Theory and Applications of Categories, Vol. 28, No. 33, 2013, pp. 1153–1169.

GALOIS THEORIES OF COMMUTATIVE SEMIGROUPS VIA SEMILATTICES

ISABEL A. XAREZ AND JOĀO J. XAREZ

Abstract. The classes of stably-vertical, normal, separable, inseparable, purely inseparable and covering morphisms, defined in categorical Galois theory, are characterized for the reflection of the variety of commutative semigroups into its subvariety of semilattices. It is also shown that there is an inseparable-separable factorization, but there is no monotone-light factorization.

1. Introduction

The present work is done at three levels, namely category theory, universal algebra and semigroup theory. The classes of morphisms we will characterize are defined generally at the level of category theory, in particular in categorical Galois theory. Some results are then stated at the level of universal algebra, concerning reflections into idempotent subvarieties of universal algebras (that is, in which every one-element subset is a subalgebra). At the level of semigroup theory, we analyse the reflection of the variety of commutative semigroups into its subvariety of semilattices, making use of the results stated for universal algebra. Some of the results included in this paper were previously obtained by the first-named author and are included in her PhD thesis [10].

A semi-left-exact reflection (in the sense of [3]) can be seen as an admissible Galois structure (in the sense of categorical Galois theory), in which the classes of admissible morphisms are the classes of all the morphisms in both category and subcategory. In the special case of the variety of semigroups, semi-left-exactness was called attainability in [8], where it was proved that the reflection of semigroups into semilattices is the unique attainable reflection of the variety of semigroups into a subvariety. Furthermore, this reflection satisfies the stronger property of having stable units, as shown in [6].

First, we state some results for admissible (= semi-left-exact) reflections of varieties of universal algebras. Second, Proposition 5.3 is stated, proving that the I-images of the projections of any pullback diagram are jointly monic, in the (sub)reflection of the variety of commutative semigroups into its subvariety of semilattices. Then, in the latter subreflection, with the help of the results for varieties and crucial Proposition 5.3, we will be able to characterize the classes of stably-vertical, covering, normal, separable, purely inseparable, and inseparable (homo)morphisms defined in categorical Galois theory. The first table at the end summarizes the characterizations of these classes. The second table addresses the existence of inseparable-separable (see [5]) and monotone-light factorizations (see [2]). In both tables the results used in each case are specified.

This work was supported in part by FEDER funds through COMPETE– Operational Programme Factors of Competitiveness (“Programa Operacional Factores de Competitividade”) and by Portuguese funds through the Center for Research and Development in Mathematics and Applications and the Portuguese Foundation for Science and Technology (“FCT–Fundação para a Ciência e a Tecnologia”), within project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER- 022690.

Received by the editors 2013-10-19 and, in revised form, 2013-12-14.

Transmitted by Walter Tholen. Published on 2013-12-21.

2010 Mathematics Subject Classification: 18C99, 08B99, 20M07.

Key words and phrases: Commutative semigroups, semilattices, admissible reflection, covering morphisms, stably-vertical morphisms, normal morphisms, inseparable-separable factorization.

\begin{matrix}c \\ ◯ \end{matrix} Isabel A. Xarez and Joāo J. Xarez, 2013. Permission to copy for private use granted.

2. Preliminaries

Admissible reflection into idempotent subvarieties

In this section we define semi-left-exact reflection, in the sense of [3], and recall a characterization of a semi-left-exact reflection of a variety of universal algebras into an idempotent subvariety (that is, a variety in which every one-element subset is a subalgebra), which follows from a more general result in [12]. A semi-left-exact reflection is also called an admissible reflection in categorical Galois theory (see [1, §5.5]).

2.1. Definition. A reflection H ⊢ I : C → M into a full subcategory M, with unit η : 1_{C} → HI, is said to be admissible if every pullback diagram

[figure: P; M; I(B) ,; B; u; η_{B}; g; v]

with M ∈ M has u ∈ E_{I}, where E_{I} is the class of morphisms f in C such that I(f) is an isomorphism.

Let us recall some data of a reflection of a variety of universal algebras into one of its subvarieties.

1. Let C be a variety of universal algebras. It is well known that there exists an adjunction U ⊢ F : Set → C between the variety C and the category of sets, with unit λ : 1_{Set} → UF and counit ε : F U → 1_{C}, such that:

(a) F is a functor which assigns to a set S the free algebra F (S) in C.

(b) U is the underlying functor which assigns to any C ∈ C its underlying set U(C) ∈ Set, satisfying the following properties:

i. U reflects isomorphisms, since an isomorphism in a variety of universal

algebras is just a bijective homomorphism; ii. U preserves finite limits, since it has a left adjoint F.

GALOIS THEORIES OF COMMUTATIVE SEMIGROUPS VIA SEMILATTICES 1155

(c) The unit λ_{S} : S → UF (S) is the inclusion map.

(d) The counit ε_{C} : F U(C) → C assigns to the letter x ∈ F U(C) the element x ∈ C, since U(ε_{C})λ_{U(C)} = 1_{U(C)}. On the other hand, ε_{C}(θ_{F U(C)}(x, y, ..., z)) = θ_{C}(x, y, ..., z), for every n-ary operation θ in the variety C, since ε_{C} is a (surjective) homomorphism in C.

2. Let H ⊢ I : C → M be a reflection of a variety C of universal algebras into its subvariety M, with unit η : 1_{C} → HI.

(a) Note that η_{A} : A → HI(A) = A/ ∼_{A} is the canonical projection of A onto the quotient of A by the congruence ∼_{A} associated by the reflection to A, hence a surjective homomorphism, for every A in C.

(b) It is easy to check that the following conditions are equivalent:

i. the subvariety M is idempotent, in the sense that every one-element set

{x} ⊆ M is a subalgebra in any M ∈ M; ii. every map U_{T,M} : C(T, M) → Set({∗}, U(M)) is a surjection, for any

object M ∈ M, with T a terminal object in C.

