Suppose (B,,,, 0, 1) is a Boolea algebra. A uary operatio : B B is called a moadic operator o B if the followig three coditios are satisfied for all a, b B: 0 =0; a a; ( a b) = a b. Defiitio. ([8, Defiitio 1.1.]). AtripleB =(B, 1, ) is called a twodimesioal diagoal-free cylidric algebra, oradf -algebra for short, if B is a Boolea algebra, ad 1, are moadic operators o B satisfyig the followig coditio for all a B: 1 a = 1 a. The variety of two-dimesioal diagoal-free cylidric algebras is deoted by Df. Suppose X is a oempty set, R is a biary relatio o X, x X ad A X. Let R(x) ={y X : xry}, R 1 (x) ={y X : yrx}, R(A) = x A R(x), R 1 (A) = x A R 1 (x).

3 Vol. 51, 004 Varieties of two-dimesioal cylidric algebras II 179 We call R(x) the R-saturatio of x, ad R(A) the R-saturatio of A. Note that if R is a equivalece relatio, the R(x) =R 1 (x) adr(a) =R 1 (A). Recall that a subset A of a topological space X is called a clope subset of X if it is simultaeously closed ad ope. Also recall that a topological space X is called a Stoe space if X is 0-dimesioal (that is clope subsets of X form a basis for the topology), compact, ad Hausdorff. Deote by CP(X) the Boolea algebra of all clope subsets of a Stoe space X. ArelatioR o a Stoe space X is said to be a clope relatio if A CP(X) implies R 1 (A) CP(X). We call R poit-closed if R(x) is a closed subset of X for every x X. Defiitio.3 ([, p.15]). Atriple(X, E 1, )issaidtobeadf -space if X is a Stoe space ad E 1 ad are poit-closed ad clope equivalece relatios o X with E 1 (x) = E 1 (x) for every x X. Give two Df -spaces (X, E 1, )ad(x,e 1,E ), a fuctio f : X X is said to be a Df -morphism if f is cotiuous ad fe i (x) =E i f(x) for every x X, i =1,. We deote the category of Df -spaces ad Df -morphisms by DS. The we have the followig represetatio of Df -algebras: Theorem.4. [, Theorem.4] Df is dual to DS. I particular, every Df - algebra ca be represeted as (CP(X),E 1, ) for the correspodig Df -space (X, E 1, ). For a Df -space (X, E 1, ), let E 0 = E 1. It is routie to check that E 0 is a equivalece relatio o X. Call the E i -equivalece classes, that is the sets of the form E i (x), E i -clusters (i =0, 1, ). A subset A of X is called saturated if E 1 (A) =A. ADf -space (X, E 1, ) is called a compoet if E 1 (x) =X for each x X. A partitio R of X is called correct if (1) From (xry) it follows that there exists a R-saturated clope A such that x A ad y/ A, () RE i (x) E i R(x) for every x X ad i =1,. The we have the followig dual characterizatio of cogrueces ad subalgebras of Df -algebras, as well as subdirectly irreducible ad simple Df -algebras. Theorem.5. [, Theorems.3,.5,.8] (1) The lattice of cogrueces of a Df -algebra (B, 1, ) is isomorphic to the lattice of ope saturated subsets of its dual (X, E 1, ). () The lattice of subalgebras of (B, 1, ) Df is dually isomorphic to the lattice of correct partitios of its dual (X, E 1, ). (3) (B, 1, ) Df is subdirectly irreducible iff (B, 1, ) is simple iff its dual (X, E 1, ) is a compoet.

4 180 N. Bezhaishvili Algebra uivers... CA. Defiitio.6 ([8, Defiitio 1.1.1]). A quadruple B =(B, 1,,d)issaidto be a two-dimesioal cylidric algebra, oraca -algebra for short, if (B, 1, )is a Df -algebra ad d B is a costat satisfyig the followig coditios for all a B ad i =1,. (1) i (d) =1; () i (d a) = i (d a). Deote the variety of all two-dimesioal cylidric algebras by CA. Sice i this paper we oly deal with two-dimesioal cylidric algebras, we will simply call them cylidric algebras. Below we will geeralize the duality for Df -algebras to CA -algebras. Defiitio.7. A quadruple (X, E 1,,D)issaidtobeacylidric space if the triple (X, E 1, )isadf -space ad D is a clope subset of X such that every E i -cluster of X cotais a uique poit from D for i =1,. A routie cosequece of this defiitio is the followig propositio. Propositio.8 (For a algebraic versio see [8, Theorem 1.5.3]). Suppose X is a cylidric space. The the cardiality of the set of all E 1 -clusters of X is equal to the cardiality of the set of all -clusters of X. Proof. Let E 1 ad deote the sets of all E 1 ad -clusters of X, respectively. Defie f : E 1 by puttig f(c) = (C D). Suppose C 1,C E 1, C 1 C, C 1 D = {x}, adc D = {y}. Sice every E i -cluster of X cotais a uique poit from D, it follows that f(c 1 )= (x) (y) =f(c ). Therefore, f is a ijectio. Now suppose C ad C D = {x}. If we let C = E 1 (x), the f(c) = (x) =C. Thus, f is a surjectio. Hece, we obtai that f is a bijectio. Give two cylidric spaces (X, E 1,,D) ad (X,E 1,E,D ), a fuctio f : X X is said to be a cylidric morphism if f is a Df -morphism ad f 1 (D )=D. We deote the category of cylidric spaces ad cylidric morphisms by CS. The we have the followig represetatio of cylidric algebras: 1 Theorem.9. CA is dual to CS. I particular, every cylidric algebra B = (B, 1,,d) ca be represeted as (CP(X),E 1,,D) for the correspodig cylidric space X =(X, E 1,,D). Proof. A routie adaptatio of Theorem.4 to cylidric algebras. 1 With regard to the extet of this beig a true represetatio theorem see the discussio i [8, Remarks.7.45,.7.46].

