21

On Semigroups Whose Proper Subsemigroups Have Lesser Power

RALPH MCKENZlE

Jensen and Miller [3] have shown that every semigroup of power N O which possesses no proper subsemigroup of the same power is a group. The object of this paper is to prove the same result for every infinite semigroup, using the generalized continuum hypothesis (G.C.H. in what follows). Actually, we shall prove without using the G.C.H. that if A is a semigroup of power ~ (where ~ is infinite), which is not a group and which contains no proper subsemigroup of power c~, then e is cofinal with co and (3fl) (fl < a A ~ <2S). (It wiII further be shown that the operation of A can not be commutative.)

Since about 1962 when B. J6nsson asked whether there may exist infinite alge- bras :) possessing no proper subalgebra of their own power, such algebras have been called J6nsson algebras. Whether or not there exists a J6nsson algebra of power is a question whose answer definitely depends upon which axioms are assumed to hold in the theory of sets. This question has been thoroughly studied; some of the results can be found in [1] and [4], although many have never been published. [We remark, for instance, that in Zermelo-Fraenkel set theory with the axiom of choice, one can prove the existence of Jdnsson algebras with one binary operation, having power ~=Nn, for n=0 , 1, 2, . . . . This was proved by Fred Galvin for n = l , and by F. Rowbottom and C. C. Chang for n > 1.]

Problems which concern the existence and the description of J6nsson algebras belonging to some restricted class of algebras were, of course, formulated by alge- braists prior to 16nsson. O. J. Schmidt formulated these problems for groups about 1930, and he must have realized that the abelian J6nsson groups are just the generalized cyclic groups, i.e. Z(p~ wherep is a prime. To this date no other J6nsson groups are known. Apparently, the best known result is that every non-abelian JSnsson group must contain two elements which generate an infinite subgroup (Strunkov [7]). Actually there are at least two J6nsson problems for groups, depending upon whether multiplication, or multiplication and inversion are the fundamental operatiorts. In this paper we reduce the problem for semigroups to the former problem for groups (assuming the G.C.H.).

We remark that J6nsson problems for rings have been discussed in [5] and [6] (every uncountable J6hnsson ring is a non-commutative division ring - it is not known if any exist); for lattices they were discussed in [8] (there is no J6nsson

:) Algebra, in this context, means an algebraic system, with the only restriction that the basic operations are fmitary (nullary, unary, binary, etc.) and that they are finite in number.

Presented by C. C. Chang. Received March 23,1970. Accepted for publication July 8,1970.

22 R. McKenzie AL~EBm~ trY~v.

lattice of power ~, if e is a regular cardinal); and H a n f has established (and an- nounced in [2]) as a corollary of a stronger theorem that, if there is a J6nsson algebra of power ~, then there is also such an algebra of power c~ which is a commutat ive loop.

O. Preliminaries

Given a semigroup A (i.e. A is a non-empty set provided with an associative binary operation, o), we employ the usual notation for the product of two subsets of A : B o C = { b o c [ b e B ^ c s C } ; and of course if x e A and B~_A then xoB={x}oB.

Note that every non-empty subset of the form A o C or B o A is a subalgebra (or subsemigroup) of A, in fact a one sided ideal oI A. The semigroup A is a group if, and only if, ( V x s A ) ( x o A = A ^ A o x = A ) . When given a non-empty set B ~ A we write [B] for the subalgebra generated by B:[B] = B u Bo B u Bo Bo B w. . . .

The left and right translations in A are the functions (t e A):l t (x )= to x, r, (x )= xo t. I f S~_domain o f f , where f is a function, then we write S ] f for the restriction o f f to S.

The letters a, fl and 6 denote infinite cardinal numbers; m and n denote finite car- dinals; and ~ denotes an arbitrary ordinal number, co ( = No) is the least infinite car- dinal, a+ is the cardinal successor of a. cf(a) is the cofinality of a, the least 5(~<a)

such that ~r ~ fl~ = a for some cardinals fie < ~. The power, or cardinality, of a set S is denoted by IS[. The basic facts and defini-

tions of cardinal arithmetic will mostly be taken for granted. Recall that if B, C~_A

then I B~ CI ~< IBI" ICI (= IBI • ICI if one of the two numbers is infinite).

T H E O R E M . Let A be an infinite semigroup which has no proper subalgebra of its own power and which is not a group. Let e = IA[. Then cf(e) = co; and there zs a car- dinal fl < c~ such that ~ < f t . Furthermore, A is not commutative.

C O R O L L A R Y . The only infinite commutative semigroups without proper sub- algebras of the same power are the generalized cyclic groups.

C O R O L L A R Y . (G.C.H.) Every infinite semigroup which is not a group possesses a proper subalgebra of the same power.

Remark. I t seems likely that the second corollary can be proved without making any questionable set-theoretic assumptions, but I have not found such a proof.

