some functorial aspects of atomic compactness

Algebra Univ. 5 (1975) 367-378 Birkh~iuser Verlag, Basel

Some functorial aspects of atomic compactness

Evelyn Nelson 1)

This paper investigates the relationship between different categories of finitary algebraic structures (i.e. structures with operations and/or relations) with regard to the behavior of atomic compactness and related concepts. Various constructions, invented by Hedrlin, Lambek and Pultr (see [5"1 and [6]) provide full embeddings of the category of all algebraic structures of any given finitary type into categories of structures with a small number of fundamental operations and relations, notably bi- unary algebras, mono-binary algebras (groupoids), mono -binary relational structures ('graphs' in their terminology) and semigroups. In this paper, these constructions are exploited to show that there are embeddings of the category of all structures of a given finitary type into the category of all groupoids, which preserve and reflect purity and atomic compactness, preserve pure-essential embeddings and atomic compact hulls and reflect the existence of atomic compact hulls. If the type is countable then these embeddings are in addition full; moreover, this is in a sense the best that one can do: we shall see that if such an embedding exists which is also full, then the given type involves fewer than 2 ~~ operations and relations, and so in the presence of the continuum hypothesis is at most countable. Since every mono-unary algebra has an atomic compact hull (Taylor [14]), it is not possible to replace 'two unary operations' by 'one unary operation' in the above statement.

Some helpful comments from Walter Taylor, especially concerning an error in a preliminary version of w and from Jan Mycielski, are gratefully acknowledged.

w 1. Preliminaries

A type is a pair (a, z) where a=(n~)a~ I and z=(n~)~j are families of non-negative integers, and n~>~ 1 for all 2~I; for each type (03 z), n,~= II1 + IJI (where, for a set X, I.t'1 is the cardinal number of X). For such a, z, a structure of type (a, ~) (or (a, z)-structure) is a triple S=(S, (Sa)~i,( f~),~j) where S is a set, S ~ _ S n~ for all 2~I and f~: S n~ ~ S for all #~J.

A relational structure of type a is a pair R=(R , (R~)a~1) such that (R, (R~)a~r, 0) is a structure of type (a, 0). An algebra o f type z is a pair A=(A, (f~),~j) where

i) Financial support from the National Research Council of Canada is gratefully acknowledged.

Presented by J. MycielskL Received May 22, 1974. Accepted for publication in final form March 14, 1975.

368 Evelyn Nelson ALGEBRA UNIV.

(A, 0, (f~)j, ~,)is a structure of type (0, z). If S = (S, (Sa)x ~x, (f~,)~ ~ ~) and T = (T, (Ta)z ~x, (g~),~s) are both structures of type (or, z), then a homomorphismfrom S to T is a set map h : S ~ T such that, for all 2e/', h"~(Sx)~ Tx and for a l l / ~ J , g~h"~=hf, (where, for any natural number n, h " : S " ~ T " is the map given by h"(s t ..... sn)=(h(sl),... , h(s.))). If in addition, h is one-one, and h"~(Sx)= Txc~h(S) "~ for all 2~I, then h is an embedding.

6a,~ is the category of all structures of type (a, z), and all homomorphisms between them. A structure is rigid iff its only endomorphism is the identity map.

For a structure S and an ultrafilter 1[, S (a~ is the ultrapower of S given by the ultrafilter l[, and ~Ss, a is the diagonal embedding S ~ S (u~. Since we will never speak of more than one ultrafilter at a time, we will write 'c5 s' for '6s. u'.

The reader is referred to the books of Gr~itzer [-4] and Mitchell [-8] for background on Universal Algebra and Category Theory respectively.

The definitions o f 'pure embedding' and 'atomic compactness' in 5ao~ can be found in Weglorz [17], Taylor ['11], or Nelson [10]. The actual definitions will not be used here; what will take their places in most proofs are the following charac- terizations, due to Weglorz [,17]:

A hom omorphism h:S --+ T in 5a,,~ is pure iff there exists a homomorphism g from T into an ultrapower of S such that gh is the diagonal embedding of S.

A structure S is atomic compact iffit is a retract of each of its ultrapowers (i.e., iff the diagonal embedding of S into any ultrapower of S has a left inverse).

A pure embedding h: S --+ T is pure-essential iff, for every morphism g in Sa,~ for which the composite gh is defined, if gh is a pure embedding then g is an embedding. T is an atomic-compact hull of S iff it is a pure-essential, atomic-compact extension. (Of course, for any subcategory ~ of 6a,~ one can define, in the obvious way, 'J('- pure-essentialness'; however it turns out that for 'decent' o~e', the two notions coin- cide. This is discussed at the end of w

PROPOSITION 1. I f F: Se --+ 5a ,~, is a full embedding which preserves atomic compactness then n,~ < 2 ~~ +"'' ' ' (and so, in the presence of the generalized continuum hypothesis, n~ <<. No + n~,~,).