5 Vol. 51, 004 Varieties of two-dimesioal cylidric algebras II 181 Remark.10. We would like to poit out a close coectio betwee cylidric spaces ad cylidric atom structures defied i [8]. We recall from [8, Defiitio.7.3] that if B = (B, 1,,d) is a cylidric algebra, where B is a complete ad atomic Boolea algebra, the the cylidric atom structure of B is defied as the quadruple At(B) =(At(B),E 1,,D), where At(B) is the set of all atoms of B; E i is defied by puttig xe i y iff i x = i y,forx, y At(B), i =1, ; ad D = {x At(B) :x d}. Suppose B =(B, 1,,d) is a cylidric algebra, B + =(B +, + 1, +,d+ )isthe caoical extesio of B, adi: B B + is the caoical embeddig, [8, Defiitio.7.4]. The it is well kow that B + is complete ad atomic. Let At(B + )bethe cylidric atom structure of B +. For a B let O a = {x At(B + ):x i(a)}. We make At(B + ) ito a topological space by lettig {O a } a B to be a bases for the topology τ. The it ca be show that (At(B + ),τ) is a cylidric space, ad that (At(B + ),τ) is isomorphic to the dual cylidric space of B. As a easy corollary of Theorem.9 we obtai that the category FiCA of fiite cylidric algebras is dual to the category FiCS of fiite cylidric spaces with the discrete topology. I particular, every fiite cylidric algebra is represeted as the algebra (P (X),E 1,,D) for the correspodig fiite cylidric space (X, E 1,,D) (see, e.g., [8, Theorem.7.34]). To obtai the dual descriptio of homomorphic images ad subalgebras of cylidric algebras, as well as subdirectly irreducible ad simple cylidric algebras, we eed the followig two defiitios. Suppose X is a cylidric space. A correct partitio R of X is called a cylidric partitio if R(D) =D. A cylidric space X is called a quasi-square if E 1 (x) =X for every x X. Theorem.11. (1) The lattice of cogrueces of a cylidric algebra B is isomorphic to the lattice of ope saturated subsets of its dual X. () The lattice of subalgebras of a cylidric algebra B is dually isomorphic to the lattice of cylidric partitios of its dual X. (3) A cylidric algebra B is subdirectly irreducible iff it is simple iff its dual X is a quasi-square. Proof. A routie adaptatio of Theorem.5 to cylidric algebras. For (3) also see [8, Theorems.4.43,.4.14]. The we have the followig corollary of Theorem.11. Corollary.1. (1) CA is semi-simple. () CA is cogruece-distributive. (3) CA has the cogruece extesio property. Proof. Follows immediately from Theorem.11.

6 18 N. Bezhaishvili Algebra uivers. E 1 E 1 E 1 (a) (b) (c) Figure 1. Some cylidric spaces ad their reducts Now defie the reduct fuctor F: CA Df by puttig F(B, 1,,d)=(B, 1, ). Thus, F forgets the diagoal elemet d from the sigature of cylidric algebras. Remark.13. Note that it follows from Theorems.5(1) ad.11(1) that for ay cylidric algebra B, the lattice of cogrueces of B is isomorphic to the lattice of cogrueces of F(B). Now we show that F is ot oto. I fact, the set Df F(CA ) is ifiite. For this, defie the reduct fuctor R: CS DS by puttig R(X, E 1,,D)=(X, E 1, ). Suppose (Y,E 1, ) DS is a compoet. Call (Y,E 1, )aquasi-square if the cardiality of the sets of all E 1 ad -clusters coicide with each other. It follows immediately from Propositio.8 that a compoet (Y,E 1, )isareductofsome cylidric space iff it is a quasi-square. Note that ot every compoet from DS is a quasi-square. The simplest examples of compoets which are ot quasi-squares are fiite rectagle Df -spaces. Sice there are ifiitely may fiite rectagle Df -spaces, the set DS R(CS) is ifiite. Now call a Df -algebra a quasi-square algebra if its dual space is a quasi-square. As follows from the above ad Theorem.11, for every simple cylidric algebra B, its Df -reduct is a quasi-square algebra. Therefore, the set Df F(CA ) is ifiite. Moreover, oe Df -algebra ca be the reduct of may o-isomorphic cylidric algebras. For istace, a Df -algebra whose dual space is show i Figure 1(a) is the reduct of the cylidric algebras whose dual cylidric spaces are show i Figures 1(b) ad 1(c), where dots represet poits of the spaces, while big dots represet the poits belogig to the (diagoal) set D. We recall from [, Defiitio 3.1] that a fiite Df -space ( m, E 1, ) is called a rectagle if, m < ω ad E 1 ad are defied i the followig way: (i 1,i )E 1 (j 1,j )iffi = j, ad (i 1,i ) (j 1,j )iffi 1 = j 1, for i 1,i <ad j 1,j <m. Note that the cocept of a rectagle Df -space is differet from the oe of a rectagular elemet defied i [8, Defiitio ].

7 Vol. 51, 004 Varieties of two-dimesioal cylidric algebras II Represetable cylidric algebras For ay cardial κ, defie o the cartesia square κ κ two equivalece relatios E 1 ad by puttig (i 1,i )E 1 (j 1,j )iffi = j, (i 1,i ) (j 1,j )iffi 1 = j 1, for i 1,i,j 1,j κ. Let also D = {(i, i) :i κ} ad call (κ κ, E 1,,D)a square. Obviously (P (κ κ),e 1,,D) is a cylidric algebra, which we call a square algebra. 3 Deote the class of all square algebras by Sq. Defiitio 3.1 ([8, Remark , Defiitio 3.1.1(vii)]). A cylidric algebra B is called represetable if B SP(Sq), where S ad P deote the operatios of takig subalgebras ad direct products, respectively. 4 It is kow that the class of represetable cylidric algebras is also closed uder homomorphic images, ad so forms a variety which is usually deoted by RCA. It is kow that RCA is a proper subvariety of CA,thatRCA is geerated by fiite square algebras, ad that RCA ca be axiomatized by addig the followig Heki axioms to the axiom system of CA (see [8, Theorem 3..65(ii)]): (H) i (a b j (a b)) j ( d i a), i j, i, j =1,. I [1, 3.5.] Veema has simplified these equatios to the followig oes: (V) d i ( a j a) j ( d i a), i j, i, j =1,. Below we will recall the dual characterizatio of represetable cylidric algebras, ad costruct rather simple fiite o-represetable cylidric algebras. Suppose (X, E 1,,D) is a cylidric space. Call x D a diagoal poit, ad x X D a o-diagoal poit. Also call a E 0 -cluster C a diagoal E 0 -cluster if it cotais a diagoal poit. Otherwise call C a o-diagoal E 0 -cluster. Lemma 3. (For a algebraic versio of Lemma 3. we refer to [8, Theorem (ii)]). Let X be a cylidric space. If a diagoal poit x D is ot a isolated poit, the E 0 (x) {x}. Proof. Suppose x D is ot a isolated poit. The x is a limit poit, ad so there exists a et {x i } i I covergig to x. 5 Sice D is a clope, we ca assume that each 3 The square algebras are defied i [8, Defiitio 1.1.5(iv)], where they are called full cylidric set algebras of dimesio with base κ. However, sice we work oly with two-dimesioal cylidric algebras the term square algebra is more coveiet. 4 The defiitio of represetability is ot quite the same as the origial oe from [8] but is equivalet to it. 5 Recall that a et is a map from a directed set (I, ) tox. If X is a Hausdorff space, the every covergig et has a uique limit (see, e.g., [6, 1.6] for details).