According to the following Lemma 2.3, stated in [12, §2], a reflection H ⊢ I : C → M into an idempotent subvariety is admissible if and only if every congruence class [x]_{∼_{C}} of the decomposition of any C(∈ C) associated to the reflection is itself I-indecomposable.

2.2. Definition. Consider any morphism μ : T → HI(C) from a terminal object T into HI(C), for some C ∈ C. The connected component associated to the morphism μ is the pullback C_{μ} in the following pullback square.

[figure: C; C_{μ}; HI(C) .; T; μ; η_{C}]

2.3. Lemma. Let H ⊢ I : C → M be a reflection into an idempotent subvariety of universal algebras. The following conditions are equivalent:

1. H ⊢ I is admissible;

2. HI(C_{μ}) ≅ T, for every connected component C_{μ}, where T is the one-element algebra.

Factorization system derived from an admissible reflection

It is well known that there is a factorization system (E_{I}, M_{I}) associated to the reflection. The class of vertical morphisms E_{I} is the class of homomorphisms f in C whose I-images I(f) are isomorphisms. The class of trivial coverings M_{I} is the class of homomorphisms f : A → B such that the commutative diagram associated to the identity HI(f) ∘ η_{A} = η_{B} ∘ f is a pullback diagram in C (see [2]). The following Proposition 2.4 states, in other words, that f ∈ M_{I} if, for every congruence class [a]_{∼_{A}} ∈ A/ ∼_{A}, its restriction f_{|_{[a]∼_{A}}} : [a]_{∼_{A}} → [f(a)]_{∼_{B}} is a bijection.

2.4. Proposition. A homomorphism f : A → B in C belongs to M_{I} if and only if the following two conditions hold:

1. for each b ∈ B and each a ∈ A such that b ∼_{B} f(a), there exists a^{∗} ∈ A for which f(a^{∗}) = b and a^{∗} ∼_{A} a;

2. for all a, a^{∗} ∈ A, if f(a) = f(a^{∗}) and a ∼_{A} a^{∗} then a = a^{∗}.

Proof. The proof follows trivially from the fact that, since the reflection is admissible (hence simple in the sense of [2]), in the pullback diagram (1)

[figure: B; B ×_{HI(B)} HI(A); HI(B) ,; HI(A); m_{f}; HI(f); η_{B}; A; f; η_{A}; e_{f}]

(1)

f ∈ M_{I} if and only if the uniquely determined homomorphism e_{f} = ⟨f, η_{A}⟩ is an isomorphism.

Stably-vertical, Covering morphisms

The class of stably-vertical morphisms E′_{I} is the class of homomorphisms f : A → B in C, whose pullback h^{∗}(f) along any homomorphism h : C → B is a vertical homomorphism. The class M_{I}^{∗} of covering morphisms (coverings, for short) is the class of homomorphisms f : A → B in C, such that the pullback p^{∗}(f) along some effective descent morphism^{1} p : E → B is a trivial covering. Note that a variety of universal algebras C is an exact category, hence an effective descent morphism p : E → B in C is just a regular epimorphism (see [2, §4.7]). On the other hand, a regular epimorphism is a surjective homomorphism in universal algebra. Then, an effective descent morphism is just a surjective homomorphism in a variety of universal algebras.

The following Lemma 2.5 states in particular that, for a reflection H ⊢ I : C → M of a variety of universal algebras C into one of its subvarieties M, a covering morphism f : A → B in C is just a homomorphism such that its pullback ε_{B}^{∗}(f) along ε_{B} : F U(B) → B is a trivial covering, where ε_{B} is a counit morphism of the adjunction U ⊢ F : Set → C.

1A morphism p : E → B in C is an effective descent morphism when the functor “pullback along p”

p^{∗} : C/B → C/E is monadic.

2.5. Lemma. Let ⟨F, U, λ, ε⟩ : S → C be an adjunction with unit λ and counit ε.

If U(p) : U(E) → U(B) is a split epimorphism in S, then there exists f : F U(B) → E in C such that ε_{B} = p ∘ f.

Proof. Since U(p) is a split epimorphism in S, there exists a morphism h : U(B) → U(E) such that

U(p) ∘ h = 1_{U(B)}.  (2)

Since λ_{U(B)} is universal from U(B) to U, there exists a unique f : F U(B) → E such that the following diagram commutes.

[figure: U(B); UF U(B); U(E); λ_{U(B)}; U(f); h]

(3)

Therefore we have,

U(p ∘ f) ∘ λ_{U(B)} = U(p) ∘ U(f) ∘ λ_{U(B)}, = U(p) ∘ h, by (3)

= 1_{U(B)}, by (2)
= U(ε_{B}) ∘ λ_{U(B)}, because λ and ε are respectively the unit and

the counit of the adjunction ⟨F, U, λ, ε⟩.

Therefore, ε_{B} = p ∘ f, since λ_{U(B)} is universal from U(B) to U.

Hence, since the class M_{I} is pullback stable, f : A → B is a covering morphism if and only if the pullback ε_{B}^{∗}(f) of f along the surjective homomorphism ε_{B} : F U(B) → B is a trivial covering.

Separable, Purely Inseparable, Normal morphisms

2.6. Definition. Consider the following pullback diagram

[figure: A; A ×_{B} A; B ,; A; u; f; f; v; A; 1_{A}; 1_{A}; δ_{f}]

(4)

where (u, v) is the kernel-pair of the homomorphism f : A → B, and δ_{f} = ⟨1_{A}, 1_{A}⟩ is the uniquely determined homomorphism.

• f is called a separable homomorphism (f ∈ Sep) if δ_{f} is a trivial covering.

• f is called a purely inseparable homomorphism (f ∈ P in) if δ_{f} is vertical.

• f is called a normal homomorphism (f ∈ Normal) if u is a trivial covering.

