iJEMONSTRATTO MATHEMATICA Vol IX No4 1976
Anna Romanowska
ON SOME EQUATIONAL CLASSES OF DISTRIBUTIVE DOUBLE P-ALGEBRAS
1. Introduction Recently, T. Katrinak [5] showed that the two, three and four element chains are the only subdirect irreducible double Stone algebras and the lattice of equational subclasses of the equational class of all double Stone algebras forms a four element chain. R. Beazer [1] characterized the simple distri- butive double p-algebras. This paper is concerned with some other subdirectly irreducible distributive double p-algebras and with equational classes generated by these algebras. As corollary, the lattice of some equational classes of regular distributive double p-algebras is presented.
2. Preliminaries A universal algebra < L; V, A,*, 0, 1 > of type <2, 2, 1, 0, is called a p-algebra iff
called a Fn~algebra (F^-algebra) iff L satisfies the identity
(xyV.. .AxQ)*V (X*AX2A. . .AXq)*V. .. V(x1 A ... Ax*)* = 1 (P*)
+ + ((x1V.. .Vxk) A(x^Vx2V. ..Vxk) A. ..A(x1 V... Vx+f =0 (Pj)).
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A ¿dp-algebr^a whic. h satisfies the identities Pn* and Pj$i. is called a P^P^-algebra. A universal algebra < Lj V, A, *,0,1> of type <2, 2, 2, 0, 0> is called a Heyting algebra iff
VV-VV1' For any class of (similar) algebras, let 7fi be the smallest equational class that contains % . Let H(3i), S(3(),
P(3(), Pg(.#), Pu(^) be the classes consisting, respectively, of all homomorphic imagis, subalgebras, direct products, sub- direct products and ultraproducts of members of di . It is known (see [j]) that = HSPOO = HP0b(3C ) = P S HSPU.n (#). Of prime importance for this paper is the known Lemma of ¿'onsson and corollaries from this Lemma (see [3]). Lemma 2. [3l If JK. is a class of algebras and for all AeCK, the lattice of all congruences relations on A is
- 594 - On some equational classes 5 distributive, then every subdirectly irreducible algebra in the equational class 3ie belongs to HSP„(Ji). Let L be a ddp-algebra. Since the lattice of dp-algebra congruences on L is a sublattice of the lattice of lattice congruences on L, it is distributive. Hence, we can use the Lemma 2 and corollaries from this Lemma for this algebra.
3. Subdirectly irreducible ddp-algebras For Boolean algebras A, B with smallest elements 0, s and units e, 1, respectively, set A@ B = AUB and let x < s for all xfcA. Similarly, set AS B = AUB and i let x < s for all x eA and x ^ e and let e = s. Moreover, let Bj j denote the lattice isomorphic to 2 © 2J and B. " i i lì denote the lattice isomorphic to 2 g 2° with i,0=0,1,2,: (Several examples are presented on the Pig. 1).
B oo Bjf 1,« Fig.1
Note that is isomorphic to B^ and is iso- morphic to B^. Again, for- x 8 A © 3 (or A Q B)
0 if x > s
x' if X eA, X ^ 0
1 if x = 0,
- 595 - 4 A.Romanowska where x' is the complement of x in A and
0 if x = 1 x+ = < x1 if x e B, x ^ 1 1 if x < e,
where x' is the complement of x in B. Let (P be the class of all ¿dp-algebras of *the form :3 A @ B, where A and B are Boolean algebras and ?a) = 3 ".