8 184 N. Bezhaishvili Algebra uivers. x i belogs to D. Moreover,sice{x i } i I coverges to x, without loss of geerality we ca assume that E 1 (x i ) (x) for every x i. Let y i E 1 (x i ) (x). Sice X is compact, {y i } i I coverges to some poit y X. Moreover,y (x) because {y i } i I (x) ad (x) is closed. Sice D cotais a uique poit from every E i -cluster,wehavethat{y i } i I D. But the y D because D is a clope. Therefore, y x. Let E 1 (y) D = {z}. Ifz x, thez is ot a limit of {x i } i I, hece there exists a clope A D such that z A ad for every j I there is j j with x j / A. Butthey j / E 1 (A), which is impossible sice E 1 (A) is a clope, y E 1 (A) ady is a limit of {y i } i I.Thus,z = x, implyig that ye 1 x. Therefore, y E 0 (x), ad so E 0 (x) {x}. Defiitio 3.3. A cylidric space X is said to satisfy ( ) if there exists a diagoal poit x 0 D such that E 0 (x 0 )={x 0 } ad there exists a o-sigleto E 0 -cluster C which is either E 1 or -related to x 0. I the termiology of [8] a cylidric space satisfies the coditio ( ) of Defiitio 3.3 iff the correspodig cylidric algebra has at least oe defective atom (for details see [8, Lemma 3..59]). Now we will give a dual characterizatio of represetable cylidric algebras. A similar characterizatio ca also be foud i [8, Lemma 3..59, Theorem 3..65]. However, our characterizatio uses Veema s axioms, while the oe i [8] uses Heki s axioms. Moreover, our proof below appears to be simpler tha the origial oe i [8]. Theorem 3.4. A cylidric algebra B is represetable iff its dual cylidric space X does ot satisfy ( ). Proof. Suppose X satisfies ( ). We show that (V) does ot hold i B, implyig that B is ot represetable. Let x 0 be a diagoal poit with E 0 (x 0 )={x 0 } ad C be a o-sigleto E 0 -cluster say E 1 -related to x 0 (the case whe C is -related to x 0 is proved similarly). It follows from Lemma 3. that x 0 is a isolated poit. Therefore, E 1 (x 0 ) is a clope. Choose two differet poits y ad z from C, ad cosider a ope set E 1 (x 0 ) {x 0,y}. Let A E 1 (x 0 ) {x 0,y} be a clope cotaiig z. The y A (A), ad so x 0 D E 1 ( A (A)). O the other had, E 1 (A) =E 1 (x 0 ). Therefore, x 0 / ( D E 1 (A)), implyig that (V) does ot hold i B. Thus,B is ot represetable. Coversely, suppose B is ot represetable. We show that ( ) holdsix. We kow that (V) does ot hold i B. Therefore, there exist a poit x X ad a clope A X such that x D E i ( A E j (A)) but x / E j ( D E i (A)) for i, j =1, adi j. Sice x D E i ( A E j (A)), the x D ad there exist poits y, z X such that xe i y, ye j z, y/ A ad z A. From y/ A ad z A it follows that y ad z are differet. Also xe i y ad ye j z imply that there exists a

9 Vol. 51, 004 Varieties of two-dimesioal cylidric algebras II 185 poit u X such that xe j u ad ue i z. If u x, theu is a o-diagoal poit, ad so u D E i (A). But the x E j ( D E i (A)), which cotradicts our assumptio. Thus, u = x ad xe i z. Therefore, ye 0 z ad both y ad z are E i - related to x. Moreover,ifE 0 (x) {x}, the by choosig a poit u E 0 (x) differet from x we obtai agai that u D E i (A), ad so x E j ( D E i (A)), which is impossible. Therefore, E 0 (x) = {x} ad E 0 (y) is a o-sigleto E 0 -cluster E i -related to x 0.Thus,( ) holdsix. Usig this criterio it is easy to see that the cylidric algebras correspodig to the cylidric spaces show i Figure 1(c) are represetable, while the cylidric algebras correspodig to the cylidric spaces show i Figure 1(b) are ot. Moreover, the smallest o-represetable cylidric algebra is the algebra correspodig to the cylidric space show i Figure 1(b), where the o-sigleto E 0 -cluster cotais oly two poits. 4. Cardiality of Λ(CA ) Deote the lattice of subvarieties of CA by Λ(CA ) ad the lattice of subvarieties of RCA by Λ(RCA ). We wat to show that the cardiality of Λ(RCA )as well as the cardiality of Λ(CA ) Λ(RCA ) is that of cotiuum. For this defie a partial order o the class of all o-isomorphic fiite simple cylidric algebras by puttig A B iff A S(B). Lemma 4.1. Every two o-isomorphic fiite square algebras are -icomparable. Proof. Let A ad B be two o-isomorphic fiite square algebras ad let X A ad X B be their dual spaces. The X A is isomorphic to (, E 1,,D)adX B is isomorphic to (m m, E 1,E,D )where m. Without loss of geerality we ca assume that >m. The obviously A ca ot be a subalgebra of B. Suppose B isapropersubalgebraofa. The there exists a cylidric partitio R of X A such that X A /R is isomorphic to X B. Therefore, R must idetify poits from differet E 1 or -clusters of X A. Without loss of geerality we ca assume that R idetifies poits from differet E 1 -clusters C 1 ad C.Letx 1 C 1 be the diagoal poit of C 1 ad x C be the diagoal poit of C. Sice R(D) =D, wehave that x 1 Rx. Let E 1 (x 1 ) (x )={y 1 }. Sice x Rx 1 ad x 1 E 1 y 1,thereexists y X A such that y 1 Ry ad y E 1 x. Cosider R(x 1 )adr(y 1 ). It is obvious that R(x 1 )E 0 R(y 1 ). Also R(x 1 ) R(y 1 )sicer(d) =D. Therefore, there exists a o-sigleto E 0 -cluster of X B, which is impossible sice X B is a square. Thus, B is ot a proper subalgebra of A, ad so every two o-isomorphic fiite square algebras are -icomparable.

10 186 N. Bezhaishvili Algebra uivers. As a immediate cosequece of Lemma 4.1 we obtai the followig theorem. Theorem 4.. The cardiality of Λ(RCA ) is that of cotiuum. Proof. Let X be the square (, E 1,,D)adB be the square algebra (P ( ),E 1,,D). Cosider the family = {B } ω. From Lemma 4.1 it follows that forms a -ati-chai. For ay subset Γ of, let V Γ deote the variety geerated by Γ, that is, V Γ = HSP(Γ). Usig the stadard splittig techique, we ca easily show that V Γ V Γ wheever Γ Γ (the fact we use here is that every fiite simple cylidric algebra is a splittig algebra; see, e.g., Kracht [9, Corollary 7.3.1]). Therefore, there exist ℵ0 -may subvarieties of RCA. For >1letY deote the fiite cylidric space obtaied from the square by substitutig a sigleto o-diagoal E 0 -cluster by a two-elemet E 0 -cluster. For example, Y is show i Figure 1(b), where the o-diagoal E 0 -cluster cotais two poits. Deote by A the cylidric algebra correspodig to Y. Obviously Y satisfies ( ), ad so A is ot represetable. Similarly to Lemma 4.1, we ca prove the followig lemma. Lemma 4.3. The family {A } ω forms a -ati-chai. As a immediate cosequece of Lemma 4.3 ad the fact that {A } ω CA RCA we obtai the followig theorem. Theorem 4.4. The cardiality of Λ(CA ) Λ(RCA ) is that of cotiuum. Fially, for Γ, Γ {A } ω it is obvious that Γ Γ implies RCA V Γ RCA V Γ. Therefore, we obtai the followig corollary. Corollary 4.5. There exist cotiuum may varieties i betwee RCA ad CA. 5. Locally fiite subvarieties of CA Recall that a variety V of uiversal algebras is said to be locally fiite if every fiitely geerated V-algebra is fiite. It is called pre locally fiite if it is ot locally fiite but all its proper subvarieties are. It is kow (see, e.g., [8, Theorem.1.11]) that RCA, ad hece ay variety i the iterval [RCA, CA ], is ot locally fiite. I this sectio, we preset a criterio for a variety of cylidric algebras to be locally fiite, ad show that there exists exactly oe pre locally fiite subvariety of CA. Let B be a cylidric algebra ad X be its correspodig dual cylidric space. We have that B is simple iff X is a quasi-square. We also have that the cardialities of the sets of E 1 ad -clusters of X coicide.

