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

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1 Volume 2, Number, July 204, ISSN Print: Online: Dully qusi-de Morgn Stone semi-heyting lgebrs II. Regulrity Hnmntgoud P. Snkppnvr Abstrct. This pper is the second of two prt series. In this Prt, we prove, using the description of simples obtined in Prt I, 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 our min theorem, we present (equtionl) xiomtiztions for severl subvrieties of RDQDStSH. The pper concludes with some open problems for further investigtion. Introduction This pper is the second of two prt series. In this Prt, we prove, using the description of simples obtined in Prt I, 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 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. 65

2 66 Hnmntgoud P. Snkppnvr 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 pper concludes with some open problems for further investigtion. 2 Preliminries In this section we recll nottions nd results from Prt I in order to mke this pper self-contined. An lgebr L = L,,,,, 0, is 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. Let L be DQDSH-lgebr. Then L is of level (DQDSH -lgebr) if L stisfies: (L) x x (x x ) (Level ). L is dully pseudocomplemented semi-heyting lgebr (DPCSH-lgebr) if L stisfies: (e) x x. L is De Morgn semi-heyting lgebr (DMSH-lgebr) if L stisfies: (DM) x x. L is regulr if L stisfies: (M) x x + y y, where x + := x. L is dully qusi-de Morgn Stone semi-heyting lgebr (DQDStSH-lgebr) if L stisfies:

3 Dully qusi-de Morgn semi-heyting lgebrs II 67 (St) x x, where x := x 0. L is dully qusi-de Morgn Boolen semi-heyting lgebr (DQDBSH-lgebr) if L stisfies: (Bo) x x where x := x 0. L is strongly blended dully qusi-de Morgn semi-heyting lgebr (SBDQDSH) if L stisfies: (SB) (x y ) x y (Strongly Blended -De Morgn lw). The vriety of DQDSH -lgebrs is denoted by DQDSH, nd similr nottion pplies to other vrieties. DQDStSH denotes the subvriety of level of DQDSH defined by (St), nd DQDBSH denotes the one defined by (Bo), while RDQDSH denotes the vriety of regulr DQDSH -lgebrs nd RSBDQDSH denotes tht of regulr, strongly blended DQDSH -lgebrs, nd so on. 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 in this sequel. Let 2 e nd 2 e be the expnsions of the semi-heyting lgebrs 2 nd 2 (shown in Figure ) by dding the unry opertion such tht 0 =, = 0. Let L dp i, i =,..., 0, denote the expnsion of the semi-heyting lgebr L i (shown in Figure ) by dding the unry opertion such tht 0 =, = 0, nd =. Let i, i =,..., 0, denote the expnsion of L i by dding the unry opertion such tht 0 =, = 0, nd =. We Let C dp 0 i =,..., 0} nd C dm 0 := {Ldm := {Ldp i : i : i =,..., 0}. We lso let C 20 := C dm 0 Cdp 0. Ech of the three 4-element lgebrs D, D 2 nd D 3 hs its lttice reduct s the Boolen lttice with the universe {0,, b, }, b being the complement of, hs the opertion s defined in Figure, nd hs the unry opertion defined s follows: =, b = b, 0 =, = 0. It ws shown in [0] tht V(D, D 2, D 3 ) = DQDBSH.

4 68 Hnmntgoud P. Snkppnvr 2 : : L : L 2 : L 3 : L 4 : L 5 : L 6 : L 7 : L 8 : L 9 : L 0 :

5 Dully qusi-de Morgn semi-heyting lgebrs II 69 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 The following two results proved in Prt I. Theorem 2.. 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 {2 e, 2 e } C 20 {D, D 2, D 3 }, up to isomorphism. Theorem 2.2. DQDStSH = SBDQDStSH. 3 Min result nd consequences Let V(K) denote the vriety generted by the clss K of lgebrs. Recll V(C 20 ) is the vriety of DQDSH-lgebrs genrerted by the twenty 3- element lgebrs mentioned erlier. Let D denote the vriety generted by the three 4-element DQDSH-lgebrs whose semi-heyting reducts re D,

