Dually quasi-de Morgan Stone semi-heyting algebras I. Regularity

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1 Volume 2, Number, July 204, ISSN Print: Online: Dully qusi-de Morgn Stone semi-heyting lgebrs I. Regulrity Hnmntgoud P. Snkppnvr Abstrct. This pper is the first of two prt series. In this pper, we first prove tht the vriety of dully qusi-de Morgn Stone semi-heyting lgebrs of level stisfies the strongly blended -De Morgn lw introduced in [20]. Then, using this result nd the results of [20], we prove our min result which gives n explicit description of simple lgebrs(=subdirectly irreducibles) in the vriety of regulr dully qusi-de Morgn Stone semi-heyting lgebrs of level. It is shown tht there re 25 nontrivil simple lgebrs in this vriety. In Prt II, we prove, using the description of simples obtined in this Prt, tht the vriety RDQDStSH of regulr dully qusi-de Morgn Stone semi-heyting lgebrs of level is the join of the vriety generted by the twenty 3-element RDQDStSH -chins nd the vriety of dully qusi-de Morgn Boolen semi-heyting lgebrs the ltter is known to be generted by the expnsions of the three 4-element Boolen semi-heyting lgebrs. As consequences of this theorem, we present (equtionl) xiomtiztions for severl subvrieties of RDQDStSH. The Prt II concludes with some open problems for further investigtion. Keywords: Regulr dully qusi-de Morgn semi-heyting lgebr of level, dully pseudocomplemented semi-heyting lgebr, De Morgn semi-heyting lgebr, strongly blended dully qusi-de Morgn Stone semi-heyting lgebr, discrimintor vriety, simple, directly indecomposble, subdirectly irreducible, equtionl bse. Mthemtics Subject Clssifiction [200]: 03G25, 06D20, 06D5, 08B26, 08B5. Dedicted to the Memory of My Sister Kshww. 47

2 48 Hnmntgoud P. Snkppnvr Introduction The concept of regulrity hs plyed n importnt role in the theory of pseudocomplemented De Morgn lgebrs (see [5]). (A similr notion ws erlier considered for double p-lgebrs in [22].) The min purpose of this pper is to crry it over to the vriety DQDSH of dully qusi-de Morgn semi-heyting lgebrs studied in [20] nd to initite the investigtion of regulrity in the level subvriety DQDStSH of DQDSH whose members hve Stone semi-heyting reducts. This pper, the first of two-prt series, is orgnized s follows: We first prove in Section 3 tht the vriety DQDStSH of dully qusi-de Morgn Stone semi-heyting lgebrs of level stisfies the strongly blended -De Morgn lw introduced in [20]. Then, using this result nd the results of [20], we prove, in Section 4, our min result which gives n explicit description of simple lgebrs(=subdirectly irreducibles) in the vriety RDQDStSH of regulr dully qusi-de Morgn Stone semi-heyting lgebrs of level. It is shown tht there re 25 (nontrivil) simple lgebrs in this vriety. In Prt II, we prove, using the description of simples, obtined in this Prt, tht RDQDStSH is the join of the vriety generted by the twenty 3-element RDQDStSH -chins nd the vriety of dully qusi-de Morgn Boolen semi-heyting lgebrs the ltter is known to be generted by the expnsions of the three 4-element Boolen semi-heyting lgebrs. Furthermore, s consequences of this theorem, we present, (equtionl) xiomtiztions for severl subvrieties of RDQDStSH. The Prt II concludes with some open problems for further investigtion. 2 Dully qusi-de Morgn semi-heyting lgebrs The following definition is tken from [8]. An lgebr L = L,,,, 0, is semi-heyting lgebr if L,,, 0, is bounded lttice nd L stisfies: (SH) x (x y) x y (SH2) x (y z) x ((x y) (x z)) (SH3) x x.

