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On Some Equational Classes of Distributive Double P-Algebras

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On Some Equational Classes of Distributive Double P-Algebras 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 <L; V , A , 0, 1 > is a bounded lattice such that for every a £ L the element a*e I jL is the pseudocomplement of a, i.e. x ^ a iff a A x = 0. A universal algebra <L; V,A,*, +, 0, 1> is called a dp-al- gebra (double p-algebra) iff V, A,*, 0, 1 > is a p-alge- bra ana <Cl; V, A, 0, 1 > is a dual p-algebra (x > a+iff x V a = 1). A distributive dp-algebra is called a ddp-algebra. A distributive p-algebra (dual p-algebra) is 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)). - 593 - 2 A.Romanowska 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 <L; V,A, 0, 1> is a bounded lattice such that for all a,beL the element a^b € L is the relative pseudocomplement, i.e. x < a^b iff aAi ^ b. Analogously we define a dual Heyting algebra <L;V ,A, 0, 1 >. It is known that a Heyting algebra is always distributive. An algebra <L;V»A,*, 0, 1> is called a dH-algebra (double Heyting algebra) iff L is a Heyting algebra and a dual Heyting algebra as well. A dH-algebra is a dp-algebra. If in any p-algebra L we write B (L) = •[ x e L; x = x**} then ,Af , 0, 1) is a Boolean algebra when a b is defi-ccl to be (a^Ab*)*. Dually, if in any dual p-algebra + ++ + + L, we write B (L) = {xeL; x = x } then <B (L);V,Of f 0, 1> is a Boolean algebra when anb is defined to be (a+ V b+)+. Lemma 1. [2~\ Let L be a distributive p-algebra. Then the following conditions are equivalent: (1) L is a P*-algebra. (2) L satisfies the implication: if x^A x^ = 0, i ^ j, = 0,...,n, then x* v V ... V x* = 1. (3) L satisfies the implication: if x^ A x^ = 0, i ^ = 0,...,n, txQ6 B (L), then 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. - 597 - 6 A.Romanowska 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.
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