6 70 Hnmntgoud P. Snkppnvr D 2 nd D 3 given in Figure. The vriety V(C 20 ) ws xiomtized in [0] nd lso it ws shown there tht D = DQDBSH. We re now redy to give the min result of this pper. Theorem 3.. We hve RDQDStSH = V((C 20 ) {D, D2, D3}) = V(C 20 ) D = RSBDQDStSH = RDmsStSH. Proof. Since 2 e nd 2 e re sublgebrs of some of the other simple lgebrs listed in Theorem 2., the first eqution is immedite from Theorem 2., using well known results from universl lgebr (see [2]). The second eqution follows from the first, using the definition of the join of two vrieties, nd the third eqution is immedite from Theorem 2.2. To prove the lst eqution, it suffices to verify tht ll 25 simple lgebrs in RDQDStSH stisfy the identity: (x y) x y. If we restrict the underlying semi-heyting lgebrs to Heyting lgebrs, Theorem 3. reduces to the following Corollry 3.2. Let RDQDStH denote the subvriety of RDQDStSH defined by: (x y) x. Then RDQDStH = V({, L dp, D 2}) = V( ) V(L dp ) V(D 2) = RSBDQDStH = RDmsStH. Proof. Verify tht 2 e,, Ldp, D 2 stisfy the identity: (x y) x, while the remining simple lgebrs mong the twenty five listed in Theorem 2. do not. Also, observe tht 2 e is sublgebr of (or L dp ). Then the corollry follows from Theorem 3.. The following corollries, which give (equtionl) bses to severl subvrieties of RDQDStSH, cn lso be similrly deduced from Theorem 2. nd Theorem 3..

7 Dully qusi-de Morgn semi-heyting lgebrs II 7 In these corollries the reder should interpret defined by s defined, modulo RDQDStSH, by. Corollry 3.3. RDQDStH is lso defined by the linerity identity: (x y) (y x). It is lso defined by: x (x y) = (x y) x. Corollry 3.4. We hve () RDMStSH = V(C dm 0 ) D (b) RDPCStSH = V(C dp 0 ). (c) RDMStH = V({, D 2}) = V( ) V(D 2) (d) RDPCStH = V(L dp ). Corollry 3.5. Let RDQDcmStSH be the subvriety of RDQDStSH defined by the commuttive lw: x y y x. Then () RDQDcmStSH = V({ 0, Ldp 0, D }) = V( 0 ) V(Ldp 0 ) V(D ) (b) RDMcmStSH = V({ 0, D }) (c) RDPCcmStSH = V(L dp 0 ) (d) RDMcmStSH RDPCcmStSH = V( 2 e ). Corollry 3.6. The vriety V({, Ldp identity: (x y) (0 y) (x y)., Ldm 3, Ldp 3, D 2}) is defined by the The vriety generted by D ws xiomtized in [0]. Here re two more bses for it. Corollry 3.7. V(D ) is defined by

8 72 Hnmntgoud P. Snkppnvr x (y z) z (x y). It is lso defined by (x y) (u w) (x u) (y w) (Medil Lw). Corollry 3.8. The vriety V({, Ldp, Ldm 2, Ldp 2, D 2}) is defined by: y x y. It is lso defined by: [(x y) y] (x y) x y. It is lso defined by x (y z) (x y) (x z) (Left distributive lw). Corollry 3.9. The vriety V({, Ldp, Ldm 2, Ldp 2, Ldm 5, Ldp 5, Ldm 6, Ldp 6, D 2}) is defined by: [x (y x)] x x. Corollry 3.0. V({L dp, Ldp 2, Ldp () [x (y x)] x x (2) x x. 5, Ldp 6 }) is defined by: Corollry 3.. V({, Ldm 2, Ldm 5, Ldm 6, D 2}) is defined by: () [x (y x)] x x (2) x x. Corollry 3.2. The vriety V({, Ldp, Ldm 2, Ldp 2, Ldm 3, Ldp D 2, D 3 }) is defined by the identity: (0 ) + (0 ) 0. Corollry 3.3. The vriety V({L dp identities: 3, Ldm 4, Ldp 4, Ldm 5, Ldm 6, Ldm 7, Ldm 8,, Ldp 2, Ldp 3, Ldp 4 }) is defined by the