Inseparable-separable factorization system

For the factorization system (E_{I}, M_{I}) we can form a derived weak factorization system (see [5]) as follows:

[figure: A; δ_{f}; A ×_{B} A; u; v; A; B ;; f; e_{δ_{f}}; m_{δ_{f}}; ē_{f}; m̄_{f}]

here, ē_{f} is just the coequalizer of (u ∘ m_{δ_{f}} , v ∘ m_{δ_{f}}), where (u, v) is the kernel-pair of f, and m_{δ_{f}} ∘ e_{δ_{f}} is the (E_{I}, M_{I})-factorization of the morphism δ_{f} of diagram (4).

2.7. Definition. A morphism f : A → B is called inseparable if, in its decomposition given in the diagram just above, m̄_{f} is an isomorphism. So, we define the class of inseparable morphisms Ins := {f ∈ C | m̄_{f} iso}.

There is a factorization system (Ins, Sep) if and only if the class Ins of inseparable homomorphisms is closed under composition (see [5, §3,§4]). The following two inclusions P in ∩ E ⊆ Ins ⊆ E_{I} ∩ E, where E is the class of surjective homomorphisms, are satisfied by the class Ins in an admissible reflection into a subvariety of universal algebras (see [11], or the original source [5, §4.2]).

3. Admissible reflections of universal algebras into subvarieties

Next Propositions 3.3 and 3.6, concerning admissible reflections into an idempotent subvariety, that is, a variety in which every one-element subset is a subalgebra, help to characterize the classes of stably-vertical and covering homomorphisms, respectively.

Stably-vertical morphisms

In order to characterize the class of stably-vertical morphisms for an admissible reflection H ⊢ I : C → M into an idempotent subvariety, in which the I-images of the projections of certain pullbacks are jointly monic, we define the class F of “almost surjective” homomorphisms.

3.1. Definition. A homomorphism f is said to be almost surjective if it belongs to the class F = {f : A → B in C | ∀_{b∈B} f(A) ∩ ⟨b⟩_{B} ≠ ∅}, where ⟨b⟩_{B} denotes the subalgebra of B generated by b. In words, a homomorphism is almost surjective if its homomorphic image intersects all the monogenic subalgebras in the codomain.

The following Proposition 3.2 states a necessary condition for a homomorphism to be stably-vertical in any reflection of a variety of universal algebras into a subvariety. Next Proposition 3.3 characterizes the class of stably-vertical morphisms for reflections into idempotent subvarieties, such that the I-images of the projections of any pullback of a homomorphism belonging to E_{I} ∩ F are jointly monic.

3.2. Proposition. The class of stably-vertical morphisms, E′_{I}, satisfies the inclusion E′_{I} ⊆ E_{I} ∩ F, for any reflection H ⊢ I : C → M of a variety of universal algebras C into a subvariety M.

Proof. Consider the pullback diagram

[figure: P; C; D ,; ⟨d⟩_{D}; π_{2}; ⊆; e; π_{1}]

(5)

where e ∈ E′_{I}. Suppose that ⟨d⟩_{D} ∩ e(C) = ∅, for some d ∈ D. Then, P = ∅ = I(P). Since ⟨d⟩_{D} ≠ ∅, I(⟨d⟩_{D}) ≠ ∅ and I(π_{1}) : I(P) → I(⟨d⟩_{D}) is not an isomorphism, which contradicts the assumption of e belonging to E′_{I}.

3.3. Proposition. Consider a reflection H ⊢ I : C → M of a variety C of universal algebras into an idempotent subvariety M. The following two conditions are equivalent:

(a) I(π_{1}) and I(π_{2}) are jointly monic for all the pullback diagrams

[figure: A ×_{C} B; B; C ,; A; π_{2}; g; π_{1}]

in C, such that g ∈ E_{I} ∩ F;

(b) E′_{I} = E_{I} ∩ F.

Proof. (a)⇒(b): Consider the pullback diagram

[figure: I(A); P; I(C) ,; I(B); p_{1}; I(g); I(f); p_{2}; I(A ×_{C} B); I(π_{1}); I(π_{2}); w]

(6)

where f is an arbitrary homomorphism, g ∈ E_{I} ∩ F, P = I(A) ×_{I(C)} I(B) and w is the unique homomorphism making the diagram commute. Since I(π_{1}) and I(π_{2}) are jointly monic, w is a monomorphism, which is an injective homomorphism in a variety of universal algebras. Since g ∈ E_{I}, I(g) is an isomorphism. Hence, p_{1} is also an isomorphism. Therefore, I(π_{1}) = p_{1} ∘ w is an injective homomorphism.

On the other hand, let [l]_{∼_{A}} be an arbitrary class of A/∼_{A} = HI(A). Since every element of any object M in M is a subalgebra and η_{⟨l⟩_{A}} : ⟨l⟩_{A} → HI(⟨l⟩_{A}) is a homomorphism in a variety of universal algebras, HI(⟨l⟩_{A}) = η_{⟨l⟩_{A}}(⟨l⟩_{A}) = ⟨η_{⟨l⟩_{A}}(l)⟩_{HI(A)} = T, the one-element algebra, which means that the elements of ⟨l⟩_{A} are in the same congruence class in A. Therefore, ⟨l⟩_{A} ⊆ [l]_{∼_{A}}. Since f : A → C is a homomorphism in a variety of universal algebras, f(⟨l⟩_{A}) = ⟨f(l)⟩_{C}. Since g ∈ F, ⟨f(l)⟩_{C} ∩ g(B) ≠ ∅, which implies ⟨l⟩_{A} ∩ π_{1}(A ×_{C} B) ≠ ∅. Hence, [l]_{∼_{A}} ∩ π_{1}(A ×_{C} B) ≠ ∅. Therefore, I(π_{1}) is also surjective and so g ∈ E′_{I}. Thus, E′_{I} ⊇ E_{I} ∩ F. By Proposition 3.2, E′_{I} ⊆ E_{I} ∩ F.

(b)⇒(a): Consider the pullback diagram (6), where g ∈ E_{I} ∩ F. Since by hypothesis E′_{I} = E_{I} ∩ F, I(π_{1}) is an isomorphism, and therefore I(π_{1}), I(π_{2}) are jointly monic.

Covering morphisms

We will show that the class of coverings is just the class of trivial coverings in admissible reflections into idempotent subvarieties, provided that the I-images of the projections of any pullback of a covering f : A → B along the counit ε_{B} : F U(B) → B (see the paragraph above Lemma 2.5) are jointly monic, and that a certain property holds for the reflection (see Definition 3.4, here it is named Property F).

Consider the reflection H ⊢ I : Band → SLat, of the variety of bands (idempotent semigroups) into its subvariety of semilattices. It is well known that every band is a semilattice of rectangular bands. This means that, in this reflection, two elements a and b in a band B are in the same congruence class, a ∼_{B} b, if and only if a = aba and b = bab. The canonical homomorphism h, from the kernel pair of the unit morphism η_{F U(B)} to the kernel pair of the unit morphism η_{B} (see Definition 3.4), is surjective, since given (a, b) in B ×_{HI(B)} B, that is, a ∼_{B} b, there exists (aba, bab) in F U(B) ×_{HIF U(B)} F U(B) such that h(aba, bab) = (a, b) (note that words in any rectangular band in F U(B) are just finite sequences of the same letters).

On the other hand, in the reflection of the variety of commutative semigroups into its subvariety of semilattices, the canonical homomorphism h is not always surjective (see Remark 5.5), but it is almost surjective (h ∈ F), as we will see in Proposition 5.4. Which is enough, together with Proposition 3.6 and Proposition 5.3, to conclude that coverings and trivial coverings coincide in this reflection.

3.4. Definition. A reflection H ⊢ I : C → M, of a variety of universal algebras into a subvariety, is said to satisfy Property F if the canonical homomorphism h = ⟨ε_{B} ∘ π_{1}, ε_{B} ∘ π_{2}⟩ : F U(B) ×_{HIF U(B)} F U(B) → B ×_{HI(B)} B, from the kernel pair of the unit morphism η_{F U(B)} to the kernel pair of the unit morphism η_{B}, is almost surjective, that is, h ∈ F, for all B ∈ C. Where π_{1}, π_{2} are the projections of the kernel pair of η_{F U(B)}, and ε_{B} : F U(B) → B is a counit morphism of the adjunction U ⊢ F : Set → C, such that U is the forgetful functor into sets.

3.5. Remark. This property means that if a ∼_{B} b, with a, b ∈ B, then there exist w_{1} and w_{2} in F U(B) such that w_{1} ∼_{F U(B)} w_{2}, and h(w_{1}, w_{2}) belongs to the subalgebra ⟨(a, b)⟩_{B×_{HI(B)}B} of B ×_{HI(B)} B generated by (a, b). Note that this implies that if a ∼_{B} b, then there exist w_{1} and w_{2} in F U(B), such that w_{1} ∼_{F U(B)} w_{2}, and ε_{B}(w_{1}) belongs to the subalgebra ⟨a⟩_{B} of B generated by a, and ε_{B}(w_{2}) belongs to the subalgebra ⟨b⟩_{B} of B generated by b.

3.6. Proposition. Let H ⊢ I : C → M be an admissible reflection of a variety of universal algebras C into an idempotent subvariety M, which satisfies Property F. Then, the following two conditions are equivalent:

(a) I(π_{1}) and I(π_{2}) are jointly monic for all pullback diagrams

[figure: F U(S) ×_{S} L; L; S ,; F U(S); π_{2}; ε_{S}; f; π_{1}]

(7)

where f ∈ M_{I}^{∗} and ε_{S} is a counit morphism of the adjunction U ⊢ F : Set → C, such that U is the forgetful functor into sets;

(b) M_{I}^{∗} = M_{I}.

Proof. Suppose that I(π_{1}) and I(π_{2}) are jointly monic, that is, the canonical homomorphism α = ⟨I(π_{1}), I(π_{2})⟩ : I(F U(S) ×_{S} L) → IF U(S) ×_{I(S)} I(L) is an injective homomorphism. First we will prove that the pullback (7) is preserved by I. It remains to show that α is also surjective.

Note that α is surjective if and only if for every t ∈ L and w ∈ F U(S), such that f(t) ∼_{S} ε_{S}(w), there exists (w^{∗}, t^{∗}) ∈ F U(S) ×_{S} L, with w ∼_{F U(S)} w^{∗} and t ∼_{L} t^{∗}.

Since the reflection satisfies Property F and f(t) ∼_{S} ε_{S}(w), there exists (w_{1}, w_{2}) ∈ F U(S) ×_{HI(F U(S))} F U(S), such that ε_{S}(w_{1}) ∈ ⟨ε_{S}(w)⟩_{S}, ε_{S}(w_{2}) ∈ ⟨f(t)⟩_{S}, by Remark 3.5.

The existence of t^{∗} ∈ L, with t ∼_{L} t^{∗} follows from:

On one hand, there exists t′ ∈ ⟨t⟩_{L}, such that ε_{S}(w_{2}) = f(t′), that is, (w_{2}, t′) ∈ F U(S) ×_{S} L, since ⟨f(t)⟩_{S} = f(⟨t⟩_{L}). On the other hand, since f ∈ M_{I}^{∗}, π_{1} ∈ M_{I}, and so the restriction of π_{1} to any congruence class is a bijection. Then, since w_{1} ∼_{F U(S)} w_{2}, there exists t^{∗} ∈ L, such that (w_{1}, t^{∗}) ∈ F U(S) ×_{S} L, that is, ε_{S}(w_{1}) = f(t^{∗}), and (w_{2}, t′) ∼_{F U(S)×_{S}L} (w_{1}, t^{∗}). Hence, t′ ∼_{L} t^{∗}. Since t′ ∈ ⟨t⟩_{L} and M is an idempotent subvariety, t ∼_{L} t′. Therefore, t ∼_{L} t^{∗}.

The existence of w^{∗} ∈ F U(S), with w ∼_{F U(S)} w^{∗} follows from:

On one hand, there exists w^{∗} ∈ ⟨w⟩_{F U(S)}, such that ε_{S}(w^{∗}) = ε_{S}(w_{1}), since ε_{S}(w_{1}) ∈ ⟨ε_{S}(w)⟩_{S} = ε_{S}(⟨w⟩_{F U(S)}). On the other hand, since w^{∗} ∈ ⟨w⟩_{F U(S)} and M is an idempotent subvariety, w^{∗} ∼_{F U(S)} w.

The fact that (w^{∗}, t^{∗}) ∈ F U(S) ×_{S} L follows from: ε_{S}(w^{∗}) = ε_{S}(w_{1}) = f(t^{∗}).

Therefore, there exists (w^{∗}, t^{∗}) ∈ F U(S) ×_{S} L, such that w ∼_{F U(S)} w^{∗} and t ∼_{L} t^{∗}. Thus, α is also surjective.

Now, we have to prove that f ∈ M_{I}.

Consider the commutative diagram

[figure: F U(S); F U(S) ×_{S} L; HIF U(S); HI(F U(S) ×_{S} L); π_{1}; η_{F U(S)×_{S}L}; η_{F U(S)}; HI(ε_{S}); HI(π_{1}); HI(L); HI(S) ,; HI(f); HI(π_{2})]

(8)

where both squares are pullbacks. The right square is a pullback, since the pullback diagram (7) is preserved, and the left square is a pullback, since π_{1} ∈ M_{I}, according to the paragraph above Proposition 2.4.

As η_{S} ∘ ε_{S} = HI(ε_{S}) ∘ η_{F U(S)} and η_{L} ∘ π_{2} = HI(π_{2}) ∘ η_{F U(S)×_{S}L}, the outside square of the following commutative diagram (9) is the same as in the former (8), and therefore a pullback.

[figure: F U(S); F U(S) ×_{S} L; S; L; π_{1}; π_{2}; ε_{S}; η_{S}; f; HI(L); HI(S); HI(f); η_{L}]

(9)

According to Lemma 4.6 in [2], since the outside square in diagram (9) is a pullback, and ε_{S} is an effective descent morphism in C, and the left square is a pullback, the right square is a pullback, too. Hence, f ∈ M_{I}, by the paragraph above Proposition 2.4.

Conversely, let M_{I}^{∗} = M_{I}. Since the reflection is admissible, it is known that I preserves the pullback (7) of ε_{S} and f because the latter is a trivial covering (see [2, 3.6]). Hence, I(π_{1}) and I(π_{2}) are jointly monic.

Separable, Purely Inseparable and Normal morphisms

The following Proposition 3.7 characterizes the classes of separable, purely inseparable and normal morphisms, in admissible reflections of varieties such that the I-images of the projections of every kernel pair are jointly monic.

3.7. Proposition. Let H ⊢ I : C → M be an admissible reflection of a variety of universal algebras C into a subvariety M. Let Δ denote the identity relation, Ker(f) the congruence associated to the kernel pair of f : A → B, ∼_{A} the congruence on A induced by the reflection, and ∘ the composition of congruences.

If, for the kernel pair (u, v) of f, I(u) and I(v) are jointly monic, then:

(a) f is separable if and only if Ker(f)∩ ∼_{A}= Δ;

(b) f is purely inseparable if and only if Ker(f) ⊆∼_{A};

(c) f is normal if and only if

∼_{A} ∘Ker(f) ⊆ Ker(f)∘ ∼_{A} and Ker(f)∩ ∼_{A}= Δ.

Proof. Consider the pullback diagram (4) in Definition 2.6 and note that I(u) and I(v) are jointly monic when, for all pairs (a, b), (c, d) in A ×_{B} A, a ∼_{A} c and b ∼_{A} d if and only if (a, b) ∼_{A×_{B}A} (c, d).

(a) A homomorphism f : A → B is separable if, for all a ∈ A, the restriction δ_{f}|_{[a]_{∼_{A}}} : [a]_{∼_{A}} → [δ_{f}(a)]_{∼_{A×_{B} A}} is a bijection. On one hand, this restriction is injective for all a ∈ A, since v ∘ δ_{f} = 1_{A}. On the other hand, this restriction is surjective if and only if, for all a, a′ ∈ A, such that f(a) = f(a′) and b ∼_{A} a and b ∼_{A} a′, for some b ∈ A, there exists c ∈ A such that (c, c) = (a, a′). That is, a homomorphism f : A → B is separable if and only if, for all a, a′ ∈ A, such that f(a) = f(a′) and a ∼_{A} a′, then a = a′. Which is to say that a homomorphism f : A → B is separable if and only if Ker(f)∩ ∼_{A}= Δ.

(b) A homomorphism f : A → B is purely inseparable if and only if I(δ_{f}) is an isomorphism. On one hand, I(δ_{f}) is always injective, since v ∘δ_{f} = 1_{A}. On the other hand, I(δ_{f}) is a surjection if and only if, for all a, a′ ∈ A, such that f(a) = f(a′), there exists b ∈ A such that b ∼_{A} a and b ∼_{A} a′. That is, a homomorphism f : A → B is purely inseparable when, for all a, a′ ∈ A, if f(a) = f(a′) then a ∼_{A} a′. Which is to say that a homomorphism f : A → B is separable if and only if Ker(f) ⊆∼_{A}.

(c) A homomorphism f : A → B is normal if and only if the next two conditions hold:

(1) The restrictions of u to the congruence classes of ∼_{A×_{B}A} are surjective. That is, for all a, b ∈ A, such that a ∼_{A} c and f(c) = f(b) for some c ∈ A, there exists d ∈ A, such that f(a) = f(d) and d ∼_{A} b, or equivalently, ∼_{A} ∘Ker(f) ⊆ Ker(f)∘ ∼_{A} .

(2) The restrictions of u to the congruence classes of ∼_{A×_{B}A} are injective. That is, for all a, b ∈ A such that a ∼_{A} b and f(a) = f(b) then, a = b, or equivalently, Ker(f)∩ ∼_{A}= Δ.

4. Inseparable-Separable factorization

In Proposition 4.1 it will be proved that there is an inseparable-separable factorization system, for admissible reflections into subvarieties, provided the class of stably-vertical morphisms is the class of almost surjective vertical homomorphisms.

4.1. Proposition. Let H ⊢ I : C → M be an admissible reflection of a variety of universal algebras C into a subvariety M. If E′_{I} = E_{I} ∩ F then there exists a factorization system (Ins, Sep), with Ins = E_{I} ∩ E.

Proof. First we will show that E′_{I} ⊆ P in. Consider the pullback diagram (4) and suppose f ∈ E′_{I}. Then v ∈ E_{I}. Since I(v) is an isomorphism and I(v)∘I(δ_{f}) = 1_{I(A)}, I(δ_{f}) is an isomorphism. Hence, δ_{f} ∈ E_{I}, thus f ∈ P in. Therefore, by the paragraph below Definition 2.7,

E′_{I} ∩ E ⊆ Ins ⊆ E_{I} ∩ E.

Now we have to prove that the class Ins is closed under composition. Since E ⊆ F, E_{I} ∩ E ⊆ E_{I} ∩ F = E′_{I}. Hence, E_{I} ∩ E ⊆ E_{I} ∩ F ∩ E = E′_{I} ∩ E. On the other hand, since E′_{I} ⊆ E_{I}, E′_{I} ∩ E ⊆ E_{I} ∩ E. Hence, E′_{I} ∩ E = E_{I} ∩ E. Therefore, Ins = E_{I} ∩ E. Since E_{I} and E are closed under composition, so it is E_{I} ∩ E. Therefore, (Ins, Sep) is a factorization system.

4.2. Remark. Under the assumptions of Proposition 4.1, the (Ins, Sep)-factorization of f : A → B given in the paragraph above Definition 2.7, could be obtained alternatively by applying the surjection-injection factorization to the homomorphism e_{f} in diagram (1). Hence, f = m̄_{f} ∘ ē_{f} = (m_{f} ∘ m) ∘ ē_{f}, with e_{f} = m ∘ ē_{f}. This means that inseparableseparable and concordant-dissonant factorizations coincide (see Proposition 2 in [13, §3], which is a generalization of the results given in [2, §3.9] or [3, Proposition 5.5]; see also [5, §2.11] in the context of pointed endofunctors, where previous references are provided).

5. The reflection of commutative semigroups into semilattices

Consider the admissible reflection H ⊢ I : CommSgr → SLat of commutative semigroups into semilattices. It is well known that a commutative semigroup C is semilattice indecomposable if and only if it is archimedean, in the sense of [9]. That is, a commutative semigroup C is archimedean if for every ordered pair (a, b) of elements in C there exist n ∈ N (positive integers) and x ∈ C, with a^{n} = bx. Hence, every commutative semigroup is a semilattice of archimedean subsemigroups.

(I) If C is a finitely generated commutative semigroup, any of its archimedean components A has some power A^{k} = {a ∈ C | a = a_{1}...a_{k}, with a_{1}, , a_{k} ∈ A} with cancellation (see Proposition 9.6 in [4, §IV]).

(II) Since the subvariety of semilattices is idempotent, each archimedean component of the semilattice decomposition of a commutative semigroup S is a subalgebra of S. Therefore, every power of each archimedean component is a subsemigroup of S.

(III) Any archimedean component of the semilattice decomposition of a commutative semigroup is semilattice indecomposable (see [8]). That is, let S be a commutative semigroup and consider an archimedean class [x]_{∼_{S}} in S : then, since I([x]_{∼S}) = 1, for every y, z ∈ [x]_{∼_{S}}, there exist m, n ∈ N and there exist c, d ∈ [x]_{∼_{S}}, such that z^{m} = yc and y^{n} = zd.

The following Proposition 5.1 will be used to prove Proposition 5.2, which in turn will be used to prove the more general Proposition 5.3.

5.1. Proposition. Consider the reflection H ⊢ I : CommSgr → SLat, and the following pullback diagram,

[figure: A ×_{C} B; B; C ,; A; π_{2}; f; e; π_{1}]

(10)

in CommSgr.

Then, I(π_{1}) and I(π_{2}) are jointly monic, provided C is a commutative semigroup with cancellation.

Proof. Let (a_{1}, b_{1}), (a_{2}, b_{2}) ∈ A ×_{C} B be such that a_{1} ∼_{A} a_{2} and b_{1} ∼_{B} b_{2}, that is, there exist m, n, p, q ∈ N and c, d ∈ A and u, v ∈ B, such that a_{1}^{m} = a_{2}c, a_{2}^{n} = a_{1}d, b_{1}^{p} = b_{2}u and b_{2}^{q} = b_{1}v.

We need (x, y), (z, t) ∈ A ×_{C} B and r, s ∈ N such that (a_{1}, b_{1})^{r} = (a_{2}, b_{2})(x, y) and (a_{2}, b_{2})^{s} = (a_{1}, b_{1})(z, t).

We take (x, y) = (a_{2}^{p−1}c^{p}, b_{2}^{m−1}u^{m}), (z, t) = (a_{1}^{q−1}d^{q}, b_{1}^{n−1}v^{n}) and r = mp, s = nq.

Clearly we have (a_{1}, b_{1})^{r} = (a_{1}^{r}, b_{1}^{r}) = (a_{2}, b_{2})(x, y) and (a_{2}, b_{2})^{s} = (a_{2}^{s}, b_{2}^{s}) = (a_{1}, b_{1})(z, t). We have to prove (1) f(x) = e(y), and (2) f(z) = e(t). The equality (1) follows from f(a_{2})f(x) = f(a_{2}^{p}c^{p}) = f(a_{1}^{mp}) = e(b_{1}^{mp}) = e(b_{2}^{m}u^{m}) =

e(b_{2})e(y), by cancellation.

The equality (2) follows from f(a_{1})f(z) = f(a_{1}^{q}d^{q}) = f(a_{2}^{nq}) = e(b_{2}^{nq}) = e(b_{1}^{n}v^{n}) = e(b_{1})e(t), by cancellation.

5.2. Proposition. Consider the reflection H ⊢ I : CommSgr → SLat, and the pullback diagram (10). Then, I(π_{1}) and I(π_{2}) are jointly monic, provided C is a finitely generated commutative semigroup.

Proof. Let (a_{1}, b_{1}), (a_{2}, b_{2}) ∈ A ×_{C} B be such that a_{1} ∼_{A} a_{2} and b_{1} ∼_{B} b_{2}. Let k ∈ N be such that [f(a_{1})]_{∼_{C}}^{k} = [e(b_{1})]_{∼_{C}}^{k} has cancellation, by (I).

Consider the following pullback

[figure: X ×_{H} Y; Y; H ,; X; p_{2}; f_{|_{X}}; e_{|_{Y}}; p_{1}]

where

• H = [f(a_{1})]_{∼_{C}}^{k} = [e(b_{1})]_{∼_{C}}^{k}, X = [a_{1}]_{∼_{A}}^{k} = [a_{2}]_{∼_{A}}^{k}, Y = [b_{1}]_{∼_{B}}^{k} = [b_{2}]_{∼_{B}}^{k},

• f_{|_{X}} : X → H and e_{|_{Y}} : Y → H are restrictions, respectively,

of f : A → C and e : B → C.

Let us prove that:

(1) (a_{1}^{k}, b_{1}^{k}), (a_{2}^{k}, b_{2}^{k}) ∈ X ×_{H} Y;

(2) a_{1}^{k} ∼_{X} a_{2}^{k} and b_{1}^{k} ∼_{Y} b_{2}^{k}.

(1) follows from: f(a_{1}^{k}) = e(b_{1}^{k}) and f(a_{2}^{k}) = e(b_{2}^{k}), since f(a_{1}) = e(b_{1}) and f(a_{2}) = e(b_{2}). Therefore f_{|_{X}}(a_{1}^{k}) = e_{|_{Y}} (b_{1}^{k}) and f_{|_{X}}(a_{2}^{k}) = e_{|_{Y}} (b_{2}^{k}), that is, (a_{1}^{k}, b_{1}^{k}), (a_{2}^{k}, b_{2}^{k}) ∈ X ×_{H} Y.

(2) follows from (III) above: by (III), there exist m, n, p, q ∈ N and c, d ∈ [a_{1}]_{∼_{A}} and u, v ∈ [b_{1}]_{∼_{B}}, such that a_{1}^{m} = a_{2}c, a_{2}^{n} = a_{1}d, b_{1}^{p} = b_{2}u and b_{2}^{q} = b_{1}v; therefore, (a_{1}^{k})^{m} = a_{2}^{k}c^{k}, (a_{2}^{k})^{n} = a_{1}^{k}d^{k}, (b_{1}^{k})^{p} = b_{2}^{k}u^{k} and (b_{2}^{k})^{q} = b_{1}^{k}v^{k}, with c^{k}, d^{k} ∈ X and u^{k}, v^{k} ∈ Y. Hence, by Proposition 5.1, (a_{1}^{k}, b_{1}^{k}) ∼_{X×_{H}Y} (a_{2}^{k}, b_{2}^{k}), since H is cancellative. Thus, (a_{1}^{k}, b_{1}^{k}) ∼_{A×_{C}B} (a_{2}^{k}, b_{2}^{k}), since X ×_{H} Y is a subsemigroup of A ×_{C} B. Therefore, (a_{1}, b_{1}) ∼_{A×_{C}B} (a_{2}, b_{2}), since (a_{1}^{k}, b_{1}^{k}) ∼_{A×_{C}B} (a_{1}, b_{1}) and (a_{2}^{k}, b_{2}^{k}) ∼_{A×_{C}B} (a_{2}, b_{2}).

5.3. Proposition. Consider the reflection H ⊢ I : CommSgr → SLat, of commutative semigroups into semilattices. The I-images of the projections of any pullback diagram in CommSgr, are jointly monic.

Proof. We have to show that, for any commutative semigroup C in the pullback diagram (10), the homomorphisms I(π_{1}) and I(π_{2}) are jointly monic.

Let (a_{1}, b_{1}), (a_{2}, b_{2}) ∈ A ×_{C} B be such that a_{1} ∼_{A} a_{2} and b_{1} ∼_{B} b_{2}. That is, there exist m, n, p, q in N, c and d in A, and u and v in B, with a_{1}^{m} = a_{2}c, a_{2}^{n} = a_{1}d, b_{1}^{p} = b_{2}u and b_{2}^{q} = b_{1}v. Let H be the subsemigroup in C generated by f(a_{1}), f(a_{2}), f(c), f(d), e(u) and e(v).

Recall that f(a_{1}) = e(b_{1}) and f(a_{2}) = e(b_{2}), and consider the pullback diagram

[figure: X ×_{H} Y; Y; H ,; X; p_{2}; f_{|_{X}}; e_{|_{Y}}; p_{1}]

in which X = f^{−1}(H), Y = e^{−1}(H), and f_{|_{X}} and e_{|_{Y}} are the obvious restrictions. Note that:

(i) a_{1}, a_{2}, c, d ∈ X and b_{1}, b_{2}, u, v ∈ Y; (ii) (a_{1}, b_{1}), (a_{2}, b_{2}) ∈ X ×_{H} Y;

(iii) a_{1} ∼_{X} a_{2} and b_{1} ∼_{Y} b_{2}, since c, d ∈ X and u, v ∈ Y.

Hence, by Proposition 5.2, (a_{1}, b_{1}) ∼_{X×_{H}Y} (a_{2}, b_{2}). Therefore, (a_{1}, b_{1}) ∼_{A×_{C}B} (a_{2}, b_{2}), since X ×_{H} Y is a subsemigroup of A ×_{C} B.

5.4. Proposition. The reflection H ⊢ I : CommSgr → SLat, of commutative semigroups into semilattices, satisfies Property F.

Proof. Remark that, in the following, referring to a word in a free semigroup, x^{k} stands for the concatenation of a positive integer number k of letters x.

Let B be a commutative semigroup and a, b ∈ B such that a ∼_{B} b. Recall that a ∼_{B} b if and only if there exist n, m ∈ N and c, d ∈ B, such that a^{n} = bc and b^{m} = ad. Then, a^{nm} = b^{m}c^{m} = adc^{m} and b^{mn} = a^{n}d^{n} = bcd^{n}.

Hence, a^{n^{2}m} = a^{n}d^{n}c^{mn} = bcd^{n}c^{mn} = bc^{mn+1}d^{n}.

Thus, a^{n^{2}m} = bc^{nm+1}d^{n} and b^{nm} = bcd^{n}.

Finally, a^{n^{2}m} = bc^{nm+1}d^{n} and b^{n^{2}m} = b^{n}c^{n}d^{n^{2}}.

Since the words in F U(B) are in the same archimedean component if and only if they have the same letters, the words w_{1} = bc^{nm+1}d^{n} and w_{2} = b^{n}c^{n}d^{n^{2}}, are in the same congruence class in F U(B), that is, (bc^{nm+1}d^{n}, b^{n}c^{n}d^{n^{2}}) ∈ F U(B) ×_{HIF U(B)} F U(B).

Hence, applying the canonical homomorphism h = ⟨ε_{B} ∘ π_{1}, ε_{B} ∘ π_{2}⟩ of Definition 3.4, we have h(bc^{nm+1}d^{n}, b^{n}c^{n}d^{n^{2}}) = (a, b)^{n^{2}m} ∈ ⟨(a, b)⟩_{B×_{HI(B)}B}.

5.5. Remark. Consider the reflection H ⊢ I : CommSgr → SLat, of the variety of commutative semigroups into its subvariety of semilattices, and the canonical homomorphism h : F U(B)×_{HIF U(B)}F U(B) → B×_{HI(B)}B of Definition 3.4, where B = F (a, b; a^{2} = (ab)^{2}) is the quotient of the free commutative semigroup on the set {a, b}, by the congruence generated by the identity a^{2} = (ab)^{2}.

The archimedean components of B are {a^{k}, a^{m}b^{n}}, with k, m, n arbitrary positive integers, since a ∼_{B} ab, and {b^{l}}, with l an arbitrary positive integer, since a and b are not in the same archimedean component of B. While the archimedean components in F U(B) are its subsemigroups whose elements have the same letters. Hence, a ∼_{B} ab, but the words w_{1} = a and w_{2} = ab are not in the same archimedean component in F U(B), and they are the unique two words such that ε_{B}(w_{1}) = a and ε_{B}(w_{2}) = ab. Therefore, the canonical homomorphism h of Definition 3.4 is not surjective.

The following first table concludes the characterizations of the classes of morphisms considered above, for the reflection of commutative semigroups into semilattices. The second table concludes that there exists an inseparable-separable factorization system, but does not exist a monotone-light factorization system, for that reflection.

Classes of homomorphisms f : A → B associated to I : CommSgr → SLat

E′_{I}  M_{I}^{∗}  Normal  P in  Sep  Ins
E_{I} ∩ F M_{I}^{∗} = M_{I} ∼_{A} ∘Ker(f) ⊆ Ker(f)∘ ∼_{A} Ker(f) ⊆∼_{A} Ker(f)∩ ∼_{A}= Δ E_{I} ∩ E
Ker(f)∩ ∼_{A}= Δ
P r. 3.3 P r. 3.6  P r. 3.7  P r. 3.7  P r. 3.7  P r. 4.1
and  and  and  and  and
P r. 5.3 P r. 5.3  P r. 5.3  P r. 5.3  P r. 5.3

and

P r. 5.4

Factorizations associated to I : CommSgr → SLat

(E_{I}, M_{I})  (Ins, Sep)  (E′_{I}, M_{I}^{∗})
Y ES  Y ES  NO
M_{I}^{∗} = M_{I}, E′_{I} = E_{I} ∩ F ≠ E_{I}
Admissible reflection P r. 4.1
(i : (N, +) ⊆ (R, +), i ∈ E_{I} but i ∉ F)

References

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[2] Carboni, A., Janelidze, G., Kelly, G. M., Paré, R. On localization and stabilization for factorization systems, App. Cat. Struct., 5(1997), 1–58.

[3] Cassidy, C., Hébert, M., Kelly, G. M. Reflective subcategories, localizations and factorization systems, J. Austral. Math. Soc., 38A(1985), 287–329.

[4] Grillet, P. A. Semigroups: an introduction to the structure theory, Marcel Dekker, Inc.(1995).

[5] Janelidze, G., Tholen, W. Functorial factorization, well-pointedness and separability, J. Pure App. Algebra, 142(1999), 99–130.

[6] Janelidze, G., Laan, V., Márki, L. Limit preservation properties of the greatest semilattice image functor, Internat. J. Algebra Comput., 18, Issue 5(2008), 853–867.

[7] Mac Lane, S. Categories for the Working Mathematician, 2nd ed., Springer(1998).

[8] Tamura, T. Attainability of systems of identities on semigroups, J. Algebra, 3(1966), 261–276.

[9] Tamura, T., Kimura, N. On decompositions of a commutative semigroup, Kodai Math. Sem. Rep., 6, Issue 4(1954), 109–112.

[10] Xarez, I. A. Reflections of universal algebras into semilattices, their Galois theories and related factorization systems, University of Aveiro, PhD thesis(2013).

[11] Xarez, J. J. Separable morphisms of categories via preordered sets, Fields Inst. Commun., 43(2004), 543–549.

[12] Xarez, J. J. Generalising Connected Components, J. Pure Appl. Algebra, 216, Issues 8-9(2012), 1823–1826.

[13] Xarez, J. J. Concordant and monotone morphisms, App. Cat. Struct., 21, Issue 4(2013), 393–415.

CIDMA - Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Portugal.

Email: isabel.andrade@ua.pt
      xarez@ua.pt

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