Theorem 3. A ddp-algebra L from the class J>w is subdirectly irreducible iff it is isomorphic to A © B or A E) B for'some Boolean algebras A and B. proof. Let L be isomorphic to A © B or A BE B for some Boolean algebras A and B and 0 be a nontrivial congruence on L. If 0 = a(9) and 0 £ a eA then 0* = 1 = a*(6). Since a*eA, then x = y(0) for all x,y in B. Let beB and b £ 1. Then b = 1(6). Hence 1+ = 0 3 b+(6), but b+€ B. Therefore 0 is the trivial congruence (the greatest congruence on L). Hence there is no a £ 0 such that a = 0(0). [O]0 = {0} and similarly, [1]0 ={1}. Therefore, if x,yeA then x = y(0) iff x = y. Analogously, if x,yeB then x = y(0) iff x = y. Hence, if e = s then L is simple. If e ^ s then © is the nontrivial congruence which identifies e and s and 0 is the unique nontrivial congruence on L. The property "Algebra L belongs to the class iP " can be expressed in the first order language. Therefore every alge- bra belonging to the class ^(«P) is isomorphic to A © B for some Boolean algebras A and B and by Lemma 2 every subdirectly irreducible ddp-algebra from the class (P^ is isomorphic to A © B or A @ B for some Boolean algebras A and B. Theorem 4. A p£,p^-algebra from the class is subdirectly irreducible iff it is isomorphic to a subalgebra of Bij or Bii1)- 1) ' I would like to thank Prof. R. Freese for the remarks re- lating to this theorem. - 596 - Oil some equational classes 5
Proof. Let L be a subdirectly irreducible pf,pt-algebra from the class . By Theorem 3, L = A © B i J or A @ B, where A and B are Boolean algebras. If A is infinite or A = 2n and n > i, then there exist x0,x/],...,xie A with XjAxk = 0, j i k, j,k = 0,1,...,i and x*v x£v ... Vx* £ 1. By Lemma 1, L does not satisfie P*. Hence A = 2n with n £ i. Similarly' B = 2m with m< j. Of course, A © B is a subalgebra of B. . and i j A 0 B is a subalgebra of B... 13 x- + Now, we show that B.XJ. and B.1J. are P.,P.-algebras1 J . B*(B..) and B*(B..) are isomorphic to 2*. Therefore there x Ax for exist xQfxyj.. in B (B^) such that jj- -j = 0 k ^ j and k,j = 0,1,...,i (similarly in B*(B. .)). x, = 0 r 1 for some k e 10,1,... ,i j, xQ ,... »x^ »xk+1 ,... ,x. are atoms. Hence V...Vx. = 1 and by Lemma 1, B^ and B.. satisfie P.*. Dually we can prove that B., . and B. . J-J i ij satisfie Pt. Any subalgebra of these algebras satisfies u P-4,P^i, too. ^ is easy to see, that the subalgebras of B. . J- j lj are B^ with k = 0,1,..., i and n = 0,1,... and sub- algebras of B^j are B^.^ for k = 0,1,...,i and n=0,...j. By Theorem 3, B^ and B^ are subdirectly irreducible. Corollary 5. A P*,P*-algebra from the class is subdirectly irreducible iff it is isomorphic to a sub- algebra of B^. Corollary 6. A ddp-algebra from the class , in which the identity P^f(Pt) holds is subdirectly irredu- " i cible iff it is isomorphic to a subalgebra of 2 © A or 21 H A (A © 2«> or A 0 ) for some Boolean algebra A. In particulary, a double P^-algebra (P^-algebra) from the class 3>w is subdirectly irreducible iff it is isomorphic to a subalgebra of 2 © A (A ©2) for some Boolean algebra A.
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4. Equational subclasses of Let 3i be the class of all ddp-algebras of the form A © 2 A or 2 © A for some Boolean algebra A and Jioj = 3J . Let ^ijkn ^ IJ-2,...; k,n = 0,1,2,...) denote the equational subclasses of the class containing all algebras B and not B B B V 1n i+1 o' o j+1' 1' 1 n+1" Lemma 7. ' The lattice P- of all equational sub- classes of 3ito satisfying the identity P^ is isomorphic to the distributive lattice on the Pig.2. Proof. A) The lattic* + e rLv (i,k = 1,2,...) of all equational classes of P^jP^-algebras from the class 3i>w is isomorphic to the lattice on the Pig.3« Since, for every ¿j = 1,2,..., B^ is a subalgebra of ^ and for every p = 1,2,... jB^ is a subalgebra of
B1 p+1' follows "bila'fc
^iklp5 ^ik2p £ ••• £ ^ikip for every p = 1,2,... ,k and ^ikjl £ ®ik;J2 £ S®ik;Jk for every j = 1,2,...,i. By Cor. 3.4 in [3], these inclu- c: sions are proper. Of course, •^i]j:00 ^ikri there axists no the equational subclass <& of JSw such that ^ C3iik1 V By Theorem 4, every subdirectly irreducible algebra in the class is isomorphic to a subalgebra. of B^ or B^^. Let CH be an equational subclass of 3Jto such that ®ik11^ ^^^ikik* i^egers such that ^^XK^^' %kl+1 K ^ aad ^ A is a subdirectly irreducible algebra in the class tK., then A is isomorphic to Bi0, Bok or Bjp (0 = 1,2,...,1, p = 1,2,...,K). Thus A.£3^ikop for some j ,p and 3i = fl^ .