9 Dully qusi-de Morgn semi-heyting lgebrs II 73 () (0 ) + (0 ) 0 (2) x x. Corollry 3.4. The vriety V({, Ldm 2, Ldm 3, Ldm 4, Ldm 5, Ldm 6, Ldm 7, Ldm 8, D 2, D 3 }) is defined by the identities: () (0 ) + (0 ) 0 (2) x x. Corollry 3.5. The vriety V({ 5, Ldm 6, Ldm 7, Ldm 8, D 3}) is defined by the identity: (0 ) + (0 ) (0 ). V(D 3 ) ws xiomtized in [0]. Here is nother bse for it. Corollry 3.6. V(D 3 ) is defined by the identities: () (0 ) + (0 ) (0 ) (2) x x. Corollry 3.7. The vriety generted by the lgebrs, Ldm 2, Ldm 3, Ldm 4, D 2, D 3 is defined by the identities: () (0 ) + (0 ) (0 ) (2) (0 ) + (0 ) (0 ) (3) x x. Corollry 3.8. The vriety generted by the lgebrs 5, Ldp 5, Ldm 6, Ldp 6, Ldm 7, Ldp 7, Ldm 8, Ldp 8, Ldp 9, Ldm D, D 3 is defined by the identity: (0 ) + (0 ) (0 ). 9, Ldp 0, Ldm 0, Corollry 3.9. The vriety generted by the lgebrs L dp 5, Ldp 6, Ldp 7, Ldp 8, Ldp 9, Ldp 0 is defined by the identities:

10 74 Hnmntgoud P. Snkppnvr () (0 ) + (0 ) (0 ) (2) x x. Corollry The vriety generted by the lgebrs 5, Ldm 6, Ldm 7, Ldm 8, Ldm 9, Ldm 0, D, D 3 is defined by the identities: () (0 ) + (0 ) (0 ) (2) x x. Corollry 3.2. The vriety generted by the lgebrs 5, Ldm 6, Ldm 7, Ldm 8, D 3 is defined by the identities: () (0 ) + (0 ) (0 ) (2) (0 ) + (0 ) (0 ). It is lso defined by (0 ) 0. Corollry The vriety generted by the lgebrs D, D 3 is defined by the identities: () (0 ) + (0 ) (0 ) (2) x x. Corollry The vriety generted by the lgebrs, Ldp, Ldm 2, Ldp 2, Ldm 3, Ldp 3, Ldm 4, Ldp 4, Ldp 5, Ldp 6, Ldp 7, Ldp 8, 9, Ldp 9, Ldm 0, Ldp 0, D, D 2 is defined by the identity: (0 ) (0 ) 0. Corollry The vriety generted by the lgebrs, Ldm 2, Ldm 3, Ldm 4, Ldm 9, Ldm 0, D, D 2 is defined by the identities: () (0 ) (0 ) 0 (2) x x. Corollry The vriety generted by the lgebrs D, D 2 is defined by the identities:

11 Dully qusi-de Morgn semi-heyting lgebrs II 75 () (0 ) (0 ) 0 (2) x x. Corollry The vriety generted by the lgebrs, Ldp, Ldm 3, Ldp 3, Ldm 6, Ldp 6, Ldm 8, Ldp 8, D, D 2, D 3 is defined by the identity: x (y (x y)) (0 x) (x y). Corollry The vriety generted by the lgebrs 2, Ldp 2, Ldm 5, Ldp 5, D 2 is defined by the identity: x (y x) [(x y) y] x. Corollry The vriety generted by the lgebrs 3, Ldp 3, Ldm 4, Ldp 4, D, D 2, D 3 is defined by the identity: x (x y) x (x (y )). Corollry The vriety generted by the lgebrs 5, Ldp 6, Ldm 7, Ldp 8, D 3 is defined by the identity: (0 ) (0 ) (0 ). Corollry The vriety generted by the lgebrs, Ldp, Ldm 2, Ldp 2, Ldm 3, Ldp 3, Ldm 4, Ldp 4, D 2 is defined by the identity: 0 (FTT identity). Corollry 3.3. The vriety generted by the lgebrs, Ldp, Ldm 3, Ldp 3, Ldm 6, Ldp 6, Ldm 8, Ldp 8, D, D 2, D 3 is defined by the identity: x (y x) (x y) x. Corollry The vriety generted by the lgebrs L dp, Ldp 3, Ldp 6, Ldp 8 is defined by the identities: () x (y x) (x y) x (2) x x.

12 76 Hnmntgoud P. Snkppnvr Corollry The vriety generted by the lgebrs, Ldm 3, Ldm 6, Ldm 8, D, D 2, D 3 is defined by the identities: () x (y x) (x y) x (2) x x. Corollry The vriety generted by the lgebrs, Ldp, Ldm 2, Ldp 2, Ldm 5, Ldp 5, Ldm 6, Ldp 6, Ldm 9, Ldp 9, D, D 2, D 3 is defined by the identity: x (x y) (x y) y. Corollry V({L dp, Ldp 2, Ldp 5, Ldp () x (x y) (x y) y (2) x x. 6, Ldp 9 }) is defined by the identity: Corollry The vriety generted by the lgebrs, Ldm 2, Ldm 5, Ldm 6, Ldm 9, D, D 2, D 3 is defined by the identity: () x (x y) (x y) y (2) x x. Corollry The vriety generted by the lgebrs 5, Ldp 5, D 2 is defined by the identity: x (0 x) (y ) x [(x ) (x y)]. Corollry The vriety generted by the lgebrs 6, Ldp 6, D 2 defined by the identity: x y (x y) x [(x y) ]. Corollry The vriety generted by the lgebrs, Ldp, Ldm 7, Ldp 7, D 2 is defined by the identity: x [(0 y) y) x [(x ) y]. Corollry The vriety generted by the lgebrs 7, Ldp 7, Ldm 8, Ldp 8, D, D 2, D 3 defined by the identity:

13 Dully qusi-de Morgn semi-heyting lgebrs II 77 x [x (y (0 y))] x [(x y) y]. Corollry 3.4. The vriety generted by the lgebrs 8, Ldp 8, D, D 2, D 3 defined by the identity: x y [y (y x)] x [x (0 y)]. It is lso defined by the identity: x [y (0 (y x))] x y (y x). Corollry The vriety generted by the lgebrs 7, Ldp 7, Ldm 8, Ldp 8, Ldm 9, Ldp 9, Ldm 0, Ldp 0, D, D 2, D 3 is defined by the identity: x (x y) x [(x y) ]. Corollry The vriety generted by the lgebrs 2 e, L dp 7, Ldp 8, Ldp 9, Ldp 0 is defined by the identities: () x (x y) x [(x y) ] (2) x x. Corollry The vriety generted by the lgebrs 7, Ldm 8, Ldm 9,, Ldm 0, D, D 2, D 3 is defined by the identities: () x (x y) x [(x y) ] (2) x x. Corollry The vriety generted by the lgebrs 9, Ldp 9, Ldm 0, Ldp 0, D is defined by the identity: 0 0. (FTF identity) Corollry The vriety generted by the lgebrs 0, Ldp 0, D is defined by the identity: x y y x. (commuttive identity) A bse for V(C dp 0 ) ws given in [0]. We give some new ones below.

14 78 Hnmntgoud P. Snkppnvr Corollry The vriety V(C dp 0 ) is defined by : x x 0 (dul Stone identity). It is lso defined by : x x. A bse for V(C 20 ) ws given in [0]. We give new one below. Corollry The vriety V(C 20 ) is defined by : x x. A bse for V(2 e, 2 e ) ws given in [0]. We give new one below. Corollry The vriety V(2 e, 2 e ) is defined by : x x. Corollry The vriety generted by the lgebrs in : i =,..., 8} { i : i =,..., 8} {D 2 } is defined by the identity: {L dp i (x y) (x y ). It is lso defined by (0 ) 0. Corollry 3.5. The vriety generted by the lgebrs in, i =,..., 8}, is defined by the identities: {L dp i () (x y) (x y ) (2) x x 0 (dul Stone identity). Corollry The vriety generted by the lgebrs, Ldp, Ldm 2, Ldp 2, D 2 is defined by the identity: x z y (y z) (strong Kleene identity). Corollry The vriety generted by, Ldp, Ldm 2, Ldp 2, Ldm 5, Ldp 5, Ldm 6, Ldp 6, D 2 is defined by the identity:

15 Dully qusi-de Morgn semi-heyting lgebrs II 79 x y (x y) y. Corollry The vriety generted by L dp, Ldp 2, Ldp 5, Ldp 6 is defined by the identity: () x y (x y) y (2) x x. Corollry The vriety generted by, Ldm 2, Ldm 5, Ldm 6, D 2 is defined by the identity: () x y (x y) y (2) x x. The vriety D = V{D, D 2, D 3 } ws xiomtized in [0]. Here re two more bses for it. Corollry The vriety D is defined by the identity: x (y z) (x y) (x z). It is lso defined by the identity: x 2( ) x. We would like to mention here tht in the cse of either of the two bses in the preceding corollry the identities (St) nd (L) re consequences of the rest of the identities nd hence re redundnt. Corollry The vriety generted by 2, Ldp 2, D 2 is defined by the identity: (x y) x x. V(D 2 ) ws xiomtized in [0]. Here re some more bses for it. This vriety hs n interesting property in tht is definble in terms of. Corollry The vriety generted by D 2 is defined by the identity: x y (x y) y.

