Transfer theorems for finitely subdirectly irreducible algebras

We show that under certain conditions, well-studied algebraic properties transfer from the class $\mathcal{Q}_{_\text{RFSI}}$ of the relatively finitely subdirectly irreducible members of a quasivariety $\mathcal{Q}$ to the whole quasivariety, and, in certain cases, back again. First, we prove that if $\mathcal{Q}$ is relatively congruence-distributive, then it has the $\mathcal{Q}$-congruence extension property if and only if $\mathcal{Q}_{_\text{RFSI}}$ has this property. We then prove that if $\mathcal{Q}$ has the $\mathcal{Q}$-congruence extension property and $\mathcal{Q}_{_\text{RFSI}}$ is closed under subalgebras, then $\mathcal{Q}$ has a one-sided amalgamation property (equivalently, for $\mathcal{Q}$, the amalgamation property) if and only if $\mathcal{Q}_{_\text{RFSI}}$ has this property. We also establish similar results for the transferable injections property and strong amalgamation property. For each property considered, we specialize our results to the case where $\mathcal{Q}$ is a variety -- so that $\mathcal{Q}_{_\text{RFSI}}$ is the class of finitely subdirectly irreducible members of $\mathcal{Q}$ and the $\mathcal{Q}$-congruence extension property is the usual congruence extension property -- and prove that when $\mathcal{Q}$ is finitely generated and congruence-distributive, and $\mathcal{Q}_{_\text{RFSI}}$ is closed under subalgebras, possession of the property is decidable. Finally, as a case study, we provide a complete description of the subvarieties of a notable variety of BL-algebras that have the amalgamation property.


Introduction
This paper studies a cluster of interrelated properties of general interest in algebra: the congruence extension property, (strong) amalgamation property, transferable injections property, and surjective epimorphisms property.The latter have each been investigated in a range of algebraic contexts, with studies spanning groups, rings, lattices, Lie algebras, and numerous other algebraic structures (see [27] for an extensive survey).In this paper, we adopt a general vantage point and develop a widely applicable toolkit for studying these properties.Our main contribution consists of transfer theorems that characterize each property for a class of algebraic structures satisfying certain hypotheses in terms of a smaller and more easily studied subclass, namely, the (relatively) finitely subdirectly irreducible members of the class.These theorems generalize other transfer results in the literature (found in, e.g., [9,10,20,26,29]), while simultaneously providing a more uniform treatment.
We freely make use of basic notions of universal algebra throughout our discussion, but first recall several key definitions, referring to [5,18] for more detailed treatments.Let Q be a quasivariety: a class of similar algebras (i.e., a class of algebras of the same signature) defined by quasiequations, or, equivalently, closed under isomorphisms, subalgebras, direct products, and ultraproducts.A congruence Θ of an algebra A ∈ Q is called a Q-congruence if A/Θ ∈ Q.When ordered by inclusion, the set of Q-congruences of A forms an algebraic lattice Con Q A. Arbitrary meets coincide in Con Q A with those taken in the algebraic lattice Con A of all congruences of A, as do joins of chains, but this may not be the case for arbitrary joins.If Q is a variety -that is, a class of similar algebras defined by equations, or, equivalently, closed under homomorphic images, subalgebras, and direct products -then every congruence of A is a Q-congruence and Con Q A and Con A coincide.
An algebra A is said to be (finitely) subdirectly irreducible if whenever A is isomorphic to a subdirect product of a (non-empty finite) set of algebras, it is isomorphic to one of these algebras.Equivalently, A is finitely subdirectly irreducible if the least congruence ∆ A := { a, a | a ∈ A} is meet-irreducible in Con A, and subdirectly irreducible if ∆ A is completely meet-irreducible in Con A. 1 When A belongs to Q, it is convenient to relativize these notions.
In this case, A is said to be (finitely) Q-subdirectly irreducible if whenever A is isomorphic to a subdirect product of a (non-empty finite) set of algebras in Q, it is isomorphic to one of them.Equivalently, A is finitely Q-subdirectly irreducible if and only if ∆ A is meet-irreducible in Con Q A, and Q-subdirectly irreducible if and only if ∆ A is completely meet-irreducible in Con Q A.