11 Vol. 51, 004 Varieties of two-dimesioal cylidric algebras II 187 E 1 E 1 E 1 = Figure. Uiform quasi-squares Defiitio 5.1. (1) A quasi-square X is said to be of depth (0 <<ω)ifthe cardiality of the set of E 1 -clusters ( -clusters) of X is equal to. () A quasi-square X is said to be of a ifiite depth if the cardiality of the set of E 1 -clusters ( -clusters) of X is ifiite. (3) A simple cylidric algebra B is said to be of depth if its dual quasi-square X is of depth. (4) A simple cylidric algebra B is said to be of a ifiite depth if its dual quasisquare X is of a ifiite depth. (5) A variety V of cylidric algebras is said to be of depth if there is a simple V-algebra of depth ad the depth of every other simple V-algebra is less tha or equal to. (6) A variety V is said to be of depth ω if the depth of simple members of V is ot bouded by ay atural umber. We ote that there exists a formula measurig the depth of a variety of cylidric algebras (see [, Theorem 4.]). Let d(v) deote the depth of the variety V. Our goal is to show that a variety V of cylidric algebras is locally fiite iff d(v) <ω. For this we eed the followig defiitio. Defiitio 5.. (1) Call a quasi-square X uiform if every o-diagoal E 0 - cluster of X is a sigleto set, ad every diagoal E 0 -cluster of X cotais oly two poits. () Call a simple cylidric algebra B uiform if its dual quasi-square X is uiform. Fiite uiform quasi-squares are show i Figure, where big dots deote the diagoal poits. Deote by X the uiform quasi-square of depth. Also let B deote the uiform cylidric algebra of depth. ItisobviousthatX is (isomorphic to) the dual cylidric space of B.LetU deote the variety geerated by all fiite uiform cylidric algebras, that is U = HSP({B } ω ).

12 188 N. Bezhaishvili Algebra uivers. Propositio 5.3. U RCA. Proof. Sice oe of the diagoal E 0 -clusters of X is a sigleto set, X does ot satisfy ( ). Therefore, each B is represetable by Theorem 3.4. Thus, {B } ω RCA, implyig that U RCA. Lemma 5.4. (1) If B is a simple cylidric algebra of a ifiite depth, the each B is a subalgebra of B. () If B is a simple cylidric algebra of depth, theb is a subalgebra of B. Proof. (1) Suppose B is a simple cylidric algebra of a ifiite depth ad X is its dual cylidric space. The X is a quasi-square with ifiitely may E 1 ad -clusters. I the same way as i the proof of Claim 4.7 of [], for every we ca divide X ito -may E 1 -saturated disjoit clope sets G 1,...,G. We let D i = D G i ad F i = (D i )fori =1,...,. Obviously each of the D i s ad F i s is clope. Defie a partitio R of X by puttig xry if x, y D ad there exists i =1,..., such that x, y D i ; xry if x, y X D ad there exist 1 j, k such that x, y G j F k. It is easy to check, either directly or by trasformig the proof of Claim 4.7 of [], that R is a cylidric partitio of X,adthatX/R is isomorphic to X. Therefore, by Theorem.11(), each B is a subalgebra of B. () Suppose B is a simple cylidric algebra of depth ad X is its dual cylidric space. The X is a quasi-square. Moreover, there are exactly -may E 1 ad - clusters of X. Obviously all of them are clopes. Let C 1,...,C be E 1 -clusters of X ad let G i = C i 1 C i for i =1,...,. Obviously every G i is E 1 -saturated clope. Now applyig the same techique as i (1) shows that B is a subalgebra of B. Theorem 5.5. For a variety V of cylidric algebras, d(v) =ω iff U V. Proof. It is obvious that d(u) = ω. So, ifu V, the obviously d(v) = ω. Coversely, suppose d(v) = ω. We wat to show that every fiite uiform cylidric algebra belogs to V. Siced(V) =ω, the depth of simple members of V is ot restricted to ay atural umber. So, either there exists a family of simple V- algebras of icreasig fiite depth, or there exists a simple V-algebra of a ifiite depth. I either case, it follows from Lemma 5.4 that {B } ω V. Therefore, U V sice {B } ω geerates U. Our ext task is to show that U is ot a locally fiite variety. For this we will eed the followig lemma. Lemma 5.6. (1) Every fiite square algebra is 1-geerated. () Every fiite uiform algebra is 1-geerated.

13 Vol. 51, 004 Varieties of two-dimesioal cylidric algebras II 189 Figure 3. Geerators of square ad uiform quasi-square algebras Proof. (1) For a fiite square X =(, E 1,,D), cosider the set g = {(k, m) : k<m}. It is well kow (see e.g., [8, p.53] or [, p.4]) that a cylidric algebra geerated by g cotais all sigleto subsets of. Hece, (P ( ),E 1,,D) is geerated by g. () is proved aalogously to (1). If B is a fiite uiform algebra ad X is its dual cylidric space, the X is obtaied from a fiite square by replacig every diagoal poit by the two poit E 0 -cluster cotaiig oe diagoal poit. The same argumets as above show that every E 0 -cluster of X belogs to the algebra geerated by the lower triagle g (see Figure 3, where big dots represet the diagoal poits ad poits i circles represet the poits belogig to g ad g, respectively). Hece it is left to be show that for every diagoal E 0 -cluster C ad x C, the sigleto set {x} belogs to the algebra geerated by g. But for ay x C, eitherx D ad hece {x} = C D or x/ D ad {x} = C D. Hece every sigleto set belogs to the cylidric algebra geerated by g ad therefore g geerates B. Remark 5.7. Note that the Df -reducts of fiite uiform algebras are ot geerated by g. Ideed, the Df -algebra geerated by g does ot cotai the sigleto sets from o-sigleto E 0 -clusters. We poit out here that o fiite uiform algebra is a 1-geerated Df -algebra sice we ca show that the followig theorem holds true: A quasi-square Df -algebra is 1-geerated iff either it is a square algebra, or every E 0 -cluster of its dual space is a sigleto set except oe E 0 -cluster that cotais exactly two poits. Sice this fact is ot importat from the poit of view of this paper we skip the details. Now i order to coclude that U is ot locally fiite all we eed is to remember the followig characterizatio of locally fiite varieties from G. Bezhaishvili [1].

