Minimal Generators In Varieties

Hans–E. Porst (porst@math.uni-bremen.de)

Department of Mathematics, University of Bremen, 28359 Bremen, Germany

(Received: . . . . . . ; Accepted: . . . . . . )

Abstract. Given a variety V, the free V-algebra F 1 on one generator represents the canonical underlying functor from V to the category Set of sets. Hence one might ask, whether F 1 is, in some sense, a “canonical” generator of V. To make this question precise the notion of “minimal varietal generator” is introduced. It is shown that in many (though not all) varieties F 1 is a generator of this kind, and often even the unique one. The question whether every variety has a (unique) minimal varietal generator, remains on open problem.

Keywords: Regular projective regular generator, retracts in varieties, equational representation of varieties.

Mathematics Subject Classification (1991): Primary 18C05, Secondary 03C05, 08B99

Horst Herrlich zu seinem 60. Geburtstag

Introduction

Varieties (finitary, one–sorted), by their very definition, are concrete categories (V, U). Their underlying functor U is representable by the free algebra F 1 on one generator. This algebra is a finitely presentable regular projective regular generator, or for short: a varietal generator. (see e.g. [1]). However, a variety has usually plenty non–isomorphic varietal generators: for example each finitely generated free algebra is one. It is therefore natural to ask whether there is something special about the varietal generator F 1; intuitively: whether it might be in some sense a minimal varietal generator. I remember Horst Herrlich asking this question some decades ago (for the special case of module categories) and the question doesn’t seem to be answered since. An answer to the affirmative would mean that the operations and equations used in the definition of V are, in a sense, a canonical representation of V.

We begin our investigation by first making precise what a minimal varietal generator should be. A discussion of a couple of varieties familiar to every mathematician then shows, that in many — though not all — cases, the free algebra on generator is in fact a minimal varietal

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generator, and even the unique one. In the final section of this note we generalize the problem of minimal varietal generators to categories more general than one–sorted varieties.

What is a minimal generator?

We call a regular generator G in a variety V, which in addition is regular projective and finitely presentable, a varietal generator. Then

– all finite copowers of G are varietal generators again;

– every varietal generator H is a retract of one of these copowers.

Note that every varietal generator G of a variety V yields a representation of V by operations and equations in the same way as F 1 does (see e.g. [9] for details): V is equivalent (as a category) to the variety V_{G} consisting of all algebras isomorphic to an algebra V_{G} (V in V) which has

– hom_{V}(G, V) as its underlying set and

– hom_{V}(nG, V) ⟶^{hom_{V}(τ,V)} hom_{V}(G, V) (τ ∈ hom_{V}(G, nG) as its n-ary operations

and satisfies all equations which are consequences of the fact that hom_{V}(−, V) is a functor.

If now H ⟶^{s} G ⟶^{r} H = 1_{H} defines a retraction between varietal generators G and H the representation determined by H is simpler than that determined by G in the following sense:

(1) V_{H} is given by at most as many operations as V_{G}. This is a consequence of the fact that the map φ: hom_{V}(G, nG) → hom_{V}(H, nH) with φ(τ) = nr ∘ τ ∘ s, due to projectivity of G, is surjective.

(2)For each V in V the underlying set of the algebra V_{H} is not larger than the one of V_{G}. This follows from the equation r∘s = 1 and functoriality of hom_{V}(−, V).

These observations suggest to consider the following preorder on (isomorphism classes of) varietal generators:

H ⊑ G iff H is a retract of G.

This brings about the following possible notions of minimality (both collapsing to the notion of minimal element in the associated order):

Definition. A varietal generator G in a variety V is called

– minimal provided none of its proper^{1} retracts is a regular generator.

– preminimal provided that for each varietal generator H with H ⊑ G one also has G ⊑ H.

Obviously every minimal generator is preminimal. Related — but stronger — notions of minimality (also based on the preorder ⊑) have been used for generators in arbitrary categories in [7] (integral object) and in [10] (basic codirect object).

Remark. Every retract of a varietal generator is regular projective and finitely presentable; hence, if a retract of a varietal generator is not a regular generator, it is not a varietal generator either.

Varieties with minimal generators

1. Abelian Groups

1.1. In the category Ab of Abelian groups the group of integers ℤ is a minimal varietal generator.

Proof. ℤ is a varietal generator, being free on one generator. Evidently the only retracts of ℤ in Ab are the groups nℤ, n ∈ ℕ which are either isomorphic to ℤ (n ≠ 0) or the zerogroup, which is not a generator. ♢

1.2. ℤ is the only minimal varietal generator in Ab.

Proof. Every other minimal varietal generator must be a retract of some ℤ^{n}, but the only retracts of ℤ^{n} are (upto isomorphism) either ℤ^{m} with m ≤ n or 0. For m > 1, however, ℤ^{m} would have ℤ as a proper retract; hence m = 1 and G ≃ ℤ. ♢

2. Groups

2.1. In the category Grp of groups the group of integers ℤ is a minimal varietal generator.

Proof. The arguments for Ab apply here as well, since ℤ is also the free group on one generator. ♢

1 We call a retract G′ of G proper, if G′ is not isomorphic to G; observe that a retract might be improper even if the retraction under consideraton is not an isomorphism.

2.2. ℤ is the only minimal varietal generator in Grp.

Proof. Every other minimal varietal generator G must be a retract (hence isomorphic to a subgroup) of the free group F n on n generators. Hence, by the Nielsen–Schreier Theorem, G is a free group. But then the codiagonal ∇: G ⟶ ℤ = F 1 is a retraction. Now minimality of G shows G ≃ ℤ. ♢

3. Commutative Rings

3.1. In the category cRing of commutative unital rings the polynomial ring ℤ[X] is a minimal varietal generator.

Proof. ℤ[X] is a varietal generator, being the free commutative unital ring on one generator.

If G is a retract of ℤ[X] in Ring, G will be isomorphic to a subring ℤ[p(x)] of ℤ[X] with some polynomial p(x) ∈ ℤ[X]. For p(x) constant this will either be the zero ring or the ring ℤ (up to isomorphism), and none of these rings is a generator. For p(x) not a constant ℤ[p(x)], hence G, is isomorphic to ℤ[X]. ♢

3.2. ℤ[X] is the only minimal varietal generator in cRing.

Proof. Every other minimal varietal generator G must be a retract of ℤ[X_{1}, …, X_{n}] — the free commutative unital ring on n generators — for some n ∈ ℕ. Hence G is isomorphic to ℤ[p_{1}, … p_{n}] for polynomicals p_{1}, …, p_{n} ∈ ℤ[x_{1}, …, x_{n}]. Chose a minimal system q_{1}, …, q_{m} ∈ {p_{1}, …, p_{n}} such that ℤ[p_{1}, …, p_{n}] = ℤ[q_{1}, …, q_{m}]; then G is isomorphic to ℤ[X_{1}, …, X_{m}]. In case m > 1 ℤ[X_{1}] would be a proper retract; hence m = 1 and G ≃ ℤ[X]. ♢

4. Modules over PID’s

4.1. Let R be a principal ideal domain. Then R is a minimal varietal generator in the category Mod-R of right R–modules.

Proof. R, being the free module on one generator, is a varietal generator. Every retract of R is (isomorphic to) a submodule of — hence an ideal — of R. Thus the only possibilities for retracts are 0 or isomorphic copies of R. ♢

4.2. R is the only minimal varietal generator in Mod-R, provided R is a principal ideal domain.

Proof. The argument is basically the same as in the case of Abelian groups. ♢

5. Modules over arbitrary rings

If R is not a PID but at least an integral domain, the existence of a minimal varietal generator can be proved as before: any retract of R in Mod-R is isomorphic to xR for some x ∈ R, hence either 0 or isomorphic to R.

5.1. For every integral domain R, the right R–modul R is a minimal varietal generator in the category Mod-R.

For more general rings the situation might change: it is well known that for any ring R the category Mod-R is equivalent to the category Mod-S_{n}, where S_{n} is the endomorphism ring Hom_{R}(R^{n}, R^{n}). The equivalence Mod-R⟶^{Φ} Mod-S_{n} is given by M ⟼ Hom_{R}(R^{n}, M) with S_{n}–action on Hom_{R}(R^{n}, M) defined by composition (see Prop. 41.16 of [4]). If G is a minimal generator in Mod-R then ΦG clearly is a minimal generator in Mod-S_{n}. However, ΦR is not isomorphic to S_{n} considered as an S_{n}–Modul (R and R^{n} are not isomorphic as R–modules, for n > 1, since R is commutative and unital, and Φ(R^{n}) ≃ S_{n}.) Moreover, since R is a retract of R^{n} in Mod-R, ΦR is a retract of Φ(R^{n}) = S_{n}. Hence, for Mod-S_{n}, we have the following statement contrasting 4.1:

5.2. Let R be a commutative unital ring and S_{n} the endomorphism ring of R^{n} in Mod-R (n ∈ ℕ, n > 1). Then the following hold

1. Mod-S_{n} has a (unique) minimal varietal generator iff Mod-R has one.

2. S_{n} fails to be a minimal varietal generator in Mod-S_{n}.

6. The category MOD of all modules

Let MOD be the category of all modules. The most canonical description of MOD is given in terms of 2–sorted algebras: objects of Mod are triples (R, M) where R is a commutative unital ring, and M an R–module; morphisms (R, M) ⟶ (S, N) are pairs (φ, f), consisting of a ringhomomorphism φ: R ⟶ S and an R-module-homomorphism f: M ⟶ N_{R} where N_{R} is the R–module obtained from N by “pulling back along φ”. It is well known that MOD is a one–sorted variety (see e.g. [2], [8]).

6.1. In the category MOD of all modules the object (ℤ[X], ℤ[X]) is a minimal varietal generator.

Proof. It is not difficult to see that the ℤ[X]–module ℤ[X] is free over a singleton with respect to the underlying functor (R, M) ⟼ R × M, which is known to be monadic; hence this object is a varietal generator in MOD (see also [3]). Since for a retract (R, M) of (ℤ[X], ℤ[X]) R has to be a retract of ℤ[X] in cRing, one has R ∈ {0, ℤ, ℤ[X]} by section 3, hence R ≅ ℤ[X] if (R, M) is a minimal generator; by (5.1) it follows that then also M ≃ ℤ[X] in Mod–ℤ[X], which clearly is a (full) subcategory of MOD. Consequently (ℤ[X], ℤ[X]) is minimal in MOD. ♢

7. Boolean algebras

7.1. In the category Bool of Boolean algebras the powerset algebra P(2) on the 2-element set 2 is a minimal varietal generator.

Proof. Since P(2) is the free Boolean algebra on one generator, only minimality is to be proved. Now the only retracts of P(2) in Bool are the 2–chain and P(2) itself, but the 2–chain is not a generator. ♢

7.2. P(2) is the only minimal varietal generator in Bool.

Proof. Let G be any generator of Bool and s: P(2) ⟶ P(2) the Boolean homomorphism exchanging the singletons {0} and {1}. Then there exists a morphism x: G ⟶ P(2) with s ∘ x ≠ s ∘ id = s. x must be surjective; choose, for {1} ∈ P(2) some g ∈ G with x(g) = {1}. Since P(2) is free over {1}, this choice extends to a morphism r: P(2) ⟶ G. Now xr({1}) = x(g) = {1}, hence xr({0}) = xr({1}′) = [xr({1})]′ = {0}, and therefore xr = id. Hence, P(2) is a retract of every generator in Bool, thus the only minimal varietal one. ♢

8. An artificial example

Let pAb be the variety of pointed Abelian groups, i.e. objects are abelian groups with a chosen additional constant (no additional equations) and morphisms are the group homomorphism respecting the chosen constants. Obviously, the free pAb–algebra on one generator is F 1 = (ℤ^{2}, (0, 1)) with (1, 0) as the free generator and (0, 1) as chosen constant.

8.1. In the category pAb of pointed Abelian groups the pointed group (ℤ^{2}, (0, 1)) is a minimal varietal generator.

Proof. If G is an Abelian group with chosen constant p ∈ G, and r: F 1 ⟶ G a retraction in pAb, G is isomorphic to a pointed subgroup of F 1 generated by one element. Hence, G ≃ nℤ × ℤ for some n > 1 with chosen constant (0, 1) (which is isomorphic to F 1), or G ≃ {0} × ℤ with constant (0, 1) which is the initial algebra in pAb , hence not a generator. ♢

8.2. (ℤ^{2}, (0, 1)) is the only minimal varietal generator in pAb.

Proof. Observe that (ℤ^{n+1}, (0, …, 0, 1)) is the free pointed Abelian group on n generators and continue argueing as in section 1. ♢

Outlook

The example of right modules over a ring which isn’t necessarily an integral domain (section 6), showing that the free algebra in a variety might not be a minimal generator, is not really surprising: for any varietal generator G in a variety (V, U) its n–th copower nG is a varietal generator, as mentioned before. By the introductory remarks this means that there exists a variety (V′, U′) which is concretely equivalent to (V, hom_{V}(nG, −)). Now the free V′–algebra on one generator is the varietal generator nG (or rather its image under the concrete equivalence (V ⟶ V′)), which clearly fails to be minimal in general (for n > 1).

Unfortunately we did not succeed in proving existence or uniqueness of (pre-)minimal generators in varieties in general. Non-existence of a preminimal varietal generator would imply that there exists a nonstationary chain of retractions (in some variety V)

F m = Q_{0} ⟶^{r_{1}} Q_{1} ⟶^{r_{2}} Q_{2} ⟶ … ⟶^{r_{n}} Q_{n} ⟶ …

where F m is the free V-algebra on m generators (m ∈ ℕ) and each Q_{i} is a proper retract of Q_{i−1}. Though this might seem unlikely to be possible, it can happen^{2}. Together with the observation that the proofs of existence of minimal generators in the examples above don’t show any common pattern this might even suggest that minimal generators need not exist in arbitrary varieties. In any case we doubt that minimal varietal generators (if they exist) will be unique in general.

It is obvious how to generalize the observations of this note to many– sorted varieties. One would have to generalize the notion of minimal varietal generator as follows: a strong generator {G_{i} | i ∈ I} (I a set)

2 This was pointed out to me by J. Rosický.

consisting of finitely presentable strong projective is minimal, provided no family {G′_{i} | i ∈ I} with proper retracts G_{i} ⟶ G′_{i} is a strong generator. Observe that in this context one cannot expect unicity of minimal generators: in the category MOD (see section 6) also {(ℤ, ℤ), (ℤ[X], 0)} would be a minimal varietal generator (see [3]).

Using the same notion of minimal varietal generator, one also could investigate quasivarieties, probably with the same results. It seems not to be clear, however, what a minimal strong generator in an arbitrary finitely presentable category should be.

References

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4. H. Herrlich and G. E. Strecker, Category Theory, 2nd ed., Heldermann Verlag 1979

5. J. R. Isbell, Subobjects, adequacy, completeness, and categories of algebras, Rozprawy Matem. 36, Warszawa, 1964

6. N. Jacobson, Basic Algebra, Vol II, Freeman 1980

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391–411

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