3 Dully qusi-de Morgn semi-heyting lgebrs I 49 Let L be semi-heyting lgebr. L is Heyting lgebr if L stisfies: (SH4) (x y) y. L is Stone semi-heyting lgebr if L stisfies: (St) x x, where x := x 0. L is Boolen semi-heyting lgebr if L stisfies: (Bo) x x (where x := x 0). Semi-Heyting lgebrs re distributive nd pseudocomplemented, with s the pseudocomplement of n element. We will use these nd other properties (see [8]) of semi-heyting lgebrs, frequently without explicit mention, throughout this pper. The following definition is tken from [20]. Definition 2.. An lgebr L = L,,,,, 0, is semi-heyting lgebr with dul qusi-de Morgn opertion or dully qusi-de Morgn semi-heyting lgebr (DQDSH-lgebr, for short) if L,,,, 0, is semi-heyting lgebr, nd L stisfies: () 0 nd 0 (b) (x y) x y (c) (x y) x y (d) x x. A DQDSH-lgebr L is dully pseudocomplemented semi-heyting lgebr (DPCSH-lgebr) if L stisfies: (e) x x. A DQDSH-lgebr L is De Morgn semi-heyting lgebr or symmetric semi-heyting lgebr (DMSH-lgebr) if L stisfies: (DM) x x.

4 50 Hnmntgoud P. Snkppnvr The vrieties of DQDSH-lgebrs, DPCSH-lgebrs nd DMSH-lgebrs re denoted, respectively, by DQDSH, DPCSH nd DMSH, while DQDStSH denotes the subvriety of DQDSH defined by the Stone identity (St), nd DQDBSH denotes the one defined by (Bo). Let L DQDSH. L is semi-heyting lgebr with blended dul qusi-de Morgn opertion or blended dully qusi-de Morgn semi-heyting lgebr (BDQDSH-lgebr, for short ) if L stisfies: (B) (x x ) x x (Blended -De Morgn lw). L is semi-heyting lgebr with strongly blended dul qusi-de Morgn opertion or strongly blended dully qusi-de Morgn semi-heyting lgebr (SBDQDSH, for short ) if L stisfies: (SB) (x y ) x y (Strongly Blended -De Morgn lw). L is semi-heyting lgebr with dul ms-opertion (DmsSH-lgebr) if L stisfies: (x y) x y ( -De Morgn lw). The vrieties of BDQDSH-lgebrs, SBDQDSH-lgebrs nd DmsSH-lgebrs re denoted, respectively, by BDQDSH, SBDQDSH nd DmsSH. It is cler tht DmsSH SBDQDSH BDQDSH. If the underlying semi-heyting lgebr is Heyting lgebr, then we replce the prt SH by H in the nmes of the vrieties tht we consider. Two importnt clsses of exmples of DQDStSH-lgebrs re Stone Heyting lgebrs with dul pseudocomplement (DPCStH) nd De Morgn Stone Heyting lgebrs (symmetric Stone Heyting lgebrs) (DMStH, for short). In the sequel, will be denoted by +, for L DQDSH. The following lemm will be used, often without explicit reference to it. Most of the items in this lemm were proved in [20] nd the others re left to the reder. Lemm 2.2. Let L DQDSH nd let x, y, z L. Then (i) =

5 Dully qusi-de Morgn semi-heyting lgebrs I 5 (ii) x y implies x y (iii) (x y) = x y (iv) x x (v) x = x (vi) (x y) = (x y ) (vii) (x y) = (x y) (viii) x (x y) x (ix) x x (x) x [(x y) z] = x (y z) (xi) x x + =. Recll tht L DQDSH is DQDSH-chin if the lttice reduct of L is chin. In wht follows we re interested in the DQDSH-chins of size 2 nd 3 which we describe below. It ws observed in [8] tht 2 nd 2, shown in Figure, re, up to isomorphism, the only two 2-element semi-heyting lgebrs. 2 : : Figure Let 2 e nd 2 e denote their expnsions obtined by dding the unry opertion defined by: 0 = nd = 0. It is cler tht 2 e nd 2 e re the only 2-element DQDStSH-lgebrs. It ws lso observed in [8] tht there re, up to isomorphism, ten 3- element semi-heyting chins, shown in Figure 2.

6 52 Hnmntgoud P. Snkppnvr L : L 2 : L 3 : 0 L 5 : L 4 : L 6 : L 7 : L 8 : L 9 : L 0 : Figure 2 Let L dp i, i =,..., 0, denote the expnsion of L i (shown in Figure 2) by dding the unry opertion such tht 0 =, = 0, nd = ; nd let, i =,..., 0, denote the expnsion of L i by dding the unry opertion such tht 0 =, = 0, nd =. We Let C dp 0 := {Ldp i : i =,..., 0} nd C dm 0 : i =,..., 0}. We lso let C 20 := C dm 0 Cdp L dm i := {Ldm i It is esy to verify tht C dp 0 DPCStSH DQDStSH nd Cdm 0 DMStSH DQDStSH. It is lso esy to see tht these 20 lgebrs re the only 3-element DQDStSH-lgebrs. 0.

7 Dully qusi-de Morgn semi-heyting lgebrs I 53 We re lso interested in the three 4-element lgebrs D, D 2 nd D 3 described below. Ech of the three lgebrs hs Boolen lttice reduct with the universe {0,, b, }, in which b is the complement of, nd the unry opertion is defined s follows: =, b = b, 0 =, = 0, nd is defined in Figure 3: 0 b D : 0 0 b 0 b D 2 : b 0 b b 0 0 b 0 0 b b b b D 3 : 0 b 0 0 b b 0 b Figure 3 It ws shown in [20] tht the lgebrs D, D 2, nd D 3 re simple nd the vriety generted by the three lgebrs is precisely the vriety DQDBSH which is the subvriety of DQDSH defined by (Bo). The semi-heyting reducts of D, D 2 nd D 3 re, up to isomorphism, 2 2, 2 2, nd 2 2 respectively. Notice lso tht {D, D 2, D 3 } = x x, nd D2 hs Heyting lgebr reduct, while the other two do not. Furthermore, D stisfies the commuttive lw: x y y x. The following definition is from [20]. Definition 2.3. Let x L DQDSH. recursively s follows: For n ω, we define t n (x) x 0( ) = x, x (n+)( ) = (x n( ) ), for n 0. t 0 (x) = x, t n+ (x) = t n (x) x (n+)( ), for n 0. For n ω, the nth level (or level n) subvriety DQDSH n of DQDSH

8 54 Hnmntgoud P. Snkppnvr is defined by the identity: t n (x) t n+ (x). For n ω, we let BDQDSH n := BDQDSH DQDSH n ; nd similrly SBDQDSH n, etc. re defined. It ws shown in [20] tht, for n ω, the vriety BDQDSH n, nd hence the subvriety SBDQDSH n, is discrimintor vriety. Note tht the level 0 (or 0th level) vriety DQDSH 0 is defined by the identity: x x x, (equivlently, x x) nd it ws shown in [20] tht this vriety is generted by {2 e, 2 e }. An interesting open problem is the problem of investigtion of the (complex) structure of the lttice of subvrieties of the vriety DQDSH which is defined by: x x x x x. In this pper we re interested in specil cse of this problem, nmely tht of describing the subvriety lttice of the vriety RDQDStSH of regulr dully qusi-de Morgn Stone semi-heyting lgebrs of level (see Section 4 for definition). The vriety V(C 20 ) generted by the twenty 3-element lgebrs (Figure 2) nd the vriety DQDBSH generted by the three 4-element lgebrs (Figure 3) re subvrieties of DQDSH tht were xiomtized in [20]. In this pper, we re lso interested in the problem of finding n equtionl bse (n equtionl xiomtiztion) for the join of the vrieties V(C 20 ) nd DQDBSH. It is somewht surprising tht these two problems re closely relted (see Prt II, Theroem 3.4). It is esy to see tht ll the twenty five lgebrs described bove in Figures -3 re ctully in SBDQDStSH. The following lemm is useful in the sequel. In prticulr, it ids us in giving n lterntive definition of DQDStSH n. Lemm 2.4. Let L DQDStSH with L 2. Then () x = x (2) x x (3) x x = (4) x x (5) x x x (x x ). Proof. From x x =, we hve x x =, implying x x =. It follows tht x x, from which we cn conclude (). Next, x x

9 Dully qusi-de Morgn semi-heyting lgebrs I 55 implies x x x, proving (2). From (2) we get x x = 0, which implies x x =, in view of () this proves (3). Now, from (3) we hve x x =, yielding x x. Replcing x by x nd using (2), we get x x x, thus proving (4). Finlly, (x x ) = x x = x x x in view of (4), which proves (5). The following theorem, which is now immedite from Lemm 2.4(5) nd the definition of t n, gives n lterntive definition of DQDStSH n. Theorem 2.5. For n ω, DQDStSH n is defined by the identity: (x x ) n( ) (x x ) (n+)( ), modulo DQDStSH. In prticulr, the vriety DQDStSH is defined by the identity: x x (x x ), reltive to DQDStSH. 3 Dully qusi-de Morgn Stone semi-heyting lgebrs of level We now focus our ttention on the (sub)vriety DQDStSH which, in view of Theorem 2.5, consists of dully qusi-de Morgn Stone semi-heyting lgebrs stisfying the identity: (L) x x (x x ) (Level ). If the underlying semi-heyting lgebr of DQDStSH-lgebr is Heyting lgebr we denote the lgebr by DQDStH-lgebr. The corresponding vrieties re denoted by DQDStSH nd DQDStH respectively. In this section our gol is to prove DQDStSH = SBDQDStSH. To chieve this, we need the following lemms. Lemm 3.. Let x L DQDStSH. Then () (x x ) x x (2) x x x = 0 (3) x x = 0. Proof. (x x ) = (x x ) = (x x ) = x x = x x, in view of Lemm 2.4() nd the xiom (L), proving (). For (2), we hve x x x = x (x x ) by ()