14 190 N. Bezhaishvili Algebra uivers. Theorem 5.8. AvarietyV of a fiite sigature is locally fiite iff for every atural umber there exists a atural umber M() such that the cardiality of every - geerated subdirectly irreducible V-algebra is less tha or equal to M(). Corollary 5.9. U is ot locally fiite. Proof. Follows from Lemma 5.6 ad Theorem 5.8. Next we show that if a variety of cylidric algebras is of fiite depth, the it is locally fiite. Theorem If d(v) <ω,thev is locally fiite. Proof. The proof is aalogous to that for the diagoal-free case (see [, Lemma 4.4]): To show V is locally fiite, by Theorem 5.8, it is sufficiet to prove that the cardiality of every -geerated simple V-algebra is bouded by some atural umber M(). Let B be a -geerated simple V-algebra. Let also B i = { i b : b B}, for i =1,. Sice d(v) <ω,wehave B 1 = B <ω. Suppose B is geerated by G = {g 1,...,g }. The as a Boolea algebra B is geerated by G B 1 B {d}. Sice the variety of Boolea algebras is locally fiite, there exists M() <ωsuch that B M() (ifact B + B 1 +1 ). Hece V is locally fiite. Fially, combiig Theorem 5.5 with Corollary 5.9 ad Theorem 5.10 we obtai the followig characterizatio of locally fiite varieties of cylidric algebras. Theorem (1) For V CA the followig coditios are equivalet: (a) V is locally fiite; (b) d(v) <ω; (c) U V. () U is the oly pre locally fiite subvariety of CA. Therefore, i cotrast to the diagoal-free case, there exist ucoutably may subvarieties of CA (RCA ) which are ot locally fiite. Sice every locally fiite variety is obviously geerated by its fiite members, we obtai from Theorem 5.11 that every subvariety of CA of a fiite depth is geerated by its fiite members. We cojecture that every subvariety of CA is i fact geerated by its fiite members. 6. Fiitely geerated ad pre fiitely geerated subvarieties of CA Recall that a variety of uiversal algebras is said to be fiitely geerated if it is geerated by a fiite uiversal algebra. We call a variety pre fiitely geerated if it is ot fiitely geerated but all its proper subvarieties are. It was show i [, Theorem 5.4] that there exist exactly six pre fiitely geerated varieties i Λ(Df ). The situatio is more complex i Λ(CA ). I this sectio, we show that there exist

15 Vol. 51, 004 Varieties of two-dimesioal cylidric algebras II 191 exactly fiftee pre fiitely geerated varieties i Λ(CA ), ad that six of them belog to Λ(RCA ). It trivially implies a characterizatio of fiitely geerated subvarieties of Λ(CA ). Cosider the fiite quasi-squares X i show i Figure 4, where i =1,...,15 ad. Agai big dots represet the diagoal poits. The patter accordig to which the quasi-squares are depicted is the followig: First come the spaces with depth 1, the the spaces with depth, ad fially the spaces with depth 3; quasi-squares with more clusters come later i the list; the first ad last quasisquares (of the same depth) do ot satisfy ( ), i.e., the correspodig algebras are represetable. As it ca be see from the figure, each E 0 -cluster of X i cosists of either oe, two or poits. Let B i deote the cylidric algebra correspodig to X i. For fixed i =1,...,15 let V i deote the variety geerated by the family {B i : }. From Theorem 3.4 it follows that oly B1 B, B3, B7, B14 ad B 15 are represetable algebras, ad so oly V 1, V, V 3, V 7, V 14 ad V 15 belog to Λ(RCA ). Now we are i a positio to prove that V 1 V 15 are the oly pre fiitely geerated subvarieties of CA. For this we eed to show that V 1 V 15 are ot fiitely geerated, which follows from their defiitio, ad that every variety of cylidric algebras which is ot fiitely geerated cotais exactly oe of V 1 V 15. Lemma 6.1. V 3 U. Proof. Suppose B is the fiite uiform algebra of depth. We show that B 3 is a subalgebra of B. Cosider the uiform square X of depth, fix a diagoal E 0 -cluster, say C, adletd C = {x 0 }. Defie a equivalece relatio R o X by puttig xry if x = y for all x, y C; xry for all x, y E 1 (C) C; xry for all x, y (C) C; xry for all x, y D {x 0 }; Let each of the remaiig 1 R-equivalece classes cosist of 1poits chose so that each class cotais exactly oe poit from each E i -cluster of X (E 1 (C) (C) D) fori =1,. It is a matter of routie verificatio that R is a cylidric partitio, ad that X /R is isomorphic to X. 3 Therefore, B 3 is a subalgebra of B for every, implyig that V 3 U. Therefore, we obtai that if d(v) = ω, thev 3 V. Suppose d(v) <ω. The V is locally fiite by Theorem Let FiV S deote the class of all fiite simple V-algebras. Sice V is locally fiite, V is geerated by FiV S. Suppose

16 19 N. Bezhaishvili Algebra uivers. E 1 = {}}{ X 1 E 1 {}}{ E 1 {}}{ {}}{ E 1 {}}{ X E 1 X 3 E 1 X 4 {}}{ E 1 X 5 X 6 }{{} X 7 {}}{ E 1 {}}{ E 1 X 8 X 9 Figure 4. Quasi-squares X 1 X 15

17 Vol. 51, 004 Varieties of two-dimesioal cylidric algebras II 193 {}}{ E 1 {}}{ E 1 X 10 X 11 {}}{ E 1 {}}{ E 1 X 1 X 13 {}}{ E 1 {}}{ E 1 X 14 X 15 Figure 4. (Cotiued) Quasi-squares X 1 X15

