alan h. mekler, saharon shelah and otmar spinas- the essentially free spectrum of a variety

8/3/2019 Alan H. Mekler, Saharon Shelah and Otmar Spinas- The essentially free spectrum of a variety

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The essentially free spectrum of a variety

Alan H. Mekler1

Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C.

V5A 1S6 CANADA

Saharon Shelah2

Institute of Mathematics, Hebrew University, Givat Ram, 91904 Jerusalem, ISRAEL

Otmar Spinas3

Department of Mathematics, University of California, Irvine, CA 92717, USA

ABSTRACT: We partially prove a conjecture from [MeSh] which says that

the spectrum of almost free, essentially free, non-free algebras in a variety is either

empty or consists of the class of all successor cardinals.

Introduction and notation

Suppose that T is a variety in a countable vocabulary . This means that isa countable set of function symbols and T is a set of equations, i.e. sentences of the

form x1, . . . , xn (1(x1, . . . , xn) = 2(x1, . . . , xn)) where i are terms. The class of all

1 The author is supported by the NSERC2 The author is supported by the United StatesIsrael Binational Science Foundation;

publication 417.3 The author is supported by the Schweizer Nationalfonds.

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models ofT will be denoted by Mod(T), and a member of Mod(T) is called an algebra in

the variety T. Let M Mod(T). For A M, A denotes the submodel of M generated

by A. Such A is called a free basis (of A) if no distinct a1, . . . , an A satisfy an

equation which is not provable from T. Moreover, M is called free if there exists a free

basis of M, i.e. one which generates M. By F we denote the free algebra with free basis

of size , where is a cardinal. For M1, M2 Mod(T), the free product ofM1 and M2 is

denoted by M1 M2. Formally it is obtained by building all formal terms in the language

with constants belonging to the disjoint union of M1 and M2, and then identifying them

according to the laws in T. For M, M : < such that M, M Mod(T) and M

is a submodel of M for all < , the free product of the Ms over M is defined

similarly, and it is denoted by M{M : < }; the intention being that distinct M ,

M are disjoint outside M except for those equalities which follow from the laws in T andthe equations in Diag(M) Diag(M). Here Diag denotes the diagram of a model. For

M, N Mod(T) we say N/M is free if M is a submodel of N and there exists a free

basis A of N over M, i.e. A is a free basis, N = M A and between members of A

and M only those equations hold which follow from T and Diag(M).

Suppose |M| = . Then M is called almost free if there exists an increasing con-

tinuous family M : < cf() of free submodels of size < with union M. Moreover,

M is called essentially free if there exists a free M

Mod(T) such that M M

is free,essentially non-free otherwise. The essentially free spectrum of the variety T which is

denoted by EINC(T), is the class of cardinals such that there exists M Mod(T) of size

which is almost free and essentially free, but not free.

In [MeSh] the essentially non-free spectrum, i.e. the spectrum of cardinalities of

almost free and essentially non-free algebras in a variety T, has been investigated, and it is

shown that this spectrum has no simple description in ZFC, in general. Here we will show

that the situation is different for EINC(T). Firstly, by a general compactness theorem dueto the second author (see [Sh]), EINC(T) contains only regular cardinals. Secondly, we

will show that EINC(T) is contained in the class of successor cardinals. Our conjecture

is that EINC(T) is either empty or equals the class of all successor cardinals (depending

on T). Motivating examples for this conjecture are among others Z/4Zmodules (where

EINC is empty) and Z/6Zmodules (where EINC consists of all successor cardinals) (see

[EkMe, p.90]). We succeed to prove the conjecture to a certain extent. Namely, we prove

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the following theorem.

Theorem. If for some cardinal , (0)+ EINC(T), then every successor cardinal

belongs to EINC(T).

For the proof we will isolate a property of T, denoted Pr1(T), which says that a

countable model of T with certain properties exists, and then show that, on the one hand,

the existence of M Mod(T) in any cardinality of the form (0)+ implies that Pr1(T)

holds, and on the other hand, from Pr1(T) an algebra M Mod(T) can be constructed in

every successor cardinality.

1. EINC(T) is contained in the class of successor cardinals

Theorem. For every varietyT, EINC(T) is contained in the class of successor cardinals.

Proof: Suppose EINC(T). By the main result of [Sh], must be regular. So

suppose is a regular limit cardinal. Let M Mod(T) be generated by {a : < } andsuppose that M is almost free and essentially free. We will show that then M must be

free, and hence does not exemplify EINC(T).

By assumption and a Lowenheim-Skolem argument, M F is free. Let {c : < },

{b : < } be a free basis ofM F, F, respectively.

Let be a large enough regular cardinal, and let C be the club consisting of

all such that for some substructure A H(), , of size < which contains

M, F, {a : < }, {b : < } and {c : < }, we have A = . Here H()

is the set of all sets which are hereditarily of cardinality < , and is a fixed well-

ordering of H(). Note that the information about M reflects to each C, especially

{c : < } = {a : < } {b : < }.

Since M is supposed to be almost free, the set

C0 = { C : {a : < } is free}

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is still a club. Let , C0 be cardinals with < . We will show that {a : use aLowenheimSkolem argument. So let {b : < } be a free basis ofF, and let {c : < }

be a free basis ofN =M F.

Let be a large enough regular cardinal, and let N, for every < , be a countable

elementary substructure of H(), , such that ,M, F,N, {a : < }, {b : } is stationary. By Fodors Lemma, for some

< , S1 = { S0 : u } is stationary. By assumption, ||0 0 < .

So by thinning out S1 further (using this assumption and the completeness of the

nonstationary ideal on ), we may find a stationary S2 S1 and u such that for

every 1, 2 S2 the following hold:

(4) u1 1 = u;

(5) o.t.(u1) = o.t.(u2), and the unique order-preserving map h = h12 : u1 u2

induces (by c ch()) an isomorphism from {c : u1} onto {c : u2}

which maps a to ah() and b to bh().

Let = min(S2 \ ), M = {a : u} and N = {a : u}.

As C, by elementarity we know that M is free.

As {c : } is a free basis, clearly {c : u} / {c : u} is free, and by

(2) and as

S2 C, also {c : u

} / M is free. Finally, {c : u

} = N F0

by (1). Hence we conclude that N F0 / M is free.

Hence, if the pair M, N does not exemplify Pr1(T), then (iii) in its definition fails.

We will use this to show that then M is free, which contradicts our assumption. Then we

conclude that Pr1(T) holds.

By induction on < we choose w such that the following requirements are

satisfied:

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(6) w0 = ;

(7) |w | < ;

(8) for limit, w =

{w : < };

(9) if () = min( \ w), then w+1 = w {()} {(, n) : n }, where the

(, n) belong to S2, and for any m, n with m < n,

{u(), u : w}

alan h. mekler, saharon shelah and otmar spinas- the essentially free spectrum of a variety

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