infinitary equational compactness

I N F I N I T A R Y E Q U A T I O N A L C O M P A C T N E S S

EVELYN NELSON 1)

Examples were given in Banaschewski-Nelson [4] of equational classes of infinitary algebras in which there is only one (up to isomorphism) subdirectly irreducible algebra, but no non-trivial equationally compact algebras at all, thus providing counterexamples, in the infinitary case, to the result of Walter Taylor [11] that if an equational class has only a set of subdirectly irreducible algebras then every algebra in the class can be embedded in an equationally compact one in the class. The definition of equational compactness used in [4] coincides with the definition for finitary algebras. It was suggested by GiJnter Bruns that things might go more smoothly in the infinitary case if the definition of equational compactness were changed to take into account the infinitariness of the operations. In this paper, such a modified definition (which reduces to the usual one in the finitary case) is adopted; it is then seen that several of the results on equational compactness for finitary algebras carry over to the infinitary case. In particular the result of Taylor [11] that, in an equational class, if every algebra can be embedded in an equationally compact one then every algebra has only a set of essential extensions, and thence that there is only a set of subdirectly irreducibles in the class, is proved for infmitary algebras. However, the converse is still not true; the equational class" of No-Boolean algebras is shown to be a counterexample.

Sections 1 to 4 of this paper are modelled after Banaschewski-Nelson [3], and most of the definitions and proofs which appear here for infinitary algebras are modifications of the corresponding ones in [3] for finitary algebras. The results of [3] which do not carry over to the infinitary case, in particular their proofs, rely heavily on either Birkhoff's Subdirect Representation Theorem or other Zorn's Lemma-type- arguments, none of which remain valid for infinitary algebras.

The author is grateful to B. Banaschewski for several helpful conversations, especially concerning the proofs in w

w 1. Equational compactness

For a type z = (rtx)x+a of infinitary algebras (the n~ arbitrary cardinal numbers), let e, be the smallest infinite regular cardinal >rtx for all 2sA. Note that if all rtx are finite then e,=No. Except in dealing with specific counterexamples, 'algebra' will mean 'algebra of type v'.

1) Research supported by the National Research Council of Canada.

Presented by G. Gr~tzer. Received September 1, 1973. Accepted for publication in final form December 6, 1973.

2 EVELYN NELSON ALGEBRA UNIV.

A partially ordered set P is called n-up-directed for a cardinal number rt iff every subset S ~ P with ISI < r t has an upper bound in P. Note that ,~0-up-directed is just up-directed.

It is well-known that if A is an algebra of type z, then the set of all subalgebras of A is closed under ~,-up-directed unions, and thus also the set of all congruences on A is closed under a,-up-directed unions. Further, if S is any subset of A, then the subalgebra generated by S is the union of the subalgebras generated by the subsets T ~ S with IZl < ~ . These facts will be used extensively, without specific

mention. Since sets with fewer than as elements play the role for algebras of type T that

fni te sets do for finitary algebras, it will be convenient to have a name for them; thus a set S will be called small iff I SI < ~,.

As in [3], for an algebra A in an equational class A, A [X] is the free extension of A in A by the set X (of indeterminates).

An extension B of A (in a fixed equational class A) is called pure iff for every X, every small subset S _ A [X] z which is contained in the kernel of a homomorphism A [X] ~ B over A is also contained in the kernel of a homomorphism A IX] ~ A over A. A homomorphism h: A ~ B is a pure embedding iff any small subset S _ A [X] 2 which is contained in the kernel of a homomorphism A IX] --* B over h is already contained in the kernel of a homomorphism A [X] -0 A over A. (As was pointed out in [3], the latter condition implies that h is an embedding). Clearly, composites of pure embeddings are pure embeddings.

An algebra A is called equationally compact iff a subset S ~ A IX] z is contained in the kernel of a homomorphism A IX] ~ A over A whenever each small subset T _ S already has this property.

Note that if ~ is finitary then these definitions coincide with the usual definitions of purity and equational compactness. Note also that although on the surface, the condition for the equational compactness of A seems to depend on the equational class in which the free extension A IX] is taken, in fact it is independent of the equational class; this is also pointed out in [1], and is a consequence of the fact that, for equational classes A and B, if A~A and A_~B then the free extension of A in A by the set X is a homomorphic image (over A) of the free extension of A in B by the set X.

Just as for finitary algebras (Mycielski, [9, Proposition 1]), the Tychonoff Product Theorem can be used to show that an infinitary compact Hausdorff topological algebra is equationally compact. However, whereas every finite finitary algebra is a compact Hausdorff topological algebra when endowed with the discrete topology (the point being that the product of finitely many discrete spaces is discrete, and hence all finitary operations are continuous with respect to the discrete topology), finite algebras with infinitary operations need not even be equationally compact; it

Vol. 4, 1 9 7 4 INFINITARY EQUATIONAL COMPACTNESS 3

will be shown in w that the finite non-trivial ~o-Boolean algebras are not equationally compact.

The definitions of essential extension, pure-essential extension, pure-injective, pure-absolute retract, equationally compact hull, read exactly the same for infinitary algebras as they do for finitary algebras in [.3] and consequently will not be repeated here.

LEMMA 1. I f B is an extension of A, and if ~ is an ct,-up-directed set of congruences on B such that A ~ B/O is a pure embedding for each 0 ~ then A ~ B~ ~_) ~ is also a pure embedding.

Proof. If S _ A IX] 2 is small, and if there is a homomorphism h: A IX] ~ B~ U over A ~ B / ~ ~ with S~_Kerh then there is a homomorphism g:A IX] ~ B over A such that h = vg where v : B ~ B / U ~ is the natural homomorphism, and consequently g2(S)~_Kerv= ~ ~. Since gZ(S) is small and since ~ is ct,-up-directed, there exists 0 e ~ with g2 ( S ) _ 0. Since A ~ B/O is pure, it follows that there exists a homomorphism A I-X] ~ A over A containing S in its kernel.

Note that Lemma 1 is a consequence of the more general observation that c~-up-directed colimits preserve pure embeddings (Banaschewski, private communi- cation).

The following proposition is the infinitary extension of the characterization theorem for equational compactness of finitary algebras essentially due to W~glorz [,12] and Taylor [11] (See [,3], Proposition I).

PROPOSITION 1. For any A ~ A the following are equivalent: ( i) A is equationally compact. (2) A is pure-injeetive in A. (3) A is a pure-absolute retract in A. Proof The proof is word-for-word the same as the proof of Proposition 1 in [,3],

except that 'finite' must be replaced by 'small' wherever it appears, and the proof of (3)=~ (1) uses Lemma 1 above instead of [3, Lemma 1].

Note that in the finitary case, there is a fourth equivalent condition, namely (See [3, Proposition 1]):

(4) A has no proper pure-essential extensions in A. In the infinitary case, the implication (3)=~ (4) is still valid (same proof as in [3])

but the reverse implication is not true; it will be seen in w that the two-element ~o-Boolean algebra has no proper pure-essential extensions in the class of all ~o- Boolean algebras, but is not equationally compact.

As an immediate corollary to Proposition 1, it follows, as in the finitary case, that absolute retracts are equationally compact, and products and retracts of equationally compact algebras are equationaUy compact.


w 2. Cardinality bounds on essential and pure-essential extensions

In [3], a proof is given for the result, essentially due to Taylor [11], that i f B is an equationally compact hull of A then IBIs< "+lal where rt is the sum of ~o and the number of operations. Although this result is true for infinitary algebras (where n is now the sum of at and the number of operations), and is a corollary of Proposition 3 of this section, it does not play the same role in later proofs as it did in the finitary case. The reason for this is the following: in the finitary case one uses the fact that if A has a pure embedding into an equationally compact algebra then it has an equationally compact hull [3, Proposition 2]: it is not known whether this remains true for infinitary algebras.

(Of course, part of [3, Proposition 2] does remain valid; the proofs for infinitary algebras that for any algebra A in an equational class A, if A has an equationally compact hull in A then A has a pure embedding into an equationally compact algebra in A, and if the latter condition holds then A has, up to isomorphism, only a set of pure-essential extensions in A, are exactly the same as those for finitary algebras. On the other hand, as will be seen in w the two-element ~qo-Boolean algebra has no proper pure-essential extensions, and is not equationally compact, and hence has no equationally compact hull, and hence (3)=~ (1) of [3, Proposition 2] is not true for infinitary algebras.)

This section is devoted to proving two results which give cardinality restrictions on essential and pure-essential extensions in the presence of suitable equationally compact extensions. As in [3] and Taylor [11], the following combinatorial result (Erd6s [5], also Erd6s-Rado [6]) is an important tool. (Where, for a set Y, yt2~ is the set of two-element subsets of Y):

L E M M A 2. l f St21= U {C,] ~eI} for infinite sets S and l such that ISl>2tXl then there exists ~ I and a subset X ~ S with Xt2a___ Ca and IXI > III.

In addition, the following characterization of principal congruence relations, which is due to B. Banaschewski, plays an important part in the proofs of the two propositions which follow (where, for elements a, b~A, 0 a (a, b) is the congruence on A generated by (a, b)):

L E M M A 3. I f B is an extension of A, E = A [ X ] [x, y] for IXl =c~=, and if c, d sB then On(c, d ) = U {h2(Kerhv OE(x,y)) l h sH} where v is the joh~ in the lattice of congruence relations on E, and H is the set of all homomorphisms h: E ~ B over A with h ( x )=c ,h (y )=a .

Proof. If hel l , then the fact that h 2 (Kerh v Or (x, y)) is a reflexive and symmetric relation on h (E), and a subalgebra of h (E)Z follows immediately from the corresponding properties for Ker h v 0 z (x, y). If (r, s), (t, u) ~ Ker h v Or (x, y) and h (s) = h (t)


then (s, t ) e K e r h and consequently (r, u )eKerh v OE(x, y); from this it follows that h 2 (Ker h v 0E (x,y)) is transitive and hence a congruence relation on h (E). Since h (x) = c and h ( y ) =d , it follows that Oh(E)(C, d)~-h 2 (Kerh v O~(x, y)); the reverse inclusion is clear, and hence the congruence on h (E) generated by (e, d) is h 2 (Kerh v 0 E (x,y)).

Since the h (E), for hell, are precisely the subalgebras of B which contain {c, d} and are generated over A by a set with at most ~, elements, they form an a,-up- directed set of subalgebras with union B. It follows from this that OB(c, d ) = = U {0h(E)(c, d) ] hell} and this yields the desired result.

PROPOSITION 2. If B is an essential extension of A and if B has an equationally compact extension then [B[ ~< 2 "+ lal where rt = ~ + [A].

Proof. Let E=A IX] [-x, y], where [Xl=cc,, and let Z be the set of all small subsets F __. E z such that there is no homomorphism g:E-~ D over A, D any extension of A, with FU {(x, y)} _ Ker g.

Let ~< be a well-ordering of B, and for each Fe Z, let Be = { (c, d) I c, deB, c < d and there exists a homomorphism h: E--* B over A with F_~ Kerh and h (x) = c, h (y) = d}.

If c, deB and c < d then, since B is an essential extension of A, there exists a, b e A with a~b and (a, b)eOs(c, d). By Lemma 3, there exists h : E ~ B over A with h ( x ) = =c, h(y)=d, and (a, b)ehZ(KerhvO(x,y)). Thus there is an F _ K e r h , [F[<~x, with (a,b)ehZ(OE(Fu{(x,y)})) where OE(Fw{(x,y)})is the congruence on E generated by F w {(x, y)}. But this implies that there are elements u, veE with h (u) = a, h(v)=b, and (u, v)eOE(Fu {(x, y)}). Let F=Fw {(a, u), (v, b)}. Then FeZ, since if there were a homomorphism g : E ~ D over A with Fu{(x,y)}~_Kerg then Fw{(x,y)}~_Kerg and hence (u, v )eKerg and this would imply a=b. Clearly (e, d)eBF.

It follows that {(c, d) ] c, deB, c<d} = U {Be [ FEZ}. Now the proof proceeds in the same way as does the proof of ['3, Lemma 3] : If IB[>2 "+lal then, since [E l=r t+]A [, and hence [Zl=rt+lAI, it follows from

Lemma 2 that there exists F e Z and Bo~_B with [Bol >r t+ lAt and {(c, d) I c, deBo, e<~/}---/~F.

Let C be an equationally compact extension of B, let Y be disjoint from C with ] Y[ > [C[ and let <~ be a well-ordering of Y.

Let Z = Yu (Xx Yx Y) and for each u, ve Y with u < v, let h,v: E ~ A [Z] be the homomorphism over A which maps x ~ u, y ~ v and s ~ (s, u, v) for s e X.

Any small subset of Q = U {h,v 2 (F) ] u, ve Y, u < v} is a subset of one of the form Q ~ = U {h,v2(F) l u, veK, u<v} where K is a small subset of Y. For any such K, since [Bol > 1t + ]At > ~,, there exists a one-one order preserving map u ~ a from K into B0, and since (~, ~)eBo for all u, veK, there exists a homomorphism f,~: E ~ B over A with f,v(x)=t2, f , o ( y ) = ~ and F_Ker f ,~ . Consequently there is a homomorphism f :A [ Z ] ~ C over A such that f (u)=~i for ueK, f(s, u, v)=f,,(s) for all s e x and


all u, v~K with u<v, and hence QK_Kerf. Since A~_C,f can be extended to a homomorphism C [Z] ~ C over C containing Q~ in its kernel.

Now by the equational compactness of (7, there exists a homomorphism g : C [ Z ] - . C over C with Q_~Kerg. Since IY[>lCI, there exist u, v e Y with u<v and g(u)=g(v). But then gh, v :E~C is a homomorphism over A containing Fw {(x, y)} in its kernel, and this is a contradiction.

PROPOSITION 3. l f B is a pure-essential extension of A and if B has an equationally compact pure extension then IBI ~< 2 "+ lal.

Proof. Let E=A[X] Ix, Y] where IXl=~, , and let ~" be the set of all small subsets F ~ E 2 such that there is no homomorphism g:E--,A over A with Fw {(x, y)} _~ Kerg.

Let ~< be a well-ordering of B, and for each F ~ let Be= {(c, d) I c, dsB, c<d and there exists a homomorphism h : E ~ B over A with h(x)=c, h(y)=d and F _ Kerh}.

For any c, deB with c<d, since A ~ B is pure-essential, there exists Fo= = {(Pi, qi) [ isJ}_~A [,,~]2 with l J[ <c~, such that F 0 is not contained in the kernel of any homomorphism A IX] --*A over A but there is a homomorphism s:A IX] --*B over A with s 2 (Fo) - 0n (c, d).

By Lemma 3, there exists h:E--*B over A with h(x)=c, h(y)=d and s2(Fo)_ ~_h2(KerhvOe(x,y)). Thus there exist (pi, 4i)sKerhvOE(x,y) for i~J with h (p i )=s (Pi), h (qi)=s (qi)- Since ]Jr <co,, there exists a small F1 ~ Kerh such that for each i, (p,, 4i)~Oe(F1 u {(x, y)}).

Now, since F1 and J are small, we may assume without loss of generality that s(p,)=h(pl) and s(qi)=h(qi) for all i~J.

Let F=Fx w {(Pi, ffl) [ isJ} tj {(q,, t]i)] isJ}. Then F_cKerh. However, if there is a homomorphism g:E--*A over A with F u {(x, y)} _ K e r 9

then F1 w {(x, y)}_~Kerg and hence g(ffi)=g(4i) for each ieJ. Since F _ K e r g it follows that g(p~)=g(q~) for each i~J and this contradicts the choice of F o.

Consequently, F~27 and (c, d)~Br. Thus we have that [,_) {BF [ VsI;} = {(c, d) [ e, dsB, c<d}. If IB[>2 "+IAI, then it follows from Lemma 2 that there exists FeZ and Bo~_B,

IBol>rt+lA l, such that {(c, d)[c, d~Bo, c<d}~Be. Now let C be an equationally compact pure extension of B. Using the same argu-

ment as in Proposition 2, we obtain a homomorphism g : E ~ C over A with Vw {(x, y)} _c Kerg. Since r w {(x, y)} is small and since B is a pure extension of A and C is a pure extension of B, this implies that there exists a homomorphism f : E ~ A over A with F u {(x, y)} _ K e r f and this is a contradiction.

COROLLARY. l f B is an equationally compact hull of A then IBI ~<2 "+lal.


w 3. Enough equationally compact algebras

Recall the following characterization, due to Taylor [11], of equational classes of finitary algebras with enough equationally compact members:

[3, PROPOSITION 3]. For an equational class A of finitary algebras, the following conditions are equivalent:

(1) Every algebra in A has an equationally compact extension in A. (2) For any essential extension E~A of any A~A,,]E[ ~<2 "+lal. (3) There exists a set B _ A of at most 2" algebras, each with at most 2" elements,

such that every algebra in A can be embedded in a product of algebras from B. (4) For any subdirectly irreducible A~A, IA[ ~<2". (5) A has, up to isomorphism, only a set of subdireetly irreducible algebras. (6) Every algebra in A has, up to isomorphism, only a set of essential extensions in A. (7) Every algebra in A can be embedded in an absohtte retract in A. It was already mentioned in the introduction that this theorem is not true for

infinitary algebras; it will be shown in w that (1) fails in the class of ~o-Boolean algebras, as does (3), even though in this class, every essential extension of an algebra A has at most 2 Ial elements, and the only subdirectly irreducible in the class is the two-element one. However, part of the proposition can be salvaged in the infinitary case; this is summed up in the following:

PROPOSITION 4. In an arbitrary equational class of (infinitary) algebras, the implications ( I ) ~ ( 2 ) ~ ( 4 ) ~ ( 5 ) , (3) ~ (2) ~ (6), and ( 7 ) ~ ( 1 ) hold, where (1)-(7) are the conditions listed above.

Proof. (1) =~ (2) follows from Proposition 2, (2) ~ (6) is trivial, (2) ~ (4) follows, as it did in the finitary case, from the fact that every subdirectly irreducible algebra is an essential extension of an algebra generated by two elements, and (4) =~ (5) and ( 7 ) ~ (1) are trivial.

(3)=~ (2). The following proof appears in Banaschewski [1, Proposition 45] and works as well for infinitary as for finitary algebras:

Let E be an essential extension of A, and let h : E ~ 17 {A, I ~ I } be an embedding where A,~B for each c~L For each pair (a, b) of distinct elements of A, choose y~ l such that p~h (a) ~p~h (b) (pr: 17A, ~ A t the projection map), and let J ~ I be the set consisting of all ~ thus chosen. Let g: 17 {A, [ ~ I ) ~ 17 {Ay [ ~,~J} be the natural homomorphism; then gh I A is an embedding, and hence by the essentialness of E, gh is an embedding. But ]Jl~<rt+lA] and latin<2" for all 7~J and hence ]E]~< <~ (2-)-+ IAI = 2-(-+ Ial) = 2" + t a l .

It might be noted that the proof of (3) :~ (2) actually shows that, in an equational class of infinitary algebras in which every algebra is the subdirect product of subdirectly irreducibles, (4) =~ (2).


This still leaves open the question whether either (1)=~ (3), or (3)=, (1) holds for arbitrary equational classes of infinitary algebras.

Remark. Call an algebra A (of any type) equationally m-compact 2) (for an infinite regular cardinal m) if each subset L'___A [X] 2 is contained in the kernel of a homomorphism A I-X] ~ A over A whenever every subset S ~ . r with ISI < m already has this property. It has been pointed out by Walter Taylor that the argument in the proof of Proposition 2 of w 1 actually shows that if B is an essential extension of A and if B has an equationally m-compact extension then I BI ~< 2 p + 1.41 where p = m + ~ + I A I.

Thus, in particular, for an equational class A of finitary algebras, if every algebra in A has an equationally m-compact extension in A for some m then every algebra in A has an equationally compact extension in A.

w 4. 'Purely enough' equationally compact algebras

In this section we will prove the 'pure' analogue of Proposition 4. The following lemma, which is the infinitary counterpart of [3, Lemma 4], will be needed:

LEMMA 4. For any extension B of an algebra A there exists an algebra C with A~_C ~_B and ]CJ <<.rt + lA[, such that B is a pure extension of C.

Proof Let E = A IX] with IX[ = ~ . If B is not a pure extension of A, then there exists a small subset K~_E 2 which is contained in the kernel of a homomorphism A IX] -~ B over A, but not in the kernel of any homomorphism A IX] ~ A over A. For any such K, let h r : E ~ B be a homomorphism over A with K_~Kerh K, and let Xr, be a small subset of X with K2~_A[.,~K] 2. The set S = U hr(Xr) has at most r t+ [A] elements, and hence the subalgebra A ~ of B generated by A u S has at most r t+ ]A[ elements.

Now iterate this procedure, by defining

A o =A Ax+I =A~* for each ordinal 2 A~=the subalgebra generated by [,.) (Aa [ 2<p} for each limit ordinal/1.

Then C=A,, is the union of an ~-up-directed set of subalgebras of B, each with at most r t+ [A[ elements, and hence C is also a subalgebra with at most r t+ [A[ elements.

Moreover, if K is a small subset of C [ X ] 2 and if there is a homomorphism h : C [ X ] ~ B over C with K _ K e r h , then there exists an ordinal 2<0~, with K ~_~A~[)(] 2, and h[Ax [X] :Az [X] ~ B is a homomorphism over Ax with K in its kernel, and consequently there is a homomorphism g:Ax [X] ~ Ax+I over Aa with

e) Other authors (see, for example, Mycielski [9]) have given a different meaning to the term 'm-compact'; the present concept does appear in the literature (for example, in McKenzie, Colloq. Math. 23(1971) 199-202) but is left nameless.


K ~ Kerg. But then g can be extended to a homomorphism C [X] ~ C with K in its kernel, and hence B is a pure extension of C.

For finitary algebras, this lemma is a consequence of the L~Swenheim-Skolem- Tarski Theorem. The finitary version of the above proof appeared in a preliminary version of [3].

Now, recall ([3] or [11]) that an algebra A is pure-irreducible iff, for any pure embedding A ~ 1-IBm, the composite with some projection l i b ~ Ba is a monomorphism.

Taylor [11] proves the 'pure' counterpart, for finitary algebras, to Birkhoff's Subdirect Representation Theorem. This theorem, like Birkhoff's Theorem, is not true for infinitary algebras; it will be seen in w 5 that 2, the two-element No-Boolean algebra, is the only pure-irreducible No-Boolean algebra; and it is well known that not every No-Boolean algebra has an No-embedding into a power of 2.

The following characterization of pure-irreducibility is the infinitary analogue of [3, Lemma 5]:

LEMMA 5. A is pure-irreducible iff there exists a small F ~_A [X] z (IXl =ce,) such

that F is not contained in the kernel o f any homomorphism A IX] --+ A over A, but for every non-trivial congruence 0 on A, there is a homomorphism h : A IX] ~ A over A with h 2 (F) =_ O.

Proof. The proof is word-for-word the same as the proof of [3, Lemma 5] except that 'finite' must be replaced by 'small' wherever it appears.

COROLLARY. Every pure irreducible algebra is a pure-essential extension of an algebra with at most rt elements.

Proof If A is pure-irreducible, then take F~_A IX] z as described in the lemma. Then there exists a subalgebra A o ~ A generated by at most ~ elements such that F - A o IX] 2. By Lemma 4, there exists a subalgebra B~_A with Ao ~ B such that A is a pure-extension of B and IBI ~< rt + IAol. Since [Ao[ ~ rt, we have I B[ ~< 11. As in the proof of [3, Corollary to Lemma 5], it follows from the properties of F and the fact that F ~ B IX] 2 that A is a pure-essential extension of B.

Recall that for finitary algebras, we have the following characterization of equational classes with 'purely enough' equationally compact members; due to Taylor [ 11] :

[3, PROPOSITION 4]. For an equational class A o f finitary algebras, the following are equivalent:

(1) Every algebra in A has an equationally compact pure extension in A.

(2) For any pure-essential extension E t A of any A cA, IEI ~< 2 "+ lal

(3) There exists a set B ~ A o f at most 2" algebras, each with at most 2" elements, such that every algebra in A can be purely embedded in a product o f algebras from B.

(4) For any pure-irreducible A ~A, IAI ~<2".


(5) A has, up to isomorphism, only a set of pure-irreducible algebras. (6) Every algebra in A has, up to isomorphism, only a set of pure-essential

extensions. This proposition also fails for infinitary algebras; it will be seen in w that (1) and

(3) fail in the class of ~o-Boolean algebras, even though the only pure-irreducible in the class is the two-element algebra. This still does not provide a counterexample for (6)=>(1).

The following is the 'pure' counterpart of Proposition 4:

PROPOSITION 5. For any equational class A of infinitary algebras, the implications (1)=*-(2)=> (4)=> (5), and (3)=> (2)=> (6) hold, where (1)-(6) are the conditions listed above.

Proof. (1)=> (2) follows from Proposition 3, (2)=> (4) follows from the Corollary to Lemma 5, (4)=> (5) and (2)=> (6) are trivial. The following is a modification of (3) => (2) of Proposition 5:

(3) => (2). Let E be a pure-essential extension of A, and let h : E ~ H {A~ [ 0t~I} be a pure embedding, where A,~B for each c~L

Since composites of pure embeddings are pure embeddings, it follows that h I A : A ~ I-IA, is a pure embedding.

Let S be the set of all small subsets Fc_A IX] 2 (IXI =~=) such that there is no homomorphism A IX] --, A over A containing F in its kernel. Since IA IX]] = n + IAI, it follows that IZl=n+lAI.

For FeS, if there exists, for each f leL fp :A ['X] ~ Aa over pah I A (pa:IIA~ ~ Ap the projection map) with F _ Ker fp, then the map f : A I-X] -~ I"IA= given by ppf=fa for each f laI is over h I A and F__Kerf; since h ] A is pure, this would imply that there is a homomorphism A IX] ~ A over A with F contained in its kernel, which is a contradiction.

Hence we may choose, for each Fear, a y ~ l such that there is no homomorphism A ['X] ~ Ar over prh I A with F contained in its kernel. Let J~_I be the set consisting of all y thus chosen; and let g:lT{A~ I I= J} be the natural homomorphism. Then it follows from the choice of ~ that gh 1 A is a pure embedding, and then, since E is a pure-essential extension of A, that gh is an embedding. Since [J[ ~< n + ]A I, and since ]AvI ~< 2" for each yaJ , we have IE[ ~< (2") p+IAI =2 "+ 1.41.

w No-Boolean algebras

An 14,o-Boolean algebra is an algebra B with the usual Boolean operations, together with two additional ~lo-ary operations which associate with each tr ~B ~~ the meet Aa and the join V~r respectively, the identities being the obvious ones. This section is devoted to showing that B, the class ofaU blo-Boolean algebras, provides the


claimed counterexamples. Note that for these algebras, ~ = N I and hence a set is small iff it is at most countable.

It is known that 2, the two-element No-Boolean algebra, is the only subdirectly irreducible algebra in B (see, for example, Banaschewski-Nelson [-4]). The proof of this fact given in [-4] proceeds by showing that if BsB, then for every asB, the mapping b -'* (a ^ b, a v b) (for b sB) is an No-embedding of B into [0, a] x [a, 1 ]. However, it is well known (and easy to see) that this embedding is actually an isomorphism of B onto [0, a] x I-a, 1] and hence in particular a pure embedding. It follows that 2 is also the only pure-irreducible in B, and hence B has, up to isomorphism, only a set of subdirectly irreducible algebras and only a set of pure-irreducible algebras.

Moreover, the proof given in Banaschewski-Bruns [2, Lemma 5] that an extension of a Boolean algebra is essential iff it is join dense works also for No-Boolean algebras, and thus in B, if B is an essential extension of A then IBI ~ 2 lal.

In the class of Boolean algebras, every embedding is pure [-7]; it is not known whether the same statement is true for No-Boolean algebras; however, we have the following partial analogue:

L E M M A 6. In B, every embedding 2 ~ B & pure. Proof. Note that F(X), the B-free algebra on the set X, is (isomorphic to) 2 [X],

and that every homomorphism F ( X ) ~ B is over 2 provided B is non-trivial. For a subset A~_F(X), let A * = { A a l a s A ~ ~ lasA~~ and define

F~ ~ F(X) for each ordinal 2 as follows: F o = X u {x' [ xsX} (where ' is Boolean complement) F~+ 1 =Eft for each 2 F~= (.J~<~ Fz for limit ordinals/~. It follows from the infinite de Morgan laws (Sikorski [,10 p. 59]) and an easy

induction argument that for each 2, Fz is closed under forming complements, and hence F(X)=F~,.

This fact, together with an induction argument, shows that, for each asF(X), there exists a countable subset Z ~_ F(X) ~~ with the following property:

[ I f f : F ( X ) ~ 2 is a mapping such that f ( x ' )= f (x ) ' for all x s X and if (*) / f ( A a ) = Afir and f ( V a ) = V f a for all a s Z then f (a )=f* (a ) where

[ f * : F ( X ) ~ 2 is the No-Boolean homomorphism extending f l X. Now suppose S ~ F ( X ) 2 is at most countable, and that h : F ( X ) ~ B (B a non

trivial algebra) is a homomorphism containing S in its kernel. Let T=pl (S)up2 (S) where Pl, P2 are the projection maps F(X)Z~ F(X); then T is at most countable. By the above remarks, there exists a countable Z c F (Z) so such that, if f : F ( X ) ~ 2 is a mapping such that f (x)' = f (x') for all x s X and i f f ( A a) = A fa, f ( V a) = V fa for all a s s then f ( a ) = f * (a) for all acT, where f * is the No-Boolean homomorphism F(X) ~ 2 extendingf [ X.


Let 27 0 = {ha [ a~27} ~_BZ. By the Rasiowa-Sikorski Lemma (Sikorski [10 p. 102]), there exists a finitary Boolean homomorphism g: B-+ 2 such that g(/~ h a ) = / ~ gha

and g ( V ha)= V gha for all ae27. Since h is an No-Boolean homomorphism, it follows that g h ( A a ) = / ~ ghtr and g h ( V a ) = V gha for all ere27. Consequently, for the homomorphism f : F ( X ) -~ 2 extending gh I X, we have f (a) =gh (a) for all a e T, and hence S _~ Kerfi

COROLLARY 1. 2 has no proper pure-essential extension in B. Proof I f B is a non-trivial N0-Boolean algebra, and B-~2 then there is a non-

trivial proper congruence 0 on B. But then by the lemma, the map 2 ~ B ~ B / 0 (there is only one) is pure, and hence 2 ~ B is not pure-essential.

This then provides an example of an algebra which is not equationally compact but yet has no proper pure essential extensions in the equational class it generates.

COROLLARY 2. There are no non-trivial equationally compact algebras in B. Proof If BEB were equationally compact then it would be pure-injective by

Proposition 1, and thus for any non-trivial C~B, since the embedding 2 ~ C is pure, there would be a homomorphism C ~ B. However, by a result of Monk 1-8], there exists an algebra C~B such that every proper homomorphic image of C has cardinality > [B[ and this is a contradiction.

It follows from Corollary 2 that B does not satisfy either (1) of [3, Proposition 3] or (1) of [3, Proposition 4].

Moreover, the result of Monk mentioned above also implies that the class B does not have a set of cogenerators, and hence neither (3) of ['3, Proposition 3] nor (3) of ['3, Proposition 4] holds in B.

REFERENCES

[1] B. Banaschewski, An introduction to universal algebra, Manuscript, I.I.T. Kanpur, Summer 1972.

[2] B. Banaschewski and G. Bruns, Categorical characterization of the MacNeille completion, Arch. Math. 18 (1967), 369-377.

[3] B. Banaschewski and Evelyn Nelson, Equational compactness in equational classes of algebras, Alg. Univ. 2 (1972), 152-165.

[4] B. Banaschewski and Evelyn Nelson, Equational compactness in ilfnitary algebras, Colloq. Math. 27 (1973), 197-205.

[5] P. Erdtis, Some set-theoretical properties of graphs, Tucuman Revista Mathematicas, Series A, 3 (1942), 363-367.

[6l P. Erd6s and R. Rado, A partition calculus in set theory, Bull. Amer. Math. Soc. 62 (1956), 427-489.

[7] McMaster Folklore. [8] J. D. Monk, Non-trivial m-injective Boolean algebras do not exist, Bull. Amer. Math. Soc. 73

(1967), 526-527. [9] J. Mycielski, Some compactifications of general algebras, Colloq. Math. 13 (1964), 1-9.

Vol. 4, 1974 INFINITARY EQUATIONAL COMPACTNESS 13

[10] R. Sikorski, Boolean Algebras, New York 1964. [t l] W. Taylor, Residually small varieties, Alg. Univ. 2 (1972), 33-53. [12] B. W~glorz, Equationally compact algebras I, Fund. Math. 59 (1966), 289-298.

McMaster University Hamilton, Ontario

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infinitary equational compactness

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