10 56 Hnmntgoud P. Snkppnvr = x [(x x ) 0] = x [x 0] by (SH2) = 0. For (3), first, we note tht x x x = 0 by (2), since x = x. Now, x x = (x x ) (x x ) by (St) = 0 by wht ws noted erlier, which proves (3). Lemm 3.2. Let x L DQDStSH. Then () x = x (2) x x = (3) x = x. Proof. x x is immedite from Lemm 3.(3), while the other inequlity follows from Lemm 2.2(iv), so () is proved. Next, x x = x x = (x x ) = by (), proving (2). Finlly, using (2), we hve x (x x ) = x, which, by distributivity, implies tht x x. Since the other inequlity is well known, proof of (3) is complete. Lemm 3.3. Let x, y L DQDStSH. Then () (x y ) (x y ) = x (2) (x y ) = x y (3) (x y ) = x y. Proof. We hve (x y ) (x y ) = (x y ) (x y ) = [x (y y )] = x by (St), proving (). Next, we hve x y = y [(x y ) (x y ) ] by () = [y (x y ) ] [y (x y ) ] = (x y ),

11 Dully qusi-de Morgn semi-heyting lgebrs I 57 s y y (x y ) nd y y (x y ). Thus (2) is proved. Finlly, Thus (3) is proved. (x y ) = (x y ) = x y by (2) = x y. We re now redy to prove our min theorem of this section. Theorem 3.4. DQDStSH = SBDQDStSH Proof. (x y ) = (x y ) by Lemm 2.4() = (x y ) by Lemm 3.2(3) = x y by Lemm 3.3(3) = x y by Lemm 3.2(3). Thus DQDStSH SBDQDStSH. Since the other inequlity is trivil, the proof is complete. We conclude this section with the remrk tht there re lgebrs in SBDQDSH in which (St) fils; thus, DQDStSH is proper subvriety of SBDQDSH. We lso observe tht there re lgebrs to show tht DQDStSH nd DmsSH re incomprble elements in the lttice of subvrieties of DQDSH. 4 Simple lgebrs in RDQDStSH We now introduce the (sub)vriety RDQDSH of regulr dully qusi-de Morgn semi-heyting lgebrs. (Recll tht x + = x.) Definition 4.. Let L DQDSH. Then L is regulr if L stisfies: (M) x x + y y. The vriety of regulr DQDSH-lgebrs will be denoted by RDQDSH.

12 58 Hnmntgoud P. Snkppnvr Remrk 4.2. The reder is cutioned here not to confuse this notion of regulrity with nother notion of regulrity defined by the identity: x x (see [9], [20]). In generl, the two notions re independent of ech other. In this pper we do not hve the occsion to use the other notion, so there is no confusion. The purpose of this section is to give n explicit description of simple lgebrs in the vriety RDQDStSH. Since RDQDStSH SBDQDSH by Theorem 3.4, we re going to obtin such description s n ppliction of the following theorem which is immedite from Corollries 7.6 nd 7.7 of [20]. Theorem 4.3. Let L BDQDSH with L 2. Then the following re equivlent: () L is simple (2) L is subdirectly irreducible (3) () Cen L = {0, } or Cen L = {0,,, } with =, nd (b) For every x L, if x, then t (x) = 0. We first show tht the condition (3)() in the bove theorem is redundnt. For this we need the following lemm. Lemm 4.4. Suppose L BDQDSH stisfies the condition (3)(b) of the preceding theorem. Let L such tht = nd / {0, }. Then () = (2) (3) = (4) = 0 or (5) =.