16 80 Hnmntgoud P. Snkppnvr It is lso defined by the identities: () x (y z) (x y) (x z) (2) (x y) x x. It is lso defined by the identity: x (x y) x ((x y) ). Corollry The vriety generted by, Ldp, Ldm 2, Ldp 2, Ldm 9, Ldp 9, D, D 2, D 3 is defined by the identity: x (y z) y (x z). Corollry The vriety generted by, Ldp, Ldm 2, Ldp 2, Ldm 5, Ldp 5, D 2 is defined by the identity: (x y) z ((y x) z) z. (Prelinerity) We conclude this section by mentioning tht one cn esily write down the bses for intersections of the vrieties mentioned in this section. Similrly, one cn lso esily determine the subvrieties of the vrieties considered in this section, obtined by dding the identity x x, or the identity: x x, to their bses occurring in the preceding corollries. 4 Conclusion nd some open problems It should be pointed out tht, bsed on the results from [0], ll vrieties ppering in Section 3. re discrimntor vrieities. It ws lso shown in [0] tht ll the twenty 3-element lgebrs in RDQDStSH re semipriml (see [2] or [0] for definition). A similr rgument proves tht D nd D 2 re semipriml s well; nd 2 e, 2 e, D3 re, in fct, priml. Thus the vriety RDQDStSH is generted by semipriml lgebrs. We would lso like to note here tht the lgebrs in {L i dm : i = 5,..., 8} {L i dp : i = 5,..., 8} re lso priml. From these observtions nd from the results of [0] we conclude tht the lgebrs in {2 e, 2 e, D3} {L i dm : i = 5,..., 8} {L i dp : i = 5,..., 8} re the only toms in the lttice of subvrieties of the vriety RDQDStSH.

17 Dully qusi-de Morgn semi-heyting lgebrs II 8 It is our view tht ech of the vrieties mentioned in Section 3 is worthy of further study, both lgebriclly nd logiclly. We conclude this pper with some more open problems for further investigtion. Problem. RDQDStSH. Find equtionl bses for the remining subvrieties of Problem 2. Give n explicit description of simple lgebrs in the vriety of regulr dully Stone semi-heyting lgebrs of level. Problem 3. Axiomtize logiclly ech of the subvrieties of RDQDStSH. (In other words, for ech subvriety V of RDQDStSH find propositionl logic P such tht V is n equivlent lgebric semntics for P.) In prticulr, the following problem is of interest. Problem 4. The 2-element, 3-element, 4-element lgebrs in Figure cn be viewed respectively s 2, 3 nd 4-vlued logicl mtrices. Axiomtize these lgebrs logiclly (with s the only designted truth vlue), using nd s impliction nd negtion respectively. (For the lgebr 2 in Figure, the nswer is, of course, well known: the clssicl propositionl logic.) Problem 5. Investigte the lttice of subvrieties of DPCStSH. Problem 6. Investigte the lttice of subvrieties of DMStSH. Problem 7. Investigte the lttice of subvrieties of the vriety of commuttive DMSH -lgebrs. Problem 8. subvrieties. Find dulity for RDQDStSH nd for ech of its References [] R. Blbes nd PH. Dwinger, Distributive Lttices, Univ. of Missouri Press, Columbi, 974.

18 82 Hnmntgoud P. Snkppnvr [2] 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. [3] B. Jónsson, Algebrs whose congruence lttices re distributive, Mth. Scnd. 2 (967), 0-2. [4] W. McCune, Prover9 nd Mce 4, [5] H. Rsiow, An Algebric Approch to Non-Clssicl Logics, North Hollnd Publ.Comp., Amsterdm, 974. [6] H.P. Snkppnvr, Heyting lgebrs with dul pseudocomplementtion, Pcific J. Mth. 7 (985), [7] H.P. Snkppnvr, Heyting lgebrs with dul lttice endomorphism, Zeitschr. f. mth. Logik und Grundlgen d. Mth. 33 (987), [8] H.P. Snkppnvr, Semi-De Morgn lgebrs, J. Symbolic. Logic 52 (987), [9] H.P. Snkppnvr, Semi-Heyting lgebrs: An bstrction from Heyting lgebrs, Acts del IX Congreso Dr. A. Monteiro (2007), [0] H.P. Snkppnvr, Expnsions of semi-heyting lgebrs. I: Discrimintor vrieties, Studi Logic 98 (-2) (20), [] H.P. Snkppnvr, Dully qusi-de Morgn Stone semi-heyting lgebrs I. Regulrity, Cteg. Generl Alg. Struct. Appl. 2() (204), Hnmntgoud P. Snkppnvr, Deprtment of Mthemtics, Stte University of New York, New Pltz, NY 256 Emil: snkpph@newpltz.edu