18 194 N. Bezhaishvili Algebra uivers. B FiV S ad X is its dual cylidric space. The X is a fiite quasi-square. Fix x X. Defiitio 6.. (1) Call the umber of elemets of E 0 (x) thegirth of x. () The maximum of the girths of all x E 0 (D) is called the diagoal girth of X. (3) The maximum of the girths of all x X E 0 (D) is called the o-diagoal girth of X. (4) The diagoal girth of B is the diagoal girth of X. (5) The o-diagoal girth of B is the o-diagoal girth of X. (6) The diagoal girth of V is said to be if there is B FiV S whose diagoal girth is, ad the diagoal girth of every other member of FiV S is less tha or equal to. (7) The diagoal girth of V is said to be ω if the diagoal girths of the members of FiV S are ot bouded by ay iteger. (8) The o-diagoal girth of V is said to be if there is B FiV S whose o-diagoal girth is, ad the o-diagoal girth of every other member of FiV S is less tha or equal to. (9) The o-diagoal girth of V is said to be ω if the o-diagoal girths of the members of FiV S are ot bouded by ay iteger. Lemma 6.3. If V is a variety of cylidric algebras of fiite depth whose diagoal ad o-diagoal girths are bouded by some iteger, the V is a fiitely geerated variety. Proof. There exist oly fiitely may o-isomorphic fiite simple cylidric algebras whose depth, the diagoal girth ad the o-diagoal girth are bouded by some iteger. Therefore, there are oly fiitely may o-isomorphic fiite simple V-algebras, implyig that V is fiitely geerated. It follows that if a variety V of a fiite depth is ot fiitely geerated, the either the diagoal girth or the o-diagoal girth of V must be ω. Lemma 6.4. If V is a variety of fiite depth whose diagoal girth is ω, theoe of V 1 V 3 is cotaied i V. Proof. Sice the diagoal girth of V is ω, foreach there is B FiV S whose diagoal girth is. Let X be the dual cylidric space of B. The X is a quasisquare. Deote by C the diagoal E 0 -cluster of X cotaiig poits. The two cases are possible. Either d(x )=1ord(X) for ifiitely may. I the former case, it is obvious that X is isomorphic to X,adsoV 1 1 V. I the latter case, defie a equivalece relatio R o X by puttig xry if x = y for ay x, y C D; xry if xe 0 y for ay x, y X (C D).

19 Vol. 51, 004 Varieties of two-dimesioal cylidric algebras II 195 Clearly R is a cylidric partitio. Deote X /R by Y. The every o-diagoal E 0 -cluster of Y is a sigleto set ad every diagoal E 0 -cluster differet from C cotais either oe or two poits. Agai there are two cases possible. Either d(y) =ord(y) > for ifiitely may. I the former case, Y is isomorphic to either X or X 3 for ifiitely may. Therefore, either V V or V 3 V. Ad i the latter case, defie a equivalece relatio Q o Y by puttig xqy if x = y for ay x, y C; xqy for ay x, y E 1 (C) C; xqy for ay x, y (C) C; xqy for ay x, y D C; xqy for ay x, y Y (E 1 (C) (C) D). It is a matter of routie verificatio that Q is a cylidric partitio, ad that Y/Q is isomorphic to X 3.Thus,V 3 V. Lemma 6.5. If V is a variety of fiite depth whose o-diagoal girth is ω, the oe of V 4 V 15 is cotaied i V. Proof. Sice the o-diagoal girth of V is ω, foreach there is B FiV S whose o-diagoal girth is. Let X be the dual cylidric space of B. The X is a quasi-square. Deote by C the o-diagoal E 0 -cluster of X cotaiig poits. Sice the o-diagoal E 0 -clusters exist oly i cylidric spaces of depth > 1, we have d(x ) > 1. Defie a equivalece relatio R o X by puttig xry if x = y for ay x, y C D; xry if xe 0 y for ay x, y X (C D). Clearly R is a cylidric partitio. Sice d(x ) > 1, there are three cases possible. Either d(x )=,d(x) = 3, or d(x ) > 3 for ifiitely may. If d(x ) = for ifiitely may, thex /R is isomorphic to oe of X 4 X 7 for ifiitely may, implyig that oe of V 4 V 7 is cotaied i V. If d(x ) = 3 for ifiitely may, thex /R is isomorphic to oe of X 8 X 15 for ifiitely may, implyig that oe of V 8 V 15 is cotaied i V. Fially, let 3 <d(x ) <ωfor ifiitely may. Deote by C the diagoal E 0 -cluster E 1 -related to C, adbyc - the diagoal E 0 -cluster -related to C. Defie a equivalece relatio R o X by puttig xry if x = y for ay x, y C ((C C ) D); xry for ay x, y D (C C ); xry for ay x, y X (D E 1 (C ) (C ) E 1 (C ) (C )); xry if xe 0 y for ay x, y ( (C ) E 1 (C )) ((C C ) D); xry for ay x, y (C) (C C ); xry for ay x, y E 1 (C) (C C ); xry for ay x, y (C ) (E 1 (C ) C );

20 196 N. Bezhaishvili Algebra uivers. xry for ay x, y E 1 (C ) ( (C ) C ). It is a matter of routie verificatio that R is a cylidric partitio. Moreover, there are four cases possible. Either both C ad C are sigleto sets, C is a sigleto set ad C is ot, C is a sigleto set ad C is ot, or either C or C are sigleto sets, for ifiitely may. I the first case X /R is isomorphic to X 11 i the secod case X /R is isomorphic to X 13, ad fially i the fourth case X /R is isomorphic to X 15 to X 1 V 1 V 15 is cotaied i V.,,ithethirdcaseX/R is isomorphic. Therefore, oe of Thus, goig through all the cases we obtai that oe of V 4 V 15 is cotaied i V. Corollary 6.6. (1) The oly pre fiitely geerated varieties i Λ(CA ) are V 1 V 15. () The oly pre fiitely geerated varieties i Λ(RCA ) are V 1, V, V 3, V 7, V 14 ad V 15. Proof. This is a immediate cosequece of Lemmas 6.1, , ad the fact that all the fiftee varieties are o-comparable. 7. Lattice structure of Λ(CA ) I order to obtai a rough picture of the lattice structure of subvarieties of CA, we eed the followig otatio: FG = {V Λ(CA ):V is fiitely geerated}; D F = {V Λ(CA ):d(v) <ωad V / FG}; D ω = {V Λ(CA ):d(v) =ω}. Let also V deote the trivial variety. Theorem 7.1. (1) {FG, D F, D ω } is a partitio of Λ(CA ). () V is a least elemet of FG. (3) FG does ot have maximal elemets. (4) D F has precisely fiftee miimal elemets. (5) D F does ot have maximal elemets. (6) U ad CA are the least ad the greatest elemets of D ω, respectively. Proof. This follows immediately from Theorem 5.5 ad Corollary 6.6. The lattice Λ(CA ) ca be roughly depicted as show i Figure 5. Now we will ivestigate the lower part of Λ(CA ) i a greater detail. It follows from Corollary 6.6 that a variety V CA (RCA ) is fiitely geerated iff V does ot cotai oe of the fiftee (six) pre fiitely geerated varieties. Aother criterio is give by the followig theorem.