13 Dully qusi-de Morgn semi-heyting lgebrs I 59 Proof. From ( ) =, we get =, implying ( ) =, nd hence, which leds us to conclude (). To prove (2), if =, then = = 0 in view of (), contrry to the hypothesis. Hence,. Then from hypothesis (3)(b) we hve ( ) =. Therefore, = ( ) = by () nd the hypothesis. So, we get, proving (2). For (3), from = 0 we get =, since =, implying ; hence. On the other hnd, we hve [( 0) 0] = (0 0) by Lemm 2.2(x), which simplifies to, proving (3). Suppose. Then = 0 by hypothesis. Hence =, which simplifies to =, in view of Lemm 2.4() nd item (3) bove, from which we conclude. Thus we hve = or, which implies = 0 or, which, in view of Lemm 2.4(), proves (4). Suppose. Then = 0 by (4), whence = 0. Then by (3) = 0, implying =, which is contrry to the hypothesis. Hence, which combined with (2) gives =, proving (5). The following corollry is now immedite from Theorem 4.3 nd (5) of Lemm 4.4. Corollry 4.. Let L BDQDSH with L 2. Then the following re equivlent: () L is simple (2) L is subdirectly irreducible (3) For every x L, if x, then x x = 0. Unless otherwise stted, in the rest of this section we ssume tht L RDQDStSH nd stisfies the following simplicity condition (which is the condition (3) of the preceding corollry): (SC) For every x L, if x then x x = 0. Lemm 4.5. Let x, y L. Then () x (x + y y ) = x (y y ) (2) x x x + =.

14 60 Hnmntgoud P. Snkppnvr Proof. We hve proving (). Next, which proves (2). x (y y ) = x [(x x + ) (y y )] by (M) = (x x + ) [x (y y )] = x (x + y y ), x x x x (x x ) x (x x ) (x x ) = [(x x ) (x x ) ] = [(x x ) (x x )] by (L) = [(x x ) (x x )] = 0, since is the pseudocomplement =, It should perhps be remrked here tht the condition (SC) is not used in the bove lemm. Lemm 4.6. Let x, y L. Then () x = or x x (2) x y = or x (x y) (3) x y or x y =. Proof. Suppose x. Then x x = (x x ) (x x ) by (SC) = x (x x ) = x (x x x ) by Lemm 4.5() = x by Lemm 4.5(2) = x, which proves (). From () we hve x y = or x y (x y). The ltter clerly implies x (x y), proving (2). From (2) we get x y = or x (x y ). But (x y ) y y. Hence (3) holds.

15 Dully qusi-de Morgn semi-heyting lgebrs I 6 Lemm 4.7. Let, b L such tht 0 < < b <. Then () b (2) = or = b (3) = (4) = (5) = 0 (6) = 0. Proof. Clim : b +. We hve, from Lemm 4.6(3), b + or b + =. Suppose the ltter holds, then b ( b ) = b, which simplifies to b, s b b b b = 0 since b. Then we hve contrdiction to the hypothesis < b, so Clim is proved. From Lemm 4.6(3), b or b + =. The former would imply b b b = 0 by (SC), contrry to the hypothesis. Hence the ltter holds, from which, using Clim, we hve b + =. Hence by (M) we get b = b = b b +, proving (). Suppose. Then from Lemm 4.6(), b b; thus b. On the other hnd, s b, b b by Lemm 4.6(), nd we lredy know b, hence b, implying = b nd thus proving (2). Now if, then = b by (2) nd by Lemm 4.6(). So b = contrdiction, so =, proving (3). For (4) we first need to prove the following Clim 2: or =. For, suppose. Then, by Lemm 4.6(3), =, whence ( ) = 0, nd so, ( ) = 0. Hence, it follows from (3) tht = 0 nd so =, proving Clim 2. Thus we hve or =, the former of which would imply, leding to contrdiction since 0. Therefore, =, which proves (4). Next, 0 = = ( ) = ( ) = by (4), thus proving (5). Finlly, observe tht = by (5), implying = 0, proving (6). Lemm 4.8. If L RDQDStSH stisfies the simplicity condition (SC), then L is of height t most 2 (tht is, no chin in L is of crdinlity 4).

