Algebra Universalis, 7 (1977) 405-407 Birkh~iuser Verlag, Basel

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Classes defined by implications

EVELYN NELSON

The purpose of this note is to provide counterexamples to the main results of Giri [2].

The implications considered are of the form

( / ~ w, = w 0 - - , w = w '

where " / ~ " denotes logical conjunction, and the w~, w~ etc. are elements of the absolutely free algebra (word algebra) of a given type on some set X, and where there is no restriction on the cardinalities of I or X. Classes defined by such implications Giri calls "varietal structures." A class defined by implications of at most countable length (i.e. implications as above where I I l < - ~ o ) are called pseudovarieties; clearly every pseudovariety K is of countable character, that is an algebra belongs to K whenever each of its countably generated subalgebras belongs to K.

The following discussion will provide a counterexample to Theorem 4.2 of Giri [2] which claims that every implicationally defined class of countable character is a pseudovariety.

Let K be the class of all algebras with ~t nullary operations (constants) ax (A < N1) and no other operations, satisfying the following implication:

/X'X (a~ = ao) ~ x = y. A < ~ 1

Then a non-trivial algebra A belongs to K iff in A, not all the constants a~ are equal. K is evidently of countable character.

Now consider an implication

wi = w ~ - - ~ w = w ' (*)

Presented by J. Mycielski. Received June I7, I976. Accepted for publication in final form October 1, 1976.

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406 EVELYN NELSON ALGEBRA UNIV.

of the type of the algebras in K, the w~'s etc. coming from F(X), the absolutely free algebra on the set X. Let 0 be the equivalence relation on F(X) generated by {(wl, w')[i ~ I}. Since there are no operations of arity I>1, O is a congruence on f(x).

If ]I]~ ~o then the blocks of the partition of F(X) induced by 8 are at most countable, and consequently there exists h < R1 with (ax, ao) ~ 0. But this implies that the quotient algebra F(X)/O belongs to K, and consequently if K satisfies (*) then (w, w') ~ 0 and so (*) is a logical tautology. Consequently K does not satisfy any non-trivial implications of at most countable length, and so in particular K is not a pseudovariety. The class K evidently also provides counterexamples to Theorem 5.1 and 5.2 of [2].

A definitive treatment of classes defined by implications has been given by Banaschewski and Herrlich [1]; among other things they prove that a class of finitary algebras is a pseudovariety (Rl-implicational in their terminology) iff it is closed under subalgebras, products and Rl-direct limits; the latter is equivalent with closure under both mono-R1-direct limits and epi-Nl-direct limits. An alternate view of the above class K is that it provides an example of a class of finitary algebras which is closed under subalgebras, products, and mono-R~-direct limits (i.e. is a quasiprimitive class of countable character in Giri's terminology) but is not closed under epi-R1-direct limits. The latter can be seen directly, as follows: Let F be the K-free algebra on one generator, and for each S _q N~, let Fs be the quotient of F by the congruence generated by {(ao, a~) [ ,~ ~ S}. Then the Fs, for S at most countable, form an epi-R~-direct system in K whose limit, which is the algebra with exactly two elements a and x, and with a~ = a for all h < R~, is not in K. This clearly works with R1 replaced by any infinite cardinal m, producing a quasiprimitive class of finitary algebras which is closed under m-direct unions, but is not m-implicational.

Remark. The class K descril~ed above has the unpleasant feature of uncounta- bly many operations; the following discussion provides an example of a class H of algebras of finite type which is closed under subalgebras, products and mono- direct limits, in fact is even Rx-implicational, but is nevertheless not a quasivariety. Specifically, among algebras with one binary operation, denoted by juxtaposition, one unary operation x~*x' and one nullary operation 0, consider the class H of those algebras satisfying the implication (Ph~n~, 0 ("~ 0 = 00) ~ x = y, where 0 (") is defined inductively by 01= 0' and 0 (n§ (0(~) '. H is evidently Nl-implicational, and closed under mono-direct limits. Let A be the absolutely free algebra of this type on the empty set of generators, and for a subset F ~ t o let OF be the congruence on A generated by St x St, where St = {00} U {0t"~0 [ n e F}. It is easy to see that St is a congruence class for OF, and consequently A/Of ~ H provided

Vol 7, 1977 Classes defined by implications 407

F~ =~o. Thus the algebras A/OF for the finite F_to, form a direct system of algebras in H whose direct limit A/O,o does not belong to H, which proves the point.

REFERENCES

[1] B. BANASCHEWSKI and H. HERRLICH, Subcategories defined by implications, Houston J. Math. 2 (1976) 149-171.

[2] R. D. GIRl, On a varietal structure of algebras, T.A.M.S. 213 (1975), 53-60.

McMaster University Hamilton, Ontario Canada