Let 0 B B 2 A ^iUpk = wtiho) { PiM ikM a » where i,k are fixed natural numbers, p = 1,2,...,i and
D0 is the class of all Boolean algebras. - 598 - On some equational classes 7
B) The lattice of all equational subclasses of Ji^ satisfying the condition ^^p^ is isomorphic to the following chain C^ ^
®ikpk C ^ik+1 pk C " ' C Bicopk '
Since, for every j = k,k+1,..., is a subalgebra of Boj+1' follows that ^ijpk5 ^ij+lpk' By Cor' in ^ this inclusion is proper. Analogously as in Th. 4 in [6] we can prove that ^^p^ = V Oi-jpk» d ai1oo C "®i2oo C • • • C \ooo • Let ^icopaj = HSP({Bi0}u (Bp1}u{2 © A, Ae»0}), where i is a fixed natural number and p = 1,2,...,i. C) The lattice Liw (i = 1,2,...) of all equational subclasses X of tisfying the condition di J5i(jir for r = 1,2,..., to is isomorphic to the chai c n c c „ Ji. „ ... ^ . iu)1 r iu2r iwir (The proof similar as in A)) and the lattice of all equational subclasses CK. of such that i(i)p1 35.iiijpit, ) for p = 1,2,...,i forms the chain c c 3biwp. „l J5.iup 2, iwpo) - 599 - 8 A.Romanowska (The proof is similar to the proof of B)). Moreover, ^i^oo0 cB ana there exists no the equational subclass of c c 33w such that ®iuoo * liia)ir Let i be a fixed natural number and let ^^o^*^ ^ ^ r , -i D) There are numbers ie{1,2,...f and j e {1,2,... 5 such that is an element of L^. and there are natural numbers p€{o,1,...,i} and r e{0,1,2,...} such that, is an element of C.^, or 3( is of the form with p = 1,2,...,i. This is easy to see from Cor. 4.1 in [3], because ^ = %;jpr ' We also got in the proof of the Lemma 7 following corol- laries* Corollary 8. All equational classes of P*«^- -algebras from the class 3JW form the lattice on the Pig.3. In particulary, all equational classes of P^,P*-algebras from the class 33w form the following chain ^iloo C ®i111 C ®i121 C •••C®i1i1 ' Corollary 9» All equational classes of double * P^1i algebras form the lattice on the Fig.4. Let Pw denote the lattice of all equational subclasses 3C of such that »WW(lMl)= (\1oo = HSP({a © 2°; A£S0}), &uwuw= HSP({A ©2; Ae»0}u{2 © A; AeRQ}). The lattice Pw can be constructed in similar way as the lattice on the Fig.2. Corollary 10. Every equational subclass of 31w is an element of the lattice P^ for some i = 1,2,...; GO. Let ^i^pp (i»k = 1,2,,..,r,p = 0,1,...) denote the equational"~iubclasses of the class iPw containing all alge- bras Br1, B1p and Bik and not Br+1 v B1p+1, B.+1 k+r Let (i,k = 1,2,..77 denote the equational subclasses of the class containing all algebras and not Bi+1 k+1 ' - 600 - On some equaticnal classes 9 Theorem 11. The lattice P^ of all equational subclasses of satisfying the identity P^ is isomorphic to the distributive lattice on the Fig.5. Proof. Let K^jj., K^, K^ denote the lattices of all equational subclasses 3C~of satisfying the conditions ^kooeX>£ *ikik Kik' »itoo"^ for Lik' B G3tS HSP 2i A i1oo ^10)1(0= '^ a ' AeB0}u{Bi,}u{2 (5) A, AeB0}) for Klt where VC ^ ^ikrp for k = 2,3,... ,cu; r = 0,1.. ,i; p = 0,1 ,...,£o and the condition ^iloo^ ^^ X S35i(0= HSP({2 © A, Ae30}) for all elements X~of the lattice K^, where ^^^ikrp k = 2,3, •••»<«>; r = 0,1,...,i; p = 0,1,...,w. It is easy to check that the mapping f: P^ " K^ defined by ^¿jj-rp-*" ®itrp is isomorPhisia (k = 1,2,... r = 0,1,...,i; p = 0,177.. and JJ^^ = HSP({2X (+] A, i AeJi0}u{Br1}u{B1p}), HSP({2 © A, A e Ji>0} U {Br1} U u|2 © A, AeBQj)). f restricted to L^ is the isomophism between L^ and (The lattice of all equational sub- classes of satisfying the identities P* and P^ is presented on the Pig.6). Moreover covers ^^rp and = HSF( B U B ^ikit { ikl { itl ^ covers 3$7^it, where k=2,3,...;oJ and t = 2,3,.77,k and 3iiicik: Corollary 12. All equational classes of double P^-algebras from the class J^ form the lattice on the Pig. Let Pw denote the lattice of all equational subclasses 3t of '^"such that B^ooS* £ ^o^oo = HSP(tA 0 2» AeRQ|)). The lattice can be constructed in a similar way as the lattice on the Fig.5. Corollary 13. Every equational subclass of iPy is an element of the lattice P^ for some i = 1,2,... ;oi. 5. Regular equational subclasses of An algebra L is said to be regular if any two con- gruences 7iith a class in common are equal. An equational - 601 - 10 A.Romanowska class, in which every algebra is regular, is said to be regular. Varlet [7] showed that a dp-algebra is regular iff it satisfies the following condition x* = y* and x+ = y+ imply x = y. (M) It is easy to see that every subalgebra of A g] B, where A and B are Boolean algebras, satisfies the condition (M) aad hence every subairect product of these algebras satisfies (M) too. therefore, we have. Corollary 14. The class 3 0„= HGP({A H B; A,BeBQ}) is the equational class of alF"regular algebras from the class . Corollary 15. All equational classes of regular algebras from the class form the lattice on the Fig,7. Lemma 16. [4] ©very regular dp-algebra is a dH-alge- bra. Corollary 17. Every algebra in the class 00 is a dH-algebra. Moreover, each equational subclass of Ji^oo is the equational clas3 of dH-algebras. Proof. Since every dH-algebra is a dp-algebra and each congruence relation of the regular dp-algebra L is a congruence of the dH-algebra L (see [4-]), we obtain the desired conclusion. - 602 - Oli some equational classes •11 - 603 - 12 A. Romano Y,'ska Fig. 4 - 604 - On some equational classes 13 - 605 - 14 A. Romanowska - 606 - On some equacional classes 15 REFERENCES [" 1 J P.. 3 e a z e r: The determination congruence oil double p-algebras, preprint. [2] G. G r a t z e r: Lattice theory. First concepts and distributive lattices, 1971. [ 3 I B. J o n s s 0 n: Algebras whose congruence lattices are "" distributive, Math. Scand. 21- (1967) 110-121. [ 4 ] ï. K a t r i n a k: The structure of distributive double p-alge^ras. Regularity and congruences. Algebra Univ. 3 (1973; 238-246. ("51 T. £ a t r 1 n a k: Infective double Stone algebras. Algebra Univ. 4 (1974) 259-267. [ 6] H. L a k s e r: The structure of pseudocomplemented distributive lattices 1.. Subdirect decomposition, Trans. Amer. Math. Soc. 156 (197'i) 335-342. [7I J. V a r 3. e t! A regular variety of type <2.2,1,1,0,0>, L J Algebra Univ. 2 (1972) 218-223. INSTITUTE OF MATHEMATICS, TECHNICAL UNIVERSITY OF WARSAW Received December 5» 1975. - 607 -