16 62 Hnmntgoud P. Snkppnvr Proof. Suppose there re, b L such tht 0 < < b <. Using Lemm 4.7(), we get b. But = 0 by Lemm 4.7(6). Thus b, which is contrdiction, proving the lemm. We re redy to prove the min theorem of this pper. Theorem 4.9. Let L RDQDStSH with L 2. Then the following re equivlent: () L is simple (2) L is subdirectly irreducible (3) For every x L, if x, then x x = 0 (4) L is of height t most 2 (5) L {2 e, 2 e } C 20 {D, D 2, D 3 }, up to isomorphism. Proof. (3) (4) by Lemm 4.8. Suppose (4) holds. Then it is cler tht the lttice reduct of L is chin of size 2, 3, or is 4 element Boolen lttice. Observe tht ll lgebrs listed in (5) re in RDQDStSH nd re the only ones of height t most 2. Hence L is isomorphic to one of the lgebrs in (5). Thus (5) holds; whence (4) (5). As it cn be esily verified tht ll lgebrs listed in (5) re simple, (5) (). The rest of the implictions follow from Corollry 4.. In concluding this section we point out tht RDQDStSH is proper subvriety of SBDQDStSH. References [] M. Abd, J.M. Cornejo nd J.P. Díz Vrel, The vriety of semi-heyting lgebrs stisfying the eqution (0 ) (0 ), Reports on Mthemticl Logic 46 (20), [2] M. Abd, J.M. Cornejo nd J.P. Díz Vrel, The vriety generted by semi-heyting chins, Soft Computing 5 (20), [3] M. Abd nd L. Monteiro, Free symmetric Boolen lgebrs, Revist de l U.M.A. 27 (976),

17 Dully qusi-de Morgn semi-heyting lgebrs I 63 [4] R. Blbes nd PH. Dwinger, Distributive Lttices, Univ. of Missouri Press, Columbi, 974. [5] S. Burris nd H.P. Snkppnvr, A Course in Universl Algebr, Springer- Verlg, New York, 98. The free, corrected version (202) is vilble online s PDF file t mth.uwterloo.c/ snburris. [6] G. Grätzer, Lttice Theory, W.H.Freemn nd Co., Sn Frncisco, 97. [7] A. Horn, Logic with truth vlues in linerly ordered Heyting lgebrs, J. Symbolic. Logic 34 (969), [8] B. Jónsson, Algebrs whose congruence lttices re distributive, Mth. Scnd. 2 (967), 0-2. [9] V.Yu. Meskhi, A discrimintor vriety of Heyting lgebrs with involution, Algebr i Logik 2 (982), [0] A. Monteiro, Sur les lgebres de Heyting symetriques, Portuglie Mthemic 39 (980), [] W. McCune, Prover9 nd Mce 4, [2] H. Rsiow, An Algebric Approch to Non-Clssicl Logics, North Hollnd Publ.Comp., Amsterdm, 974. [3] H. Rsiow nd R. Sikorski, The Mthemtics of Metmthemtics, Wrszw, 970. [4] H.P. Snkppnvr, Heyting lgebrs with dul pseudocomplementtion, Pcific J. Mth. 7 (985), [5] H.P. Snkppnvr, Pseudocomplemented Okhm nd De Morgn lgebrs, Zeitschr. f. mth. Logik und Grundlgen d. Mth. 32 (986), [6] H.P. Snkppnvr, Heyting lgebrs with dul lttice endomorphism, Zeitschr. f. mth. Logik und Grundlgen d. Mth. 33 (987), [7] H.P. Snkppnvr, Semi-De Morgn lgebrs, J. Symbolic. Logic 52 (987), [8] H.P. Snkppnvr, Semi-Heyting lgebrs: An bstrction from Heyting lgebrs, Acts del IX Congreso Dr. A. Monteiro (2007), [9] H.P. Snkppnvr, Semi-Heyting lgebrs II. In Preprtion. [20] H.P. Snkppnvr, Expnsions of semi-heyting lgebrs. I: Discrimintor vrieties, Studi Logic 98 (-2) (20), 27-8.

18 64 Hnmntgoud P. Snkppnvr [2] H.P. Snkppnvr, Expnsions of semi-heyting lgebrs. II. In Preprtion. [22] J, Vrlet, A regulr vriety of type 2, 2,,, 0, 0, Algebr Universlis 2 (972), [23] H. Werner, Discrimintor Algebrs, Studien zur Algebr und ihre Anwendungen, Bnd 6, Acdemie Verlg, Berlin, 978. Hnmntgoud P. Snkppnvr, Deprtment of Mthemtics, Stte University of New York, New Pltz, NY 256 Emil: snkpph@newpltz.edu