21 Vol. 51, 004 Varieties of two-dimesioal cylidric algebras II 197 CA varieties of ifiite depth RCA U o-fiitely geerated varieties of fiite depth pre fiitely geerated varieties fiitely geerated varieties V Figure 5. Rough picture of Λ(CA ) Theorem 7.. For a variety V CA the followig coditios are equivalet: (1) V is fiitely geerated. () V has oly fiitely may subvarieties. (3) V cotais oly fiitely may o-isomorphic simple algebras (ad all of them are fiite). Proof. (1) () is straightforward sice CA is cogruece-distributive. () (3). First ote that if V cotais a ifiite simple algebra, the it cotais ifiitely may o-isomorphic fiite simple algebras. To see this, let B be a ifiite simple V-algebra. If d(b) ω, the by Lemma 5.4 B has ifiitely may o-isomorphic simple subalgebras. If d(b) <ω, the either the diagoal or o-diagoal girth of B is ifiite ad the same argumets as i the proofs of Lemmas 6.4 ad 6.5 show that there are ifiitely may o-isomorphic simple subalgebras of B. Now suppose that V cotais a ifiite family {B i } i I of oisomorphic simple algebras. The V cotais a ifiite family {B i } i I of fiite o-isomorphic simple algebras. By Jósso s Lemma {HSP(B i )} i I is a ifiite family of distict subvarieties of V, which is a cotradictio. (3) (1). Let {B i } i=1 be the family of all simple o-isomorphic V-algebras. It follows from the above that they are all fiite. The i=1 B i geerates V ad therefore V is fiitely geerated.

22 198 N. Bezhaishvili Algebra uivers. By a immediate successor of a variety V CA we mea a immediate successor i the lattice Λ(CA ). Corollary 7.3. (1) Every immediate successor of a fiitely geerated variety of cylidric algebras is fiitely geerated. () A fiitely geerated variety of cylidric algebras has oly fiitely may immediate successors. Proof. (1) Let V be a immediate successor of a fiitely geerated variety V. Sice V is fiitely geerated V = HSP(B) for a fiite cylidric algebra B. SiceV V, there is a simple cylidric algebra B V with B / V. TheV HSP(B B ) ad because V is a immediate successor of V we have that V = HSP(B B ). Moreover, if B is ifiite, the the same argumets as i the proof of Theorem 7. show that B has ifiitely may o-isomorphic subalgebras, which is impossible sice V is a immediate successor of V ad V is fiitely geerated. Hece B is fiite ad therefore V is fiitely geerated. () The proof is aalogous to the stadard proof that a fiitely geerated variety of iterior algebras has oly fiitely may immediate successors (see, e.g., Blok [4, Theorem 7.5]) Varieties of cylidric algebras of depth oe. I this subsectio we give a complete characterizatio of the lattice structure of the varieties of cylidric algebras of depth oe. I the diagoal free case, the lattice of varieties of Df - algebras of E 1 ad -depth oe is relatively simple. It is isomorphic to the lattice of varieties of moadic algebras ad is a (ω + 1)-chai (see [10, Theorem 4] ad [8, Theorem 4.1.]). As we will see below, the structure of the lattice of varieties of cylidric algebras of depth oe is more complex. Let deote the -elemet Df -algebra, where 1ad { 0 if a=0, i (a) = 1 otherwise, for i =1,. Let also d be a atom of.the(,d) is a cylidric algebra. It is obvious that (,d) is simple ad has depth oe. Observe that the dual space of (,d) is isomorphic to X 1 defied i Sectio 6. Hece (,d) RCA for every ω. It is also clear that up to isomorphism X 1, ω, are the oly fiite quasisquares of depth oe. Thus (,d) are the oly fiite simple cylidric algebras of depth oe. We recall that i the diagoal-free case the two-elemet Df -algebra is a subalgebra of every o-trivial Df -algebra. For CA the situatio is differet. Propositio 7.4. Suppose B is a o-trivial simple cylidric algebra. (1) (, 1) is a subalgebra of B iff B is isomorphic to (, 1).

23 Vol. 51, 004 Varieties of two-dimesioal cylidric algebras II 199 (, 1) (,d) ( 3,d) ( 4,d) Figure 6. The poset of simple cylidric algebras of depth oe () If B is ot isomorphic to (, 1), the(,d) is a subalgebra of B. Proof. (1) If (, 1) is a subalgebra of B =(B,d), the d = 1. Suppose there is a elemet a B with 0 <a<1. The, by Defiitio.6(), i a a for i =1,. Hece the ideal ( a] correspods to a o-trivial proper cogruece, i.e., B/ ( a] is a proper o-trivial homomorphic image of B, which is impossible sice B is simple. Therefore, B = ad B is isomorphic to (, 1). ()Itiskowthat 1 d = 1 d = d (see, e.g., [8, Theorem (ii)]). Sice B is ot isomorphic to (, 1), we have d 1. Hece, d 0. So, 1 d =1 sice B is simple. Thus, {1, 0,d, d} is a cylidric subalgebra of B. Let Var(, 1) ad Var(,d) deote the varieties geerated by (, 1) ad (,d), respectively. Corollary 7.5. (1) The varieties Var(, 1) ad Var(,d) are the oly atoms i Λ(CA ). () If a variety V of cylidric algebras cotais the two-elemet cylidric algebra (, 1), thev is geerated by a simple algebra iff V = Var(, 1). Proof. (1) It follows from Propositio 7.4(1) that Var(, 1) ad Var(,d)are icomparable. Now let V be a o-trivial subvariety of CA,adBasimple V-algebra. By Propositio 7.4 either (, 1) or (,d) is a subalgebra of B. Thus Var(, 1) V or Var(,d) V. () Suppose (, 1) V ad V is geerated by a simple V-algebra B. Usig the stadard splittig techique (see, e.g., Kracht [9, 7.3]) we obtai that (, 1) S(B), ad applyig Propositio 7.4 we get that B is isomorphic to (, 1). Let V 1 CA be the variety of all cylidric algebras of depth oe. It is kow from [, Theorem 4.] that V 1 = CA +( 1 a = 1 a)=rca +( 1 a = 1 a).

24 00 N. Bezhaishvili Algebra uivers. V (,1) V (,1) (,d) V (,1) ( 3,d) V 1 V V (,d) V (3,d) V 1 Figure 7. The lattice of varieties of cylidric algebras of depth oe Let (F, ) deote the partially ordered set of all o-isomorphic fiite cylidric algebras of depth oe. We recall from Sectio 3 that is defied o F by puttig B B iff B S(B ). As we poited out above, F = {(,d): ω}. It follows from Propositio 7.4 that (F, ) is isomorphic to the disjoit uio of the set of atural umbers (N, ) with the set cosistig of oe reflexive poit (see Figure 6). Recall that G Fis called a dowset of F if A Gad B A imply B G. Sice every variety of cylidric algebras of fiite depth is locally fiite, applyig [5, Theorem 3.3] we obtai the followig represetatio of the lattice of varieties of cylidric algebras of depth oe. Theorem 7.6. The lattice of varieties of cylidric algebras of depth oe is isomorphic to the lattice of dowsets of (F, ). The lattice of varieties of cylidric algebras of depth oe is show i Figure 7. To explai the labelig, with each dowset of (F, ) oftheform (,d)={( k,d): 1 < k < } we associated the variety V (,d) geerated by (,d); ad with each dowset of the form (,d) {(, 1)} we associated the variety V (,1) (,d) geerated by (, 1) (,d); furthermore, V 1 = HSP({(,d):>1}). Theorem 7.7. Every subvariety of V 1 is fiitely axiomatizable. Proof. A proof similar to the oe i Scroggs [10, p.119] shows that the iequality +1 (S ) 1 a k 1 (a k a j ) k=1 1 k,j +1 k j holds true i a simple cylidric algebra of depth oe iff the correspodig quasisquare cotais poits. Therefore, the varieties V (,1) (,d) are axiomatized by the idetities of V 1 plus (S ). O the other had, the idetity 1 d =1 holds true i (,d)iff>1. Therefore, the variety V 1 is axiomatized by the idetities of V 1 plus 1 d = 1, while the varieties V (,d) are axiomatized by addig 1 d = 1 to the idetities of V (,1) (,d).

