pseudocomplemented distributive lattices begun in Part I, [8}. Using the description of congruences given in Part I, we verify the Congruence. Extension Property ...
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 156, May 1971
THE STRUCTURE OF PSEUDOCOMPLEMENTED DISTRIBUTIVE LATTICES, II: CONGRUENCE EXTENSION AND AMALGAMATION BY
G. GRATZER AND H. LAKSER(l) Abstract, This paper continues the examination of the structure of pseudocomplemented distributive lattices. First, the Congruence Extension Property is proved. This is then applied to examine properties of the equational classes iii., -1~n~w, which is a complete list of all the equational classes of pseudocomplemented distributive lattices (see Part I). The standard semigroups (Le., the semigroup generated by the operators H, S, and P) are described. The Amalgamation Property is shown to hold iff n ~ 2 or n = w. For 3 ~ n < ro, iii. does not satisfy the Amalgamation Property; the deviation is measured by a class Amal (iii.) (s; iii.). The finite algebras in Amal (iii.) are determined.
O. Introduction. This paper continues the examination of the structure of pseudocomplemented distributive lattices begun in Part I, [8}. Using the description of congruences given in Part I, we verify the Congruence Extension Property in §1. This, in effect, states that a *-congruence on a subalgebra can be extended to a *-congruence on the algebra. This property is applied in §§2 and 3. In §2 we determine the standard semigroups " of the equational classes of pseudocomplemented distributive lattices, which is, roughly speaking, the semigroup generated by the operators H, S, and P in the sense of [5}. In §3 it is shown that the Amalgamation Property holds in f!J n (notation of Part I) if and only if n= -1, 0, 1, 2, or w, and that the subalgebra theorem for free products of B. Jonsson [7} holds for exactly the same equational classes. Since the Amalgamation Property fails to hold for -f!J3 , f!J 4 , ••• , we introduce a concept attempting to measure the extent of this failure. This concept is the amalgamation class of Amal (Jf'). The Amalgamation Property holds in Jf' if and only if Amal (Jf')=~ §4 contains results on Amal (f!J,,) for 2 and let I be the dual ideal in S(B) generated by II' In view of the one-to-one correspondence for Boolean algebras between dual ideals and congruences, (e 1 )S(A)= e1> where €II = ell]. Now let J ={x E D(B) I x ~y for some y E I}. We establish condition (2) for the lattices D(A), D(B) and the pair J, e2 • Let u E J and let W E D(A), w;fz u. In view of the definitions of J and I, there is an a E II such that a;fz u. Since a E A and WE D(A) it follows that a V W E D(A). a E II implies that as 1 (e 1) and so aV ws 1 (e 2 ). Thus condition (2) is established with v=aV w. Consequently there is a congruence e2 on D(B) such that (e 2 )D(A) = e 2 and such that xs 1 (e2 ) whenever x E J. In view of the definition of 01' condition (1) thus applies to the pair ;
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2. The standard semigroup. We apply Theorem I to calculate the standard semigroup of operators in the various equational classes of pseudocomplemented distributive lattices. Let X" be an equational class of algebras. If X"':;: X" let H(X"'), S(X"'), and P(X"') denote the class of all homomorphic images, isomorphs of subalgebras, and isomorphs of direct products respectively of algebras in X"'. With the set of symbols {H, S, P} we associate a partially ordered monoid @)(X"). Let ~ be the free monoid on {H, S, P} with the identity O. If X"':;:X" and U is a word in ~ we define the class U(X"') by requiring that O(X"')=X"' and UV(X"') = U(V(X"'». Since X" is an equational class U(X"'):;:$"' for all X"':;:~ U E~. The standard semigroup of operators of ~ @)(X"), is the quotient monoid of ~ where U, V E ~ are identified if U(X"') = V(X"') for every subclass X"' of $'. The partial order is determinell by setting U~ Vif U(X"'):;: V(X"') for all X"':;:$'. Don Pigozzi [10] announced that, for each equational class ~ 6($"') is a quotient of the 18element partially order monoid 6 depicted in Figure 1, and showed the existence of a class X" of groupoids for which 6 (X") = 6.
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Let B be a Boolean algebra and let B be the pseudocomplemented distributive lattice obtained by adjoining a new greatest element to B. Then the B are exactly the subdirectly irreducible pseudocomplemented distributive lattices (Theorem 2 of Part I). Let B" denote the n-atom Boolean algebra and let f1#n denote the equational class of pseudocomplemented distributive lattices determined by Bn• Let f1#(JJ denote the class of all pseudocomplemented distributive lattices. It was shown by K. B. Lee [9] that the f1#", n finite, and f1#(JJ are all distinct and are all the nontrivial equational classes of pseudocomplemented distributive lattices (see also Part I). Observe that f1#o is the class of all Boolean algebras. THEOREM 2. (i) If 0 ~ n ~ 2 then e(f1#n) has 11 elements and is the quotient of e under the relations HS=SH, SP=HSP. (ii) If2 ifI2 provides the embedding of B 1 in D, and the various homomorphisms ifIl> 'P2 provide the embedding of B 2 in D, proving the lemma.
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To proceed further we need a universal algebraic lemma. Let B be an algebra and let A be a subalgebra of B. We say that B is an essential extension of A if any congruence on B whose restriction to A is trivial is itself trivial. LEMMA 3. (a) An essential extension of a subdirectly irreducible algebra is itself subdirectly irreducible. (b) Let A be a subalgebra of an algebra B, and let 0 be a congruence on B. Then there is a congruence on B such that 0 ~ , A = 0 A, and the extension BI of AI 0 A is essential.
Proof. To prove (a), let A be subdirectly irreducible with smallest nontrivial congruence 0 0 , Then there are a, bE A, ai=b, such that a=b(0 0 ); thus 6 0 = 6ia, b), the smallest congruence of A identifying a and b. Let B be an essential extension of A and let be a nontrivial congruence on B. Then A is nontrivial implying that a=b( 2, we have found a contradiction to the requirement that y={31>p. Thus a = XfPfJ for all x E A, contradicting the definition of a and fJ. On the other hand, 0 1 = t implies that ~ 0 since 0 is an extension of Y=fPf3 and fJ is one-to-one. This conclusion contradicts the requirements that #: wand A 0 = w. We thus conclude that y can have no extension to AI x B", and so that A ¢ Amal (til,,) by Lemma 5. The proof of the theorem is thus concluded. In the proof of Theorem 6 the finiteness of A is essential. It seems reasonable, though, that Theorem 6 holds for infinite algebras as well. We thus conclude this paper with the following problem. Does Theorem 6 hold for infinite pseudocomplemented distributive lattices? If not, what is an intrinsic characterization of the algebras in Amal (til,,) for 2