1. Proof of the Results

Since the corollaries are immediate we restrict ourselves to proving the theorem. Let A be as in the theorem, with e = [AI. Let 6 = cf(e). First observe that since any

Vol. 1, 1971 On Semigroups Whose Proper Subsemigroups Have Lesser Power 23

subset S~_A such that AoA~S is a subalgebra, we must have AoA=A. We now

define (50 to be the least fl such that there is a set B~_A with IBI =/~ and BoA=A. Likewise, let (51 be the least fl such that A .B=A for some set B having power /L We also define, for all xeA:

L ( x ) = { y e A [ 3 z y o z = x } ,

R ( x ) = { y ~ A [ 3 z zoy=x}. IfB~_A has power a, then Bo A = A = Ao B. (1)

For if IBI = a , then [ B ] = A , and it is clear that [B]oA=BoA and Ao[B]=AoB on general grounds. Thus the result follows f rom the fact that AoA =A. F r o m (I) and the fact that x ~ (A - L (x)) o A u A o (A - R tx)) we get:

IA - L(x) [ u [A - R (x)l < e (whenever xeA) . (2)

Next we observe: (50, (51 e{1, (5, ~}. (3)

To see this, let BoA=A where [B[ =(50. Assume that 1 <(5o<e . Then for all xeA, IxoAI <~ , since IxoAI = ~ implies xoA=xoAoA=A by (1) - but (50 was defined to be

minimal. N o w it is known f rom set theory that f rom the expression of A as the union o f < e sets o f power < e - i.e. A=L.){xoA [ x e B } - one may extract a subfamily of

cf(c 0 of the sets whose union has power c~. Thus there exists B1 _ B , IBll =(5 and IBlOA[ = e . F r o m (1) we have A =BxoAoA =BloA. By the minimality o f (50 this gives

(5 = IBl[ = (50. A dual p r o o f gives (51 e { 1, (5, c~}.

(50 # 1 # (51 . (4)

Suppose for instance that 6o-- 1. Then we have an element a e A such that aoA = A. Clearly, for any x, x ~ A = A ~ a e x o A. Therefore {x [ xo A = A} = L (a). N o w the former

set is a subalgebra, so it follows f rom (2) that L(a)=A; so (VxeA) (xoA = A ) . Since A is no t a group we can thus find an element beA such that Aobr Then [Aob[ < e so there can be found two elements e and d so that cCd and cob=dob. By(2) we can

no t have R(c)nR(d)=O; it is possible to write e=c'ou, d=d'ou for some u. Finally, since uoA =A we can get uov=u for some v. Then coo=c and dov=d. But veboA~ =, c ~ v = d ~ v; a contradict ion. By a dual argument, (51 = i leads to a contradict ion.

(50 = (51 = (5. ( 5 )

Assume that this is not true, say (51 ~ (5. Then by (3) and (4), (5 < (51 = e. We con-

elude f rom this that for some ]~<ce, IAoxl ~</~ for all xeA. (Otherwise there would exist a set B of power (5 such that IAoBI = e , i.e. AoB=A and this would contradict (5 < (51.) We can assume that (5,N</~, obviously.

Let (S , < ) be a well-ordered subset o f A isomorphic to /~§ Fo r each seS let

3~={xeAlAoxc~S<s}. Because IAoxI~</L we see that U{~s[seS}=A. Since

24 R. McKenzie ALGEBRa UNIV.

c~ is s ingular we have fl+ <ce, and hence f rom the family o f sets {S~} we can extract a subfamily {St ] teT} whose union has power c~ and where IT] = 6 . Since 6<fl + =[S] and fl+ is regular, there is an element g e S such tha t T < $ . Then U { S t ] t~T}~_F.~, giving tha t ]~[ =~ . Clearly, A o f f ~ S ~ follows f rom the definition, so by (I) ~ = A - - =AoS~. On the other hand, gCAo.7~; and this is a contradict ion.

Let B~_A. Then B~ = A ~:~ Ao B= A. (6)

Suppose for instance tha t BoA=A and AoB#A. Let fl=IAoBI. We have fl<ce and by (5) (since AoBoA =AoA =A) , 5 ~<fl. By (5), and the fact tha t B~ =A, there is a set C~_A such tha t [C1=6 and Ao(BoC)=A. But [AoBoC] <<.]AoBI.IC]=fl.6=fl. This is a contradict ion.

( V x e A ) (3yeA) (L (y) ~_ R (x)). (7)

This just re formula tes the implicat ion Bo A = A =~ A o B = A o f (6).

We are now able to obta in the first o f the conclusions stated in the theorem, viz. :

a = 02. (8)

Proof of(8): Let aeA. We use (7) and the fact tha t A o A = A to cons t ruc t three sequences o f elements o f A - {a.}, {b.} and {c.} (n<co) - satisfying: (i) ao = a ; (ii) L(b.)~_R(a.) fo r alt n<02; and (iii) b.=a.+toe . for all n<eo. In fact we put ao=a , choose bo so tha t L(bo)~_R(ao), and factor bo=aaor arbi t rar i ly; then repea t the process beginning with a~ to get bi, a2, e, . . . . . N o w put

s = N (R (a.) l" < a (h {L (e.)t n < o2}.

I claim tha t [S] _ ("){R (a.) [ n < 02}. To establish the claim, since S itself is included in the intersection, it is sufficient to show that i f t~R(a.) for all n, and if s~S , then t o s ~ R (a , ) for all n. In fact, if t ~R (a,, + 1) and s ~ L (c,,) then t o s ~ R (a=). (Say a= + x = uo t and c,, = s ~ v. Then b,, = uo t o so v by (iii) and so u o to s ~ L (b,,). Hence by (ii), u o t ~ s ~ R (am); and this implies toseR (a=).)

Thus the claim is established. In part icular, IS] _~R (a). N o w if 5 > 02 then (by (2)) we see tha t IS[ = a and consequent ly [ S ] = A , giving R(a)=A. Since a was arbi t rary we would have R(x)=A for all x~A, i.e. Aoy=A for all yeA. This cont radic ts (4). Thus 5 = 02.

a > 02. (9)

We can take this result f rom [3], or prove it directly as fol lows: Assume tha t a=02. Then by (4) each one o f the sets xoA is finite. Suppose tha t xooA = {xooao,. . . , xooa,}. By (2) there exists Co,..., e,, Yo so tha t a,, --- c,,oy o (m <~ n). Thus xooAoyo =xooA. N o w R(yo) is cofinite and ~_{YlXooAoy=xooA} - which is a subalgebra. Thus we see t ha t (Vy) (xooAoy=xooA). As x o was arbi trary, and a dual result can be p roved ; we

Vol. 1, 1971 On Semigroups Whose Proper Subsemigroups Have Lesser Power 25

find that ('r y) (xo A = xo A o y = Ao y). This certainly contradicts the fact that Ao A = A (since Ao y # A).

(VxEA) (Vfl) (Ix oAI = fl--*fl ~< o v c~ < 2a). (10)

Proofof(lO): Let b~A and IboAl =ft. We consider an equivalence relation (con- gruence) on A : x ~ Y r176 1 G =b~ ~ ry (where boA ~ ={b)uboA). The number o f equivalence

classes is no more than 2aw co. Suppose that this number is <~. Then there must be

a set T~_A with ]TI = 3 = co and I{Y l (3t~T) ( y ~ t))l =~ . This set o f power c~ generates A, and so (Vx~A) (3y~ [T]) (x,,,y). But that implies that b~ ~ bo [T], and hence that

IboAI ~<co as desired.

(3fl < ~) (~ < 2P). (11)

To show this, we note that since o is cofinal with ~, there can not exist a cardinal fl satisfying ~ = 2 p. Suppose then that ( l l ) fails. Then, by (4) and (10) and the last remark, every set xoA is countable. By (8) and (5) there is a countable set Bc_A with A =B~ So we derive [A[ ~<~{[boA[ [ b~B} <~co, contradicting (9).

A is not commutative. (12)

Suppose that A is commutat ive. Let xeA . We know that lxoA[<c~ (by (4)).

Clearly, there are sets B and C~_A such that IBl=3=co, ICl=c~ and xoB=xoC. Since A is commutat ive, it follows f rom this that xo [B]=xo [C]=xoA. So we have

shown that IxoAI ~<o9 for arbitrary x~A. This leads to the same contradict ion met in proving (11).

REFERENCES

[1] P. ERDOS and A. HAJNAL, On a Problem ofB. J6nsson, Bulletin de L'Acad6mie Polonaise des Sciences, S~rie des sciences math., astr. et phys. 14, 19-23 (1966).

[2] W. HANF, Representations of Lattices by Subalgebras (preliminary report), Bull. of the Amer. Math. Soc. 12, 373-374 (1965).

[3] B. A. JENSEN and D. W. MILLER, Commutative Semigroups which are Almost Finite, Pac. J. of Math. 27, 533-538 (1968).

[4] H. J. KEISLER and F. R.OWBOTTOM, Constructible Sets and Weakly Compact Cardinals, Notices Amer. Math. Sue. 12, 373-374 (1965).

[5] A. KOST/NSKY and R. McKENzIE, Cardinality Problems for Rings (preliminary report), Notices Amer. Math. Sue. 15, 389 (1968).

[6] A. KOSTINSKY, Some Problems for Rings and Lattices within the Domain of General Algebra (Thesis), Univ. of California at Berkeley (1969).

[7] S. P. STRONKOV, Subgroups of Periodic Groups, Soviet Mathematics - Doklady 7, 1201-1203 (1966).

[8] T. P. WI~ALEY, Algebras Satisfying the Descending Chain Condition (Thesis), Vanderbilt Univ. (1968).

University of California, Berkeley, California, U.S.A.