25 Vol. 51, 004 Varieties of two-dimesioal cylidric algebras II 01 Remark 7.8. I fact, usig the Jakov-Fie type formulas, the techique aalogous to [3] shows that every subvariety of CA of fiite depth is fiitely axiomatizable. (Note that the proof of this fact is much simpler tha the origial oe from [3] for the diagoal free case sice, i cotrast with Df -algebras, every cylidric algebra has the same E 1 ad -depth.) Nevertheless, the cardiality of Λ(CA ) is that of cotiuum, which meas that there exist ucoutably may o-fiitely axiomatizable subvarieties of CA of ifiite depth. 7.. Reduct fuctors. Suppose B = (B, 1,,d) is a cylidric algebra. I Sectio we deoted its Df -reduct by F(B) = (B, 1, ) Df. If K is a subclass of CA,letF(K) ={F(B) :B K}. AlsoifM is a subclass of Df,let F 1 (M) ={B CA : F(B) M}. Lemma 7.9. Suppose K CA ad M Df. The the followig hold. (1) HF(K) =FH(K). () SF(K) FS(K). (3) PF(K) =FP(K). (4) HF 1 (M) F 1 H(M). (5) SF 1 (M) F 1 S(M). (6) PF 1 (M) =F 1 P(M). Proof. (1) This claim follows immediately from Theorems.5(1) ad.11(1) (see Remark.13). () It is obvious that if B is a cylidric subalgebra of A, thef(b) isadf - subalgebra of F(A). Hece, FS(K) SF(K). To see that the coverse iclusio does ot hold, let d(k) ad cosider B K with d(b). Deote by X =(X, E 1,,D) the dual cylidric space of B. Defie a partitio R o X by puttig xry if x y. The R is a correct Df -partitio ad the Df -algebra A correspodig to the Df -space X/R belogs to SF(K). O the other had, the E 1 -depth of X/R is 1 ad the -depth of X/R is. Therefore, X/R has differet E 1 ad depths, which by Propositio.8 implies that A ca ot be the reduct of ay cylidric algebra. Thus, SF(K) FS(K). (3) Follows from the fact that for ay family {B i } i I of cylidric algebras we have F( i I B i)= i I F(B i). (4) That HF 1 (M) F 1 H(M) follows from the fact that every cylidric homomorphism is a also a Df -homomorphism. To show that this iclusio is proper, cosider a cylidric algebra B ad let A be a Df -algebra such that d 1 (A) d (A). The F(B) is a homomorphic image of F(B) A, but sice d 1 (A) d (A), F(B) A is ot the reduct of ay cylidric algebra. Hece, B F 1 H({F(B) A}), but HF 1 ({F(B) A})isempty.

26 0 N. Bezhaishvili Algebra uivers. (5) That SF 1 (M) F 1 S(M) follows from the fact that if B is a cylidric subalgebra of A, thef(b) isadf -subalgebra of F(A). To see that this iclusio is proper, suppose the two-elemet Df -algebra does ot belog to M. The the two-elemet cylidric algebra, (, 1) does ot belog to F 1 (M). By Propositio 7.4 (, 1) / SF 1 (M). O the other had, is a Df -subalgebra of every Df -algebra. Therefore, S(M) ad(, 1) F 1 (S(M)). (6) That PF 1 (M) F 1 P(M) follows from the defiitio of the product of cylidric algebras. To see the coverse, suppose B F 1 P(M). The B =(B,d), where B = i I B i for some Df -algebras B i M. Let(B i,d i )bethei-th projectio of B. Sicethei-thprojectioisaotoDf -homomorphism, by Remark.13 it is also a cylidric homomorphism. Hece each (B i,d i ) is a cylidric algebra ad d = d i i I. The B is isomorphic to i I (B i,d i ). Now every (B i,d i ) belogs to F 1 (M). Hece, F 1 P(M) PF 1 (M). Theorem (1) If K is a subvariety of Df,theF 1 (K) is a subvariety of CA. () For a o-trivial subvariety V of CA, F(V) is a subvariety of Df iff V = V (,1) (,d) for some ω. Proof. (1) By Lemma 7.9 we have HSPF 1 (K) F 1 (HSP(K)) = F 1 (K). Hece, F 1 (K) is a variety of cylidric algebras. () Suppose V is a subvariety of CA. If d(v) > 1, the it follows from the proof of Lemma 7.9() that F(V) is ot closed uder subalgebras, hece is ot a variety. Thus, if F(V) is a variety, the V V 1. If (, 1) V, thef(, 1) / F(V) ad agai F(V) is ot a variety sice every otrivial variety of diagoal-free cylidric algebras cotais = F(, 1). We ow show that F(V 1 ) is ot a variety. Let C deote the Cator space. The X = (C,E 1, )isadf -space, where E 1 (c) = (c) =C for ay c C. IfX were the reduct of a cylidric space, the X would cotai a isolated poit. Sice C is dese i itself, it follows that X is ot the reduct of ay cylidric space. Let {y} be a sigleto topological space. The Y =(C {y},e 1,, {y}) is a cylidric space, where E 1 (x) = (x) =C {y} for ay x C {y}. Moreover,B =(CP(Y),E 1,, {y}) is a ifiite simple cylidric algebra of depth 1, ad so B V 1. Now cosider R(Y) =(C {y},e 1, ). Fix ay poit c C ad let R be the smallest equivalece relatio which idetifies y ad c. It is easy to check that R is a correct Df -partitio, ad that R(Y)/R is isomorphic to X. So, A =(CP(X ),E 1, ) is isomorphic to a Df -subalgebra of F(B), but it is ot the reduct of ay cylidric algebra. Hece, A does ot belog to F(V 1 ), ad so F(V 1 ) is ot a variety. Therefore, if V V (,1) (,d) for ay ω, thef(v) is ot a variety. Fially, oe ca easily verify that for ay ω, HS({(, 1) (,d)}) ={( m,d):m }. ThisimpliesthatF(V (,1) (,d)) =