Proof. We may assume n~, is infinite. If a=(n~)x~r and z=(n~)~s where IJl~< I then III =n,~, and there are 2 "~ non-isomorphic (or, z)-structures with singleton underlying sets. If IJl~> 2 there are 2 "'" non-isomorphic rigid (a, z)-structures with doubleton underlying sets. In either case, the images of these structures under F provide 2 "~" non-isomorphic, rigid, atomic compact members of 5e ,~,. By Proposition 6.19 of Verney [,15] (which is the analogue for structures of [2, Prop. 5]), or by Taylor [11, Theorem 5.1], every rigid, atomic compact member of 5a,,~, is the atomic compact hull of a structure with at most N0 +n,,~, elements. Since there are at most 2 t%§ such structures, it follows that n~ < 2"~'~< 2 ~~176

Vol 5, 1975 Some ftmctorial aspects of atomic compactness 369

COROLLARY. I f there is a full embedding of Se into a category of structures of finite type (and so in particular, into the category of bi-unary algebras) which preserves atomic compactness, then n , , < 2 s~ (and, assuming the continuum hypothesis, we have

n~,<~r Let 0,, be the initial object (= conull object in ['8]) of the category 5eo. ,. (0~ has

all its relations empty, and as underlying algebra the free algebra of type ~ generated by the empty set; in particular if z = (n~)u~j with nu >t I for all # e J then the underlying set of 0o, is 0.) For Se 5a,,, let ffs be the unique homomorphism from 0, , to S.

A functor F: 5e,~ -~ Se, , , is O-full iff, whenever f : F(S) ~ F(T) is a homomorphism such that fF(~s) = F(~a-) then f = F(g) for some homomorphism g: S ~ T.

Although the hypotheses of the following proposition seem very restrictive, they are nevertheless common features of all the functors we will consider in w and w and so this proposition will be the main tool in later proofs.

PROPOSITION 2. I f F: 5a~, ~ 5e .,, is an O-full, faithful functor with the following property:

S For all S~ 5a,, and all ultrafilters 1~, there is a retractable (*) ~ embedding q :F(S ~u)) ~ F(S) (u) such that ~lF(bs) = fie(s)

then F preserves and reflects purity and atomic compactness. I f in addition F preserves embeddings then it also preserves pure-essential embed-

dings and hence atomic compact hulls; i f F has the further property that ISI ~< IF(S)I for all (~, z)-structures S, then F reflects the existence of atomic compact hulls.

Proof. Suppose h: S ~ T in .Y',, is a pure embedding. Then there exists an ultrafilter and a homomorphism g: T ~ S (u) with gh = fis. But then if r/: F(S (u)) --* F(S) (u) is

the embedding given by (*), then ~F(g):F(T)--*F(S) (u) and ~F(g)F(h)=qF(fs) = fir(s), and thus F(h) is a pure embedding.

Conversely if F(h) is a pure embedding, then there exists an ultrafilter 1~ and a homomorphism g: F ( T ) ~ F(S) (a) with gF(h)= fir(s)- Let ~ be the left inverse of the embedding r/:F(S(U))~F(S) (a) as given in (*); then ~gF(h)=F(fs). It follows that ~gr((r)=~gr(h)F((s)=r(fs)V((s)=r((s(U)), and hence by the 0-fullness of F, there is a homomorphism f : T ~ S (u) with F ( f ) = ~g. But then F( f h) = ~gF(h) = F(fs) and so it follows from the fact that F is faithful that fh = 6 s. This implies that h is a pure embedding.

If Se 5a~, is atomic compact, and !.~ is any ultrafilter, then there exists a homomorphism f : S(u)--* S with ffis the identity map on S. Let ~:F(S)~U)~ F(S (a)) be the left inverse of the embedding F(S r ~ F(S) r as given in (*); then F ( f ) ~ is a left inverse of the diagonal map F ( S ) ~ F(S) (u) and thus F(S) is atomic compact.

Conversely, if F(S) is atomic compact, and lI is an ultrafilter, then there is a


homomorphism h:F(S) (u) ~ F(S) with h6r(s) the identity map on F(S). But then if r/:F(S(U))--*F(S) (u) is the embedding given in (,), then h~lF(fs(U))=h~lF(~s)F((s) =F((s ) and hence by the 0-fullness of F there is a homomorphism g:S(t[)--* S with hr/=F(g). But then F(gbs) is the identity map on F(S) and hence by the faithfulness of F it follows that g6 s is the identity map on S. This shows that S is atomic compact.

I f F preserves embeddings and h:S--* T in S e is pure-essential, then by the above, F(h) is pure. If g: F(T) --* U is a homomorphism with gF(h) pure, then there exists an ultrafilter 1L and a homomorphism f : U ~ F(S) (u) with fgF(h)= fir(s). Let r (u) ~ F ( S (u)) be the left inverse of r/ as given in (.); then ~fgF((T)=r = F((Ss) F( (s )= F(~s(u)) and hence by the 0-fullness of F it follows that there is a homomorphism g: T --, S (u) with F(g) = ~fg. Since F(gh) = F(~Ss) it follows from the faithfullness of F that gh = 3 s and hence gh is a pure embedding; since h is pure-essential, this implies that ~ is an embedding, and consequently F(~)=~fg, and so also g, are embeddings.

If [TIc<IF(T)! for all T ~ , and if F(S) has an atomic compact hull, U, then suppose f : S --* T is a pure-essential embedding. Then F ( f ) : F(S) ~ F(T) is pure, and U, being atomic compact, is pure-injective (see, for example, Nelson [10]); it follows that there is a homomorphism g: F(T) ~ U with gF(f): F(S) -o U the natural embedding. But F( f ) is actually pure-essential, and gF(f) is pure; thus g is an embedding. From this it follows that IWl ~< IF(T)[ ~< IUh and thus S has only a set, up to isomorphism, of pure-essential extensions. It follows from Verney [15, Proposition 6.14] (which is the analogue for structures of [2, Proposition 2]) that S has an atomic compact hull.

w Some embeddings

All of the functors considered in this section were defined in Hedrlin-Pultr [6-1, and shown there to be full embeddings. We will obtain our result by verifying that each functor satisfies the hypotheses of Proposition 2. Before we begin, some nota- tional definitions:

~ is the category of all relational structures of type r I ~ is the category of all (binary) relational structures of type (nx)x, I where nx = 2

for all ).eL .~ is the category of all relational structures with one binary relation (i.e. mono-

binary relational structure). is the category of all groupoids, i.e. algebras with one binary operation.

I d is the category of all (unary) algebras of type (nj,),~r where n~,= 1 for all # e L In particular, 2 d is the category of all bi-unary algebras.

S ~ ' ~ . ~ t , ] : Suppose a= (nx )~ i and z=(n~),~j where nz>~l for all 2~/, and suppose for convenience that I n J = 0. Let tr Iv] = (mz)x ~ t~s where mz = nx for 2~I and

Vol 5, 1975 Some funetorial aspects of atomic compactness 371

m~=n~+ I for #eJ . Then F,, : 5 " - -+ ~-t,~ is defined as follows: for S = ( S , (&)~z, (f~)~s)eSe,, , F, , (S)=(S , ( S a ) a ~ ) where, for #e J, Su is the graph off~, i.e.

S~--- {(s, ..... s. . + ,) [ f~(s~ ..... s . . )=s . .+ , } ,

and F, , is the 'identity map' on morphisms. Then F, , is a full embedding (and so in particular 0-full and faithful) and, since it

preserves underlying sets, it is easy to see that F~ actually preserves ultrapowers, and hence trivially satisfies the condition (.) of Proposition 2. It is easy to check that F,~ preserves embeddings, and obviously ISI ~< [F,,(S)I for all S e Sa,~. Thus F~, preserves and reflects purity and atomic compactness, preserves pure-essential embeddings and atomic compact hulls, and reflects the existence of atomic compact hulls.

Handy Trivial Observation. Suppose that R = (R, (R~)~,t) is a relational structure of type (n~)z~x, and that K is an ultrafilter on a set M. For each 2e L let (RU)~ and (R(U))~ be the 2th relation of the relational structures of R u and R (a) respectively. For fe(Ra) M, let fe(RM) "* be given by f(i)(m)=f(m)(i) for i<nz, meM. Then fe(R~t)a, and it is easy to check that the map q~:(Rz)M ~ (R~t)~ given by ff ( f ) = f is one-one and onto. Let v: R M --* R (u) and ~: R~ ~t ~ Rz Ca) be the quotient maps, and let v"*:(RU)"~(R(U)) "~ be the map induced by v. Then, for f , germ, (f , g)eKer~ iff (m f (m)=g(m)}a l I iff ( - ) ,<, ,{mlf (m)( i )=g(m)( i )}eY~ iff for each i<n~, {m f ( i ) (m)= g(i) ( m ) } ~ iff ( f , g )eKerv" ~ iff (f,g)eKerv"*~b. Thus Ker~= Ker v"*~. Since ~ maps onto Ra ca) and v"~q~ maps onto (RtU))a, it follows that there is a one-one, onto map ~z:(Ra~ ~ Ra (u) such that ~= ~z"*vqS.

~ , ~ I , z r Let o-=(nz)~,t where nx>~l for all 2~/, and assume without loss of generality that 0, 1 ~L Let

I,,= {(i, 2) I ;tel, i <na) vo {1, 2, 3}.

The functor F . : ~ . - ~ I . d is defined as follows: for R=(R,(Rx)z~t )e~. , the underlying set of F.(R) is

({0, 1} x {0}) u (R x {1}) vo ~ (Rax {2}), J.eA

and the operations in F,,(R) are defined as follows:

k,,.>(x)=(o, o) f, (x)-- (o, o) f,(x)=(1, o) A(x)=(0, o)

for aeRz for x~Rzx {2} for all x for xr 0),f2(1, 0)=(0, 0). for x4=(0, 0),f3(0, 0)=(1, 0).


For a homomorphism h : R - o S in 9~,, F~(h):Fr F~(S) is given by:

F,(h) [ {(0, 0), (1, 0)} is the identity map,

F~(h) (r, l)=(h(r), 1) for reR, and F~(h) (~, 2)=(h"*(~), 2) for aeRx.

(Similar constructions are found in J6nsson [7 p. 52], Taylor [12] and Bulman- Fleming and Taylor [3].)

It was shown by Hedrlin and Pultr [6"] that F~ is a full embedding; F~ trivially preserves embeddings, and [R[ ~< IF~(R)I for all Re ~ .

If 1.I is an ultrafilter on a set M, and Re ~ . , then defne r/:F.(R(U))--+ F.(R) (a) as follows:

r/I {(0, 0), (1, 0)} =Sr.(ml {(0, 0), (1, 0)}

I R(u) 1/ x {1} is the natural map given by the inclusion R x {I}c_Fr ~1 (R(U))z x {2} =P~xPl where Pt :(RtU))a x {2} ~ (R(U))z is the projection map,

~x:(R(U))z~R~m is the map given in the Handy Trivial Observation above, and p: R~ u) --* F~ (R) tu) is the map given by the embedding Rx ~ Rx x {2} _ F, (R).

It is easy to check that ~/is a homomorphism, the point being that all the operations on F~(R) are unary.

It is also a consequence of the Handy Trivial Observation that the image of ~/is precisely the image under the quotient map v:F,(R)M--.F~(R) (u) of the set of all g e F, (R) M such that either

{m I g(m)e {(0, o), (1, o)}}e12 o r

{m [g(m)eRx {1)}eR o r

{m [g(m)eRax {2}}eR for some 2eI .

(In particular, if I is finite then r/maps onto F.(R) {u) and we are finished.) Consequently, if xeF.(R) (u) is not in the image of ~/, then x = v(g) for some

geF. (R) u with {m I g(m)eU,o, R,x {2}}~12 but for all 2eI, {m I g(m)eR, x {2}} r It follows thatfj(x)=ae.(s)(O, 0) for all jeI with j # 2 , andf2(x)=6p.(s)(1, 0). Using this, it is easy to see that any set map F.(R)(n)--+ F.(R {u)) which is a left inverse of r/ and maps F~(R)(U)-im(r/) into R(U)x {1} is a homomorphism, and this then verifies (.) of Proposition 2. It follows that F.: .~.-~ I.ag preserves and reflects purity and atomic compactness, preserves pure-essential embeddings and atomic compact hulls, and reflects the existence of atomic compact hulls.

1again: This is a special case of the functor F. , described at the beginning of this section.

.~ -~ 2ag: Let ~: ~ + 2ag be the functor described in Hedrlin-Pultr I-6, Theorem 3].

Vol 5, 1975 Some functorial aspects of atomic compactness 373

Then for (X, R)Egt, the underlying set of r R) is XUR[A {0, 1} (where ' U ' denotes disjoint union). It follows that if 1i is an ultrafilter on M, then ~(X, R) ~u) is the image under the quotient map �9 (X, R)~t --* �9 (X, R) (u) of the set of all g E 4~ (X, R) u such that either g(m)eXfor all m~M or g(m)6R for all rn~M or g(m)~ {0, 1} for all m~M. This, together with the Handy Trivial Observation, implies that there is an isomorphism ~I:~((X,R)(U))~(X,R) (u) with ~l~(6(X,R))=64,(X,R), and thus satisfies all the hypotheses of Proposition 2.

2~r ~ (r Let T : 2 d ~ ff be the functor described in Hedrlin-Pultr [-6, Theorem 4]. Then for AE2~ r the underlying set of 7t(A) is the disjoint union of {0, 1} with the underlying set of A; from this it is easy to see that for any ultra filter ~ , there is an isomorphism ~I:T(A(U))~T(A) (u) with qT(6A)=f~,(A ) and hence we may again apply Proposition 2.

w 3. Binary relational structures

In the last section, we saw that for any (a, z) there is a full embedding S,'~,, ~ I~(,)~ and full embeddings ~ ~ 2~ r 2 d ~ if, which preserve and reflect purity and atomic compactness, preserve pure-essential embeddings and atomic compact hulls, and reflect the existence of atomic compact hulls. In this section we will fill the obvious gap by discussing functors I ~ ~ R.

For any I = (/, RI)~ R, we define a functor FI:I~ ~ ~g as in Hedrlin-Lambek [5, Proposition 2], that is, for

where R = (X, (R,),~,)~I~, Ft(R) = (X*, R*),

and X * = X U X x X I I I U (0, 1, 2, 3}

R*=R, U {(0, 1), (1, 2), (2, 3), (3, 1), (1, 0)} U {(1, x), (x, 2) [ x~X} [...1 {(2, i), (i, 3) I isI} U ((x,p), (p,y)]p=(x,y)EXx.V} U ((p, i)[p~Ri}.

(The reader is referred to [5] for a very descriptive 'picture' of FI(R). ) For a homo- m o r p h i s m f : R ~ S in IN, FI(U):FI(R)~FI(S) is given by F~(f) lILJ {0, 1, 2, 3} is the identity map, F I ( / ) [ X '=f , and FI(U) IXxX=T2.

Recall that an n-cycle in a structure (X, R) ~ R is an n-tuple (xl ..... x,) of elements of X such that (xl, xi+l)eR for 1 ~<i ~ n - 1 and (x,, xl)~R.

If I is rigid and without two-cycles then Fl is a full embedding [5, Propo- sition 2].

We will sometimes follow the notation of Hedrlin and Lambek, and write 'x ~ y' instead of '(x, y)eR'.


Note that if FI preserves atomic compactness, then I is atomic compact; one sees this as follows: Let 0 be the structure in I,~ with empty underlying set; if F I preserves atomic compactness then F~ (0) is atomic compact. But

F , (0 )= { I l l {0, 1, 2, 3}, RI II {(0, 1), (1, 2), (2, 3), (3, 1), (1, 0)},

and thus if l.~ is any ultrafilter, the underlying set of FI(0) is in a natural one-one correspondence with I (a) II {0, 1, 2, 3}. Since for xeFl(O), x~ I iff 2 ~ x and x ~ 3, it is easy to see that the retraction FI(O) {u) ~Fs(O) given by the atomic compactness of FI(0) provides a retraction I{a)--+ I, and hence I is atomic compact.

However, atomic compactness of I is not enough to ensure that F~ preserves atomic compactness; the following counterexample is due to Walter Taylor: Let I = ([0, 1 ], ~< ), the real unit interval with the usual order; then I is (topologically) compact and hence atomic compact. Moreover, the structure R=({a},(R,),~tO,ll)~[0, 1] ~ , where .~o=0 and R t= {(a, a)} for 0 < t ~< 1, is atomic compact since its underlying set is a singleton. Now let 1~ be any non-principal ultrafilter on N, the set of natural numbers, and let v:FI(R)N~F~(R) (a) be the quotient map. If, for each x~F~(R), . '~F~(R) N is the constant map with value x, and if dsFs(R) N is given by d ( n ) = 2 - " for each n~N then v(d)-4 v (t)for all t~ [0, 1], t # 0, and in F,(R), (a, a)---, Z -" fo r all n ~ N, and thus v((a, a) ̂ )~ v(d). If there were a homomorphism ~:F~(R)(U)~ FI(R) which was a left inverse of the diagonal map, then the image of d under ~v would provide an element xeF , (R) such that (a, a ) ~ x and x-~ t for all t~[0, 1], t:~0. Since this is impossible, it follows that F~ does not preserve atomic compactness.

However , / f I isfinite, then for any R = ( X , (R~)~)s I~ and any ultrafilter K, the underlying set of F~(R) {a) is in a natural one-one correspondence with XCU) l t X Ca) x X ~u) II ILl {0, 1, 2, 3}, and from this it is easy to see that FI(R)(U)"~F~(R~U)). Thus

Fi has property (.) of Proposition 2, and we have:

PROPOSITION 3. I f I ~ ~ is finite, rigid, and without two-cycles then FI : I ~ ~ is a full embedding which preserves and reflects purity and atomic compactness, preserves pure-essential embeddings and atomic compact hulls, and reflects the existence of atomic compact hulls.

Of course, it is well-known that there is a rigid binary relation without two-cycles on every finite set; the example given in Hedrlin-Lambek of such a relation on (1 ,2 ..... n} is l ~ 2 ~ 3 ~ . . . ~ n .

One comment in passing: it is known that every rigid, atomic compact structure of finite type has at most 2 ~~ elements (see Taylor [11, p. 433] or Verney [15, Proposi- tion 6.19]). On the other hand, we have the following example which is due to Walter Taylor: The algebra A with underlying set 2 ~~ and two unary operations f and g given by f(xo, xx,...)=(O, x0, Xx .... ) and g(xo, Xa .... )=(1, Xo, xl .... ) is atomic compact (since it is topologically compact) and rigid. Now, starting with this algebra A, a


suitable combination of applying the above full embeddings (including of course the ones in w and enlarging the type by adding empty relations and trivial operations, produces, for any type a=((nx)z~i, (n~)i,~s) where either nx~2 for some 2 or Znl,>~2, a rigid atomic compact a-structure with 2 ~~ elements.

Here we give an example of a rigid, compact, mono-binary relational structure with 2 ~~ elements, which is due to A. Ehrenfeucht. The underlying set is Z• {a}, where Z is the set of all functionsf:co ~ {0, 1} such that no three consecutive values of f a r e equal, and aCZ, and the single binary relation is

R = {(a,f) ]fEZ, f(O)=0} w {(f, a) [ feZ , f ( 0 ) = I} w H

where H is the graph of the function h:Z ~ Z given by h ( f ) ( i ) = f ( i + 1). Z w {a}, with a as an isolated point and on Z the subspace topology inherited from the og-th power of the discrete topology on {0, 1}, is a compact topological space, and R is a closed subset of (Zw {a}) 2, so this structure is a compact (topological) structure. Moreover it is not difficult to check that it is also rigid: since a is the only element for which there exist elements x, y with (a, x), (x, y), (y, x) and (y, a)~R, every endomorphism ~b maps a identically, and then it follows by induction, using the definition of R, that for all f E Z , ~b ( f ) ( i )= f ( i ) , and hence ~b is the identity map.

Next on our program is the required full embedding I ~ ~ ~ where I is countable. Let N4=(N4, R4) where N4 is the set of natural numbers greater than or equal

to 4, and R4 = {(6, 4), (5, 7)} w {(n, n + 1) [ n>~4}. N4 is diagrammed below:

4 ~ 5 ~ 6 ~ 7 ~ 8 ~ 9 . . .

It is easy to see that N 4 is rigid, and it obviously has no two-cycles; consequently F N , : N 4 ~ ~ is a full embedding.

The proof of the following lemma is a modification of the methods used in Taylor ]-13, w

LEMMA. For every ultrafilter 1.[, there is a homomorphism h: N~ u) ~ N4 such that hgN, is the identity map on N4 and h(U(4U)-im(6u,))~_ {4, 5, 6}.

Proof. Since N~ u) is an elementary extension of N4, and since every element of N4 has finite 'valence' and N4 contains only one finite cycle, it follows that N(4 u) is (isomorphic to) N4 t__l R where R has no cycles. In fact, since all elements x of N4 with the exception of 4, 5, 6 and 7, have the property that there is exactly one y and exactly one z with y ~ x and x ~ z, it follows that N~4 u) has the same property, and hence R is the disjoint union of infinite chains .----+o~o ~o ~ . . . . Since each such chain can be mapped homomorphically onto a three-cycle, and since (4, 5, 6) is a three-cycle in N4, the result follows.

Now, let N 7 be the set of all natural members greater than or equal to 7, and

376 Evelyn Nelson ALO~BRA UNIV.

let H:Nv91~N491 be the functor such that H((X, (R,),>.7))=(X, (R,),~>4) where R4 = Rs = R6 = X x X. H is clearly a full embedding.

PROPOSITION 4. I f 1I[ = No then there is a full em bedding I91 --* 91 which preserves and reflects purity and atomic compactness, preserves pure-essential embeddings and atomic compact hulls, and reflects the existence of atomic compact hulls.

Proof. It is enough to verify that FN,H: N791-0 91 satisfies all the hypotheses of Proposition 2. We already know that it is a full embedding, and it obviously preserves embeddings and does not decrease cardinality of underlying sets.

Suppose R=(X, (R,),>7)eN791 and lI is any ultrafilter. Then the fact that the underlying set of FN,H(R) (m is in a natural one-one correspondence with X (m k_l X r x X Cu) I 1 N [ m k_l {0, 1, 2, 3} provides a natural embedding ~/: FN,H(R (m) ~ FN,H(R) (m. By the above lemma, there is a homomorphism h: N (u) ~ N , which is a left inverse to 6N,, and maps every element of N Cu~ which is not in the image of 6N, into {4, 5, 6}. From this, we obtain a set map ~:FN,H(R)(U)-~ FN,H(R (u~) which is a left inverse of t/; using the fact that in FN,H(R), (x, y) ~ n for all x, y e X and n = 4, 5, 6, it is easy to check that ~ is an 91-homomorphism, and this completes the proof.

COROLLARY. There are 2 ~~ non-isomorphic countable rigid atomic compact structures in 91.

Proof. These are provided by the images, under FN,H, of the structures in N791 with singleton underlying sets.

It still remains to discuss functors I91 ~ 91 where I is uncountable. For any I = ( / , R)e91, let I # be the 'one-point atomic compact extension of I',

i.e., I # = ( 1 # , R # ) where I # =I ra {z} (with z4:I) and R # =Rvo {(x, y) ] x, y ~ I # and {x, y} c~ {z} #O}. Then the functor H , : I 9 1 ~ I # ~ given by HI((:(, (R,)i,,)) =(X, (Rl)i~t#) where R==X x X, is a full embedding.

The functor FI#Hx:I91~ 91 is not a full embedding, since for every R~I91, the map of Fx#HI(R) into itself which is constant with value z is an endomorphism. However, we do have the following

PROPOSITION 5. For all I=(L R)~ 91, the functor Fl#Hz : I91~ 91 is O-full, and satisfies condition (*) of Proposition 2.

Proof. If R=(X, (R,),~x) and S=(Y, (S,),~z)EI~, and f:F~H~(R)--*F~oHx(S ) is a homomorphism such thatf:Fx#H1(~R)=Fx,Hi(~s), t h e n f I I # t__l {0, 1, 2, 3} is the identity map. But now we may apply the argument appearing in Hedrlin-Lambek I-5, p. 201] : since, for x~FI#Hx(R ), 1 ~ x and x ~ 2 iff x~X, it follows thatf(X)___ Y; moreover, since for each x, y~X', x--* (x, y) ~ y , it follows t h a t f ( x ) ~ f ( ( x , y)) ~ f ( y ) and thus f ( ( x , y ) )=( f ( x ) , f ( y ) ) . Also, (x,y)r implies (x, y ) ~ i , and thus ( f (x) , f ( y ) ) ~ i, and hence ( f ( x ) , f ( y ) ) eS , . It follows thatf I X is a homomorphism from R to S, and &#H,(f I X)=f. Thus F,#H, is O-full.


Since for any ultrafilter R, the map I # <u)~ I # which is a left inverse to ~ , and maps everything outside im(6x,) to z is an ~-homomorpkism, a similar argument to the one in the proof of Proposition 4 shows that FI,Hz satisfies condition (,) of Proposition 2.

COROLLARY. For every set I, there is an embedding I ~ ~ ~ which preserves and reflects purity and atomic compactness, preserves pure-essential embeddings and atomic compact hulls and reflects the existence of atomic compact hulls.

Of course, by Proposition 1, if there is a full embedding I ~ ~ ~ which preserves atomic compactness then lit < 2 ~~

This still leaves open the following

PROBLEM. Determine, (without using the continuum hypothesis) the cardinals < 2 ~~ for which there is afullembedding ~ --* ~ which preserves atomic compactness.

If No < 0~ < 2 ~~ and there is a full embedding 0r ~2 preserving atomic compactness, then it is not at all similar to the functors Ft, since by a result of McKenzie and Shelah (which is to appear in the Proceedings of the Tarski Symposium), every rigid atomic compact member of ,~, with fewer than 2 ~~ elements is at most countable.

w Concluding remarks

Combining all the functors in w and w we now have, for a=(n~)~i , T=(n~,)~,~.r (n~>~l for all 2 ~ I ) a n embedding 6 a ~ t ~ ] ~ I ~ t ~ ] d ~ I ~ t ~ ] ~ ? ~ 2 ~ r if, which preserves and reflects purity and atomic compactness, preserves pure-essential embeddings and atomic compact hulls, and reflects the existence of atomic compact hulls. Moreover if I and J are at most countable then this composite embedding is full.

Let F: Sa,~ ~ 2~ r be the composite embedding indicated above. A careful inspec- tion of the individual functors described in w and w will immediately yield the fact that for any infinite structure S~ S t , IF(S)[ ~< ISl + ]II + IJI. This then provides a proof of the fact, essentially due to Taylor [11], that if S and T are (a, -c)-structures and T is the atomic compact hull of S, then [ Tl ~< 2 tsl + 111 + IJI + ~o, as a corollary of the corresponding result for algebras (see Banaschewski-Nelson [2] or Nelson [9]).

Moreover, we can make the following rather curious observation: it is an easy consequence of G. Wenzel's characterization [18-1 of the equationally compact mono-unary algebras that for every S~ 6a,,, both of the mono-unary reducts of F(S) are equationally compact.

Finally, as was mentioned parenthetically in w 1, one might wish to consider the notions of pure-essentialness and atomic compact hulls relative to some subcategory

of o~',~. However, if ~ is any full subcategory of 6a,~ which is closed under the formation of substructures, products, and up-directed colimits, (in the case a = 0 ,

378 Evelyn Nelson

such JY" are precisely the full subcategories of the category of all algebras of type which are quasi-varieties, see Banaschewski-Herrlich [1]), t h e n X is also closed under the formation of ultrapowers (since for any ultrafilter ~ and any S e 6a,,~, S tu) is the (up-directed) colimit of the S x (xsl .D). Using this fact, it is easy to see that such a X is closed under the formation of pure-essential extensions (since any pure-essential extension of S can be embedded in an ultra power of S) and hence also closed under the formation of atomic compact hulls. Moreover, this same fact implies that a homomorphism f : S ~ T in X is pure-essential iff it is X-pure-essential.

Thus we obtain the following sharpening of the result stated in the introduction: For any (a, z), and any full subcategory X of Y,~ which is closed under the formation of products, substructures, and up-directed colimits, there is an embedding of X into the category of all algebras with two unary operations (and into the category of all algebras with one binary operation) which preserves and reflects purity and atomic compactness, preserves (X)-pure-essential embeddings and (~<')-atomic compact hulls, and reflects the existence of atomic compact hulls.

REFERENCES

[1] B. Banaschewski and H. I-Ierrlich, Subcategories defined by Implication. Manuscript (McMaster) 1975.

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

[3] S. Bulman-Fleming and W. Taylor, On a question ofG. H. Wenzel, Alg. Univ. 2 (1972), 142-145. [4] G. Gr~tzer, Universal Algebra, Van Nostrand, 1968. [5] Zdenek Hedrlin and Joachim Lambek, How comprehensive is the category ofsemigroups? J. Alg.

11 (1969), 195-212. [6] Z. Hedrlin and A. PuRr, Onfidl embeddings into categories of algebras, I11. J. Math. 10 (1966),

392-405. [7] Bjarni J6nsson, Some Topics in Universal Algebra, Springer 1972. [8] Barry Mitchell, Theory of Categories, Academic Press. [9] Evelyn Nelson, Infinitary equational compactness, Alg. Univ. 4 (1974), 1-13.

[10] Evelyn Nelson, On the adjointness between operations and relations, Colloquium Mathematicum (to appear)

[11] W. Taylor, Some constructions of compact algebras, Ann. Math. Logic 3 (1972), 395-436. [12] Walter Taylor, A note on pure-essential extensions, Alg. Univ. 2 (1972), 234-237. [13] Walter Taylor, Atomic compactness andgraph theory, Fund. Math. 65 (1966), 139-145. [14] Walter Taylor, Pure-irreducible mono-unary algebras, Alg. Univ. 4 (1974), 235-243. [15] Patricia Verney, Atomic compactness in quasi-primitive classes of structures, M.Sc. Thesis,

McMaster, April, 1974. [16] P. Vopenka, A. PuRr, and Z. Hedrlin, A rigid relation exists on any set, Comm. Math. Univ,

Carolina 6 (1965), 149-155. [17] B. W~glorz, Equationally compact algebras I, Fund. Math. 59 (1966), 289-298. [18] G. Wenzel, Subdirect irreducibility and equational compactness in unary algebras ( A , f ) , Arch.

Math. (Basel) 21 (1970), 256-264. McMaster University

Hamilton, Ontario

Canada

some functorial aspects of atomic compactness

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