For any A ∈ Q and Θ ∈ Con Q A, it follows from the correspondence theorem for universal algebra that the quotient algebra A/Θ ∈ Q is finitely Q-subdirectly irreducible if and only if Θ is meet-irreducible in Con Q A, and Q-subdirectly irreducible if and only if it is completely meet-irreducible.Clearly, if Q is a variety, the properties of being (finitely) Q-subdirectly irreducible and (finitely) subdirectly irreducible coincide.When Q is clear from the context, we call a (finitely) Q-subdirectly irreducible algebra A relatively (finitely) subdirectly irreducible.
Let Q FSI , Q SI , Q RFSI , and Q RSI denote the classes of finitely subdirectly irreducible, subdirectly irreducible, relatively finitely subdirectly irreducible, and relatively subdirectly irreducible members of Q, respectively.A sizeable number of results in the universal algebra literature state that, under certain conditions, well-studied algebraic properties transfer from Q RSI to Q, at least when Q is a variety (see, e.g., [9,10,20,26,29]).The aim of this paper is to determine conditions under which these properties transfer from Q RFSI to Q and, in some cases, back again.A key motivation for considering Q RFSI rather than Q RSI is that it is often easier to state or check conditions for the larger class.Notably, if Q has equationally definable relative principal congruence meets (a common property for quasivarieties corresponding to non-classical logics), then Q RFSI is a universal class [8,Theorem 2.3].Moreover, if V is a variety such that V FSI is a universal class, then V FSI is a positive universal class if and only if Con A is a chain (totally ordered set) for each A ∈ V FSI . 2  An algebra Q is said to be relatively congruence-distributive and in this case, as shown in [12,Theorem 2.3], Q RFSI = Q FSI .Clearly, a variety is relatively congruenceand this is true of any greatest element ⊤ of L; however, a is completely meet-irreducible if a = B implies a ∈ B for any B ⊆ L, which is not the case for ⊤ = ∅.
2 Suppose that V FSI is a universal class.Then it is a positive universal class if and only if for any A ∈ V FSI and Θ ∈ Con A, also A/Θ ∈ V FSI , i.e., Θ is meet-irreducible in Con A. But a lattice is a chain if and only if all its elements are meet-irreducible, so V FSI is a positive universal class if and only if Con A is a chain for each A ∈ V FSI .distributive if and only if it is congruence-distributive in the usual sense.An algebra When Q is a variety, the Q-CEP is the usual congruence extension property (for short, CEP) and if it is congruence-distributive, Q has the CEP if and only if Q FSI has the CEP (Corollary 2.4).This result yields also the known fact that a congruence-distributive variety V such that V SI is elementary has the CEP if and only if V SI has the CEP [10, Theorem 3.3].Note, however, that the requirement that V SI is elementary may fail or be difficult to establish, and this result may therefore be significantly harder to apply than Theorem 2.3.
When Q FSI is closed under subalgebras, Theorem 2.3 can be reformulated in terms of commutative diagrams.Let K be any class of similar algebras.A span in K is a 5-tuple A, B, C, ϕ B , ϕ C consisting of A, B, C ∈ K and homomorphisms ϕ B : A → B, ϕ C : A → C. We call this span injective if ϕ B is an embedding, doubly injective if both ϕ B and ϕ C are embeddings, and injective-surjective if ϕ B is an embedding and ϕ C is surjective.The class K has the extension property (for short, EP) if for any injective-surjective span A, B, C, ϕ B , ϕ C in K, there exist a D ∈ K, a homomorphism ψ B : B → D, and an embedding In Section 4, we consider consequences of our results for three further properties.First, a class K of similar algebras is said to have the transferable injections property (for short, TIP) if for any injective span A, B, C, ϕ B , ϕ C in K, there exist a D ∈ K, a homomorphism ψ B : B → D, and an embedding 1(iv)).It is well-known that a variety has the TIP if and only if it has the CEP and AP ([3, Lemma 1.7]); more generally, as we show here, a class of similar algebras that is closed under subalgebras has the TIP if and only if it has the EP and 1AP.It then follows from our previous results that when Q is relatively congruence-distributive and Q RFSI = Q FSI is closed under subalgebras, Q has the TIP if and only if Q RFSI = Q FSI has the TIP (Theorem 4.3).
Next, let K be a class of similar algebras and A, B ∈ K.A homomorphism ϕ : A → B is an epimorphism in K if for all C ∈ K and all homomorphisms ψ 1 , ψ 2 : B → C, if ψ 1 ϕ = ψ 2 ϕ, then ψ 1 = ψ 2 .Surjective homomorphisms are always epimorphisms, but the converse does not hold in general.If all epimorphisms in K are surjections, K is said to have surjective epimorphisms (for short, SE).In [6,Theorem 22], it was proved that an arithmetical (i.e., congruence-distributive and congruence-permutable) variety V such that V FSI is a universal class has SE if and only if V FSI has SE.
For a class of algebras K ′ of the same signature as K, an amalgam The class K is said to have the strong amalgamation property (for short, SAP) if every doubly injective span in K has a strong amalgam in K. Using the fact that a quasivariety has the SAP if and only if it has SE and the AP [22], it follows from our previous results and [6,Theorem 22] that an arithmetical variety V with the CEP such that V FSI is a universal class has the SAP if and only if V FSI has SE and the 1AP (Corollary 4.6).We also show that such a variety has the SAP if every doubly injective span in V FSI has a strong amalgam in V (Theorem 4.8).
In Section 5, we conclude that possession of all the properties mentioned above is decidable for certain finitely generated varieties.More precisely, we obtain effective algorithms to decide if a congruence-distributive variety V that is finitely generated by a given finite set of finite algebras, such that V FSI is closed under subalgebras, has the CEP, AP, or TIP (Theorem 5.1).In the case where V is arithmetical, we obtain also effective algorithms to decide if V has SE or the SAP.Finally, in Section 6, we provide a complete description of the subvarieties of a notable variety of BL-algebras (those generated by a class of "one-component" totally ordered BL-algebras) that have the AP.

The Congruence Extension Property
We first recall some basic facts about extending congruences, denoting the We also make use of the following consequence of the correspondence theorem of universal algebra.Lemma 2.2 (cf.[4, Lemma 2]).Let Q be any quasivariety and let A, B ∈ Q.For any surjective homomorphism ϕ : , where the join on the right hand side is taken in Con , it follows using the correspondence theorem that . We now establish the first main result of this section, recalling from the introduction that if Q is a relatively congruence-distributive quasivariety, then Proof.Suppose for the non-trivial direction that Q RFSI has the Q-CEP.Let A be a subalgebra of some B ∈ Q and let Θ ∈ Con Q A. We assume towards a contradiction that Cg Q B (Θ) ∩ A 2 = Θ; that is, there exists an ordered pair Then ∆ B ∈ T , so T = ∅.Moreover, every chain in T, ⊆ has an upper bound (its union) in T , so, by Zorn's Lemma, T, ⊆ has a maximal element Ψ * . Claim.
Hence Ψ 1 ∈ T or Ψ 2 ∈ T and, by the maximality of Ψ * in T, ⊆ , either Observe next that A/(Ψ * ∩ A 2 ) embeds into B/Ψ * and can be identified with a subalgebra of B/Ψ * with universe A/Ψ * .We consider the congruence (( . By the second isomorphism theorem of universal algebra, ∨ Θ and ϕ be the canonical homomorphism from B to B/Ψ * with ker(ϕ) = Ψ * , an application of Lemma 2.2 yields ).But, by Lemma 2.1, this contradicts the assumption that B/Ψ * ∈ Q RFSI has the Q-CEP.
Corollary 2.4.Let V be any congruence-distributive variety.Then V has the congruence extension property if and only if V FSI has the congruence extension property.
Note also that every finitely subdirectly irreducible member of a variety V embeds into an ultraproduct of subdirectly irreducible members of V, by the Relativized Jónsson Lemma [8, Lemma 1.5].Hence, if V SI is closed under ultraproducts, each A ∈ V FSI embeds into some B ∈ V SI , and the fact that the CEP is preserved under subalgebras yields the following known result.
Corollary 2.5 ([10, Theorem 3.3]).Let V be any congruence-distributive variety such that V SI is an elementary class.Then V has the congruence extension property if and only if V SI has the congruence extension property. 3e now turn our attention to the relationship between the Q-CEP and the EP, establishing first a simple lemma and useful corollary for investigating relatively finitely subdirectly irreducible algebras.
Lemma 2.6.Let Q be any quasivariety and let A be a subalgebra of some and, since every chain in T, ⊆ has an upper bound (its union) in T , by Zorn's Lemma, T, ⊆ has a maximal element Φ.
It remains to show that Φ is meet-irreducible in Corollary 2.7.Let Q be any quasivariety and suppose that A ∈ Q RFSI is a subalgebra of some B ∈ Q.Then there exists a meet-irreducible Φ ∈ Con Q B such that Φ ∩ A 2 = ∆ A , and hence there exist also a C ∈ Q RFSI and a surjective homomorphism ϕ : B → C such that ker(ϕ) ∩ A 2 = ∆ A .
The following result provides a general sufficient criterion for a subclass K of a quasivariety Q to have the EP.Proposition 2.8.Let K be a subclass of a quasivariety Q satisfying (i) K is closed under isomorphisms; (ii) for any B ∈ Q and subalgebra A ∈ K of B, there exists a Φ ∈ Con Q B such that B/Φ ∈ K and Φ ∩ A 2 = ∆ A ; (iii) K has the Q-congruence extension property.
Then K has the extension property.
Hence we obtain an embedding Next, we provide a general sufficient criterion for a subclass K of a quasivariety Q to have the Q-CEP.
Proposition 2.10.Let K be a subclass of a quasivariety Q satisfying (i) K is closed under subalgebras; (ii) every relatively subdirectly irreducible member of Q belongs to K; (iii) K has the extension property.
Then K has the Q-congruence extension property.
Proof.Consider any B ∈ K. To show that B has the Q-CEP, it suffices, by Lemma 2.1(a), to prove that for any subalgebra A of B and completely meetirreducible Θ ∈ Con Q A, there exists a Ψ ∈ Con Q B such that Ψ ∩ A 2 = Θ.Note first that A ∈ K, by (i), and A/Θ ∈ Q RSI ⊆ K, by (ii).Now let ϕ C : A → A/Θ be the canonical homomorphism mapping a ∈ A to [a] Θ ∈ A/Θ and let ϕ B : A → B be the inclusion map.Since K has the EP, by (iii), there exist a D ∈ K, a homomorphism ψ B : B → D, and an embedding Then, by the homomorphism theorem, B/Ψ is isomorphic to a subalgebra of D ∈ K, so Remark 2.14.Even for a congruence-distributive variety V, it is possible for V FSI to have the EP but not the CEP.For example, let V be the variety generated by the lattice-ordered monoid C 4 = {−2, −1, 1, 2}, min, max, •, 1 with multiplication table The proper subuniverses of , and C * 3 = {−2, −1, 1}, up to isomorphism, and any homomorphic image of C 4 is isomorphic to one of its subalgebras.As shown in [32], the algebra C 4 , and hence V FSI , does not have the CEP: just observe that Θ : On the other hand, using the fact that C * 3 ∈ V FSI , it is easy to confirm that V FSI has the EP.

The Amalgamation Property
We first recall a useful necessary and sufficient condition for the existence of amalgams.A slightly weaker version (applying only to varieties) was first proved in [29] and used to establish a special case of Theorem 3.3 where V is a variety of semilinear residuated lattices with the CEP and V FSI is the class of totally ordered members of V. Proposition 3.2 (cf.[29,Theorem 9]).Let K be a subclass of a quasivariety Q satisfying (i) K is closed under isomorphisms and subalgebras; (ii) every relatively subdirectly irreducible member of Q belongs to K; then there exists a Φ ∈ Con Q B such that Φ ∩ A 2 = Θ and B/Φ ∈ K; (iv) every doubly injective span in K has an amalgam in Q.
Then Q has the amalgamation property.
Proof.Let A, B, C, ϕ B , ϕ C be a doubly injective span in K, assuming, by (i), without loss of generality, that A is a subalgebra of B and C, and ϕ B and ϕ C are inclusion maps.We check condition (a) of Lemma 3.1 for the existence of an amalgam, condition (b) being completely symmetrical.Consider any distinct x, y ∈ B and let Ψ be a Q-congruence of B that is maximal with respect to x, y / ∈ Ψ.Then B/Ψ is a relatively subdirectly irreducible member of Q and belongs to K, by (ii).Define Θ := Ψ ∩ A We now prove the main result of this section.
Theorem 3.3.Let Q be any quasivariety with the Q-congruence extension property such that Q RFSI is closed under subalgebras.The following are equivalent: (1) Q has the amalgamation property.
(2) Q has the one-sided amalgamation property.
(4) Every doubly injective span in Q RFSI has an amalgam in Q RFSI × Q RFSI .
(5) Every doubly injective span in Q RFSI has an amalgam in Q.
(2) ⇒ (3).Suppose that Q has the 1AP and let A, B, C, ϕ B , ϕ C be a doubly injective span in Q RFSI .By assumption, there exist a D ′ ∈ Q, a homomorphism ψ ′ B : B → D ′ , and an embedding We may assume without loss of generality that C is a subalgebra of D ′ .By Corollary 2.7, there exist a D ∈ Q RFSI and a surjective homomorphism χ :  4) ⇒ (5).Immediate.( 5) ⇒ (1).Suppose that every doubly injective span in Q RFSI has an amalgam in Q.Since Q RSI ⊆ Q RFSI and, by assumption, Q RFSI is closed under subalgebras, it suffices to observe that condition (iii) of Proposition 3.2 is satisfied for K = Q RFSI by Lemma 2.6.
Since a variety V has the amalgamation property if and only if every doubly injective span of finitely generated algebras in V has an amalgam in V [19], small adjustments to the proofs in this section yield the following variant of Theorem 3.3 for varieties.subalgebra of C) such that ψ B is the identity on A. Let ψ C be the identity map on C. Clearly, Let us conclude this section by remarking that by considering completely meet-irreducible Q-congruences in the proof of Theorem 3.3, we obtain the same result with Q RFSI replaced by the class Q + RSI of trivial or Q-subdirectly irreducible members of Q.In particular, we obtain [20, Theorem 3], which states that a variety V with the CEP such that the class V + SI of trivial or subdirectly irreducible members of V is closed under subalgebras has the AP if and only if every doubly injective span in V + SI has an amalgam in V. Note, however, that while the property that Q RFSI is closed under subalgebras follows from the existence of equationally definable relative principal congruence meets (satisfied by many quasivarieties serving as algebraic semantics for non-classical logics), a similarly general condition guaranteeing closure under subalgebras is not known for Q + RSI .

Transferable Injections and Strong Amalgamation
We first establish a generalization for classes of similar algebras closed under subalgebras of the well-known fact that a variety has the TIP if and only if it has the CEP and AP [3, Lemma 1.7].
Proposition 4.1.Let K be a class of similar algebras that is closed under subalgebras.Then K has the transferable injections property if and only if it has the one-sided amalgamation property and extension property.
Proof.The left-to-right direction is immediate.For the converse, suppose that K has the 1AP and EP and consider any injective span A, B, C, Combining now Proposition 4.1 with our earlier results for the 1AP and EP, we are able to transfer results for the TIP from a quasivariety to the class of its relatively finitely subdirectly irreducible members, and back again.We now recall a useful characterization of the SAP and a transfer theorem for SE.Theorem 4.4 ([22]).Let Q be a quasivariety.Then Q has the strong amalgamation property if and only if Q has the amalgamation property and surjective epimorphisms.Corollary 4.6.Let V be an arithmetical variety with the congruence extension property such that V FSI is a universal class.Then V has the strong amalgamation property if and only if V FSI has the one-sided amalgamation property and surjective epimorphisms.These results do not provide a transfer theorem for the SAP, however.To obtain such a theorem, at least in one direction, we make use of a well-known characterization of the SE property.Let K be a class of similar algebras and consider any A ∈ K.We say that A is a K-epic subalgebra of B ∈ K if A is a subalgebra of B and for every C ∈ K and all homomorphisms ψ 1 , ψ 2 : B → C, if ψ 1 and ψ 2 are equal on their restriction to A, then ψ 1 = ψ 2 .It is easy to see that A is a K-epic subalgebra of B if and only if A is a subalgebra of B and the inclusion homomorphism of A into B is an epimorphism in K.The following result is folklore (see, e.g., [31,Lemma 3.1]), but we include a proof for completeness.For the converse, let B ∈ K and suppose that A is a proper K-epic subalgebra of B. Then A ∈ K since K is closed under subalgebras.Hence the inclusion map of A into B is a non-surjective K-epimorphism between members of K.
The following theorem strengthens a result from [16].
Theorem 4.8.Let V be an arithmetical variety with the congruence extension property such that V FSI is a universal class.If every doubly injective span in V FSI has a strong amalgam in V, then V has the strong amalgamation property.
Proof.Suppose that every doubly injective span in V FSI has a strong amalgam in V. Then V has the AP, by Corollary 3.4.Hence, by Theorems 4.4 and 4.5, it is enough to show that V FSI has SE, or, equivalently, by Lemma 4.7, that no member of V FSI has a proper V FSI -epic subalgebra.Let B ∈ V FSI , let A be a proper subalgebra of B, and let ι : A → B be the inclusion map.By assumption, the span (A, B, B, ι, ι) has a strong amalgam (D, ψ 1 , ψ 2 ) in V. Let χ : D → i∈I D i be a subdirect representation of D, so that in particular D i ∈ V FSI for each i ∈ I. Since D is a strong amalgam, there exists an i ∈ I such that χψ 1 (i) = χψ 2 (i).On the other hand, since D is an amalgam, it follows that χψ 1 ι = χψ 2 ι.This implies that ι is not a K-epimorphism, proving the theorem.

Decidability
To fix terminology, let us call a variety V finitely generated if it is generated as a variety by a given finite set of finite algebras of finite signature, and residually small if there exists a bound on the size of its subdirectly irreducible members.It is known that if a residually small congruence-distributive variety V has the AP, then it has the CEP [25, Corollary 2.11].
Consider any finitely generated congruence-distributive variety V such that V FSI is closed under subalgebras.By Jónsson's Lemma [24], there exists and can be constructed a finite set V * FSI ⊆ V FSI of finite algebras such that each A ∈ V FSI is isomorphic to some A * ∈ V Theorem 5.1.Let V be a finitely generated congruence-distributive variety such that V FSI is closed under subalgebras.There exist effective algorithms to decide if V has the congruence extension property, amalgamation property, or transferable injections property.If V is also arithmetical, then there exist effective algorithms to decide if V has surjective epimorphisms or the strong amalgamation property.

A Case Study: Varieties of BL-Algebras
BL-algebras, introduced by Hájek in [21] as an algebraic semantics for his basic fuzzy logic of continuous t-norms, have been studied intensively over the past twenty five years, largely in the framework of substructural logics and residuated lattices (see, e.g., [1,2,14,15,23,29,30]).In this section, we use the tools developed in previous sections to contribute to the development of a (still incomplete) description of the varieties of BL-algebras that have the AP.
A BL-algebra is an algebra A = A, ∧, ∨, •, →, 0, 1 satisfying (i) A, ∧, ∨, 0, 1 is a bounded lattice with order a ≤ b :⇐⇒ a ∧ b = a; The class of BL-algebras forms a congruence-distributive variety BL with the CEP, and BL FSI is the positive universal class consisting of all totally ordered BL-algebras (i.e., such that A, ≤ in the preceding definition is a chain).BL-algebras hence form a subvariety of the variety of semilinear integral bounded residuated lattices (also known as MTL-algebras) [13,17].The non-trivial subvarieties of MV, G, and P that have the AP are precisely the varieties generated by S n (n ∈ N >0 ), S ω n (n ∈ N >0 ), MV, G, G 3 , and P [11, Theorem 13].Note also that BL has the AP [30, Theorem 3.7], and [2, Theorem 6] gives a complete description of the varieties of BL-algebras having the AP that are generated by a totally ordered BL-algebra built as an ordinal sum of finitely many members of MV FSI and P FSI .Here, we consider subvarieties of BL 1 := MV ∨ G ∨ P, each of which is generated by a class of "one-component" totally ordered BL-algebras, i.e., members of MV FSI , G FSI , and P FSI .Lemma 6.1.Let V be a subvariety of BL 1 satisfying V ⊆ MV and S n ∈ V for some n ∈ N >1 .Then V does not have the amalgamation property.
Proof.Consider a doubly injective span A, B, C, ϕ B , ϕ C in V FSI , where A is the two-element Boolean algebra, B is S n ∈ V for some n ∈ N >1 , and C ∈ MV.Suppose that there exist a D ∈ V FSI and an embedding ψ C of C into D. Then also D ∈ MV, so D ∈ G FSI ∪ P FSI .Since B is a finite totally ordered MV-algebra, it is simple.Hence any homomorphism ψ B : B → D is either trivial (i.e., maps all elements of B to one element in D), which is not possible, since D is non-trivial, or injective, which is not possible, since B does not embed into any totally ordered Gödel algebra or product algebra.So V FSI does not have the 1AP and it follows, by Theorem 3.3, that V does not have the AP.Theorem 6.2.In addition to the varieties of MV-algebras generated by one totally ordered MV-algebra, there are precisely ten non-trivial subvarieties of BL 1 that have the amalgamation property: G, G 3 , P, G ∨ P, G ∨ C, G 3 ∨ P, G 3 ∨ C, P ∨ C, G 3 ∨ P ∨ C, and G ∨ P ∨ C.
Proof.Let V be any non-trivial subvariety of BL 1 .Then V FSI consists of members of G FSI , MV FSI , and P FSI .The cases where V FSI is included in one of these classes are clear from the previous remarks.Moreover, it follows from [2, Theorem 3.3] that if V contains G n for some n > 3 but not G, then V does not have the AP.Suppose now that V ⊆ MV and V ∩ MV is non-trivial.Then V ∩ MV is either MV or generated by a non-empty finite set of algebras of the form S n or S ω n (n ∈ N >0 ).If V ∩ MV ∈ {G 2 , C}, then, since S n is a subalgebra of S ω n for each n ∈ N, it follows that S n ∈ V for some n ∈ N >1 , and hence, by Lemma 6.1, that V does not have the AP.
It remains therefore to show that each of G ∨ P, G ∨ C, G 3 ∨ P, G 3 ∨ C, P ∨ C, G 3 ∨ P ∨ C, and G ∨ P ∨ C has the AP.Clearly, the two-element Boolean algebra G 2 is the only non-trivial algebra common to the finitely subdirectly irreducible members of the varieties in these joins.Hence, by Proposition 3.6, it suffices to observe that G 2 is a retract of any non-trivial member of G FSI , P FSI , and C FSI .This follows directly from a general result of [7,Theorem 4.5] describing BL-algebras with a Boolean retract, but we may also define a suitable retraction explicitly.Given any non-trivial A ∈ G FSI ∪ P FSI ∪ C FSI , define ϕ : A → {0, 1} by mapping a ∈ A to 0 if and only if a n = 0 for some n ∈ N, where a 0 := 1 and a k+1 := a k • a (k ∈ N).It is straightforward to verify that ϕ is a retraction from A onto G 2 .

Lemma 2 . 1 .
Let Q be any quasivariety and let B ∈ Q.(a) (cf.[26, Lemma 1.3]) Suppose that for any subalgebra A of B and completely meet-irreducible Θ ∈ Con Q A, there exists a Ψ ∈ Con Q B with Ψ ∩ A 2 = Θ.Then B has the Q-CEP.

Proof.
Consider any A, B, C ∈ K, embedding ϕ B : A → B, and surjective homomorphism ϕ C : A → C. Since K is closed under isomorphisms, by (i), we may assume without loss of generality that A is a subalgebra of B and that C = A/Θ for some Θ ∈ Con Q A. Since K has the Q-CEP, by (iii), there exists a Ψ ∈ Con Q B such that Ψ ∩ A 2 = Θ.Let D := B/Ψ ∈ Q and let ψ B be the canonical homomorphism from B to D mapping each b ∈ B to [b] Ψ ∈ B/Ψ.Observe also that for any a, b ∈ A, Finally, by (ii), there exist a D * ∈ K, a surjective homomorphism ψ * B : D → D * , and an embedding ψ *C : C → D * such that ψ * B ψ B ϕ B = ψ * C ϕ C.So K has the EP.In particular, combining Proposition 2.8 with Corollary 2.7, we obtain the following result for the class of relatively finitely subdirectly irreducible members of a quasivariety.Corollary 2.9.Let Q be any quasivariety.If Q RFSI has the Q-congruence extension property, then Q RFSI has the extension property.

Lemma 3 . 1 (
[20,  Lemma 2]).Let A, B, C, ϕ B , ϕ C be a doubly injective span in a class of similar algebras K and suppose that (a) for any distinct x, y ∈ B, there exist a D xy B ∈ K and homomorphisms ψ xy B : B → D xy B and ψ xy C : C → D xy B satisfying ψ xy B ϕ B = ψ xy C ϕ C and ψ xy B (x) = ψ xy B (y); (b) for any distinct x, y ∈ C, there exist a D xy C ∈ K and homomorphisms χ xy B : B → D xy C and χ xy C : C → D xy C satisfying χ xy B ϕ B = χ xy C ϕ C and χ xy C (x) = χ xy C (y).Then A, B, C, ϕ B , ϕ C has an amalgam D, ψ B , ψ C , where D is the product of the algebras in the set {D xy B | x, y ∈ B, x = y} ∪ {D xy C | x, y ∈ C, x = y}.The following result will play a key role in the proof of our Theorem 3.3.

2 .
The map ϕ ′ B sending [a] Θ ∈ A/Θ to [a] Ψ is an embedding of A/Θ into B/Ψ, so A/Θ ∈ K, by (i), and Θ ∈ Con Q A. Hence, by (iii), there exists aΦ ∈ Con Q C such that Φ ∩ A 2 = Θ and C/Φ ∈ K.Moreover, the map ϕ ′ C sending any [a] Θ ∈ A/Θ to [a] Φ is an embedding of A/Θ into C/Φ.So A/Θ, B/Ψ, C/Φ, ϕ ′ B , ϕ ′ C isa doubly injective span of members of K and, by (iv), has an amalgam D xy , χ B , χ C in Q.We define homomorphisms ψ xy B : B → D xy ; b → χ B ([b] Ψ ) and ψ xy C : C → D xy ; c → χ C ([c] Φ ).Then ψ xy B (x) = ψ xy B (y) (as χ B is injective and [x] Ψ = [y] Ψ ) and for any a ∈ A, ) ⇒ (4).Suppose that Q RFSI has the 1AP and let A, B, C, ϕ B , ϕ C be any doubly injective span in Q RFSI .By assumption, there exist a D C ∈ Q RFSI , a homomorphism ψ C B : B → D C , and an embedding ψ C C : C → D C such that ψ C B ϕ B = ψ C C ϕ C .However, A, C, B, ϕ C , ϕ B is also a doubly injective span in Q RFSI , so there exist a D B ∈ Q RFSI , a homomorphism ψ B C : C → D B , and an embedding ψ B B :

Lemma 4 . 2 .Theorem 4 . 3 .
If a quasivariety Q has the transferable injections property, then Q RFSI has the transferable injections property.Proof.Suppose that Q has the TIP and let A, B, C, ϕ B , ϕ C be an injective span in Q RFSI .Then there exist a D ∈ Q, a homomorphism ψ B : B → D, and an embedding ψ C : C → D such that ψ B ϕ B = ψ C ϕ C .Without loss of generality, we may assume that C is a subalgebra of D and ψ C is the inclusion map.By Corollary 2.7, there exist a D * ∈ Q RFSI and a surjective homomorphism χ : D → D * such that ker(χ) ∩ C 2 = ∆ C .Let ψ * B := χψ B and ψ * C := χψ C .Then ker(ψ * C ) = ker(χ) ∩ C 2 = ∆ C , so ψ * C is an embedding, and ψ * B ϕ B = χψ B ϕ B = χψ C ϕ C = ψ * C ϕ C .Hence Q RFSI has the TIP.Let Q be a relatively congruence-distributive quasivariety such that Q RFSI = Q FSI is closed under subalgebras.Then Q has the transferable injections property if and only if Q RFSI = Q FSI has the transferable injections property.Proof.The left-to-right direction follows directly from Lemma 4.2.For the converse, suppose that Q RFSI has the TIP.Then Q RