Algebra Universalis, 24 (1987) 289-291 0002-5240/87/030289-03501.50 + 0.20/0 �9 1987 Birkh~iuser Verlag, Basel

A note on interpretations of Heyting algebras

WALTER TAYLOR

The theorem of this note follows also from the main results of Renato Lewin (1988) on interpretations of Heyting algebras. Nevertheless, this special case has a very simple proof, which is in a sense more 'structural' than Lewin's.

We use the same notion of interpretability that is found in Lewin (1988) and in O. C. Garcia and W. Taylor (1984). For varieties V and 7r (not necessarily of the same similarity type), we say that V is interpretable in 7//, and write V _< cW, iff for each W-operation F,(xl . . . . . xn) there exists a 74r-term f ( x l . . . . . x~) such that if A = (A;Gs)~seTK, then ( A , f A) e V. As first proved by W. D. Neumann [1974], and as described further by Garcia and Taylor [1984], the class of all varieties forms a lattice under the order relation which arises naturally from the quasi-order - .

For n =2 , 3 , . . . we let H, denote the n-element chain considered as a Heyting algebra. That is, Hn = {0, ea . . . . . en_2, 1}, with the elements listed in increasing order, and H~ = (H~; A, v, 0, 1, -->), where /x and v are the usual lattice operations, and --> is the 'relative pseudocomplement, ' which is defined as follows: x--+y = 1 if x - y , and x--->y =y otherwise. Moreover, Vn is defined to be the variety generated by Hn, for n =2 , 3 , . . . . It is evident that Vn is a subvariety, and hence that V~+t _< 7/'n (in the sense defined above) for every n, via the interpretation f = F, for each t. Our theorem asserts that we cannot reverse these inequalities, and hence that we have a decreasing sequence V2 > 7/3 > . . . in the lattice of interpretability types.

T H E O R E M . For n >- 2, Vn is not interpretable in Vn+l.

Proof. H = n n + 1 is an algebra in C~n+ 1 which has the following two properties:

(i) IHI = n + 1; (ii) every subset of H containing 0 and 1 is a subuniverse of H.

Now if there were an interpretation of V~ in V,+I, then this interpretation would

Presented by Alan Day. Received July 6, 1986. Accepted for publication in final form May 4, 1987.

289

290 WALTER TAYLOR ALGEBRA UNI~.

yield an algebra H ' in W;, that also has properties (i) and (ii). The impossibility of such an interpretation therefore follows from Lemma 2 below.

Obviously, our first lemma fails for ILj = 4.

L E M M A i. Let L be a distributive lattice with 0 and 1, such fhat eueey subset containing 0 and 1 is a sublattice, and such that ILl -> 5. Then L is a chain.

Proof. The hypothesis about sublattices can be thought of as saying that every element of L other than 0 and 1 is both /x -irreducible and v -irreducible. If L is not a chain, then L has two incomparable elements a and b. Their join a v b is

not v -irreducible, and hence can only be 1. Similarly, a A b = 0. Since iLl >- 5, there exists c e L with c :/= 0, 1, a, b. By distributivity, we have

c = c A ! = c A ( a v b ) = ( c A a ) v ( c A b )

and

c = c v0=c v(a Ab)=(c va) A(cvb).

By the double irreducibility of c we know that

C = C A a o r c a b

and

c = c v a o r c v b .

Since a r c, the alternative involving a cannot occur in both lines, and similarly, the alternative involving b cannot occur in both lines. Hence, without loss of generality, we have

C = C A a = c v b .

In other words, a-> c - - b , which contradicts our assumption that a and b are incomparable. This contradiction establishes our claim that L is linearly ordered.

L E M M A 2. For n >-2, the Heyting variety 7/'~ contains no algebra which satisfies the following two properties:

(i) IHI = n + 1; (ii) every subset of H containing 0 and 1 is a subuniverse of H.

Vol 24, 1987 A note on interpretations of Heyting algebras 291

Proof. Suppose, by way of contradiction, that H satisfies (i) and (ii) for some H e ~V~. We first need to see that H is ordered as a chain. This is clear if n = 2, and it follows from Lemma i if n -> 4. The only remaining possibility is that n = 3

and that (as a lattice) H is the four-element lattice that is not linearly ordered. It is easy to check that (as a Heyting algebra) this H does not satisfy condition (ii),

and therefore we know that H is a chain. Of course the chain of n + 1 elements is unique as an ordered set, and hence

unique (up to isomorphism) as a Heyting algebra. Therefore H ~H n +l , and so we have Hn+~ �9 ~/]. Since H,,+~ is subdirectly irreducible, J6nsson's lemma yields Hn+l �9 which is of course impossible on cardinality ground. This contradiction completes the proof of Lemma 2.

REFERENCES

O. C. GARCIA and W. TAYLOR (1984), The lattice of interpretability types of varieties, AMS Memoirs #305, 125 pages.

R. LEWIN (1988), Interpretations into Heyting algebras, Algebra Universalis 24, 149-166. W. D. NEUMANN (1974), On Malcev conditions, J. Australian Math. Soc. (A) 17, 376-384.

University of Colorado Boulder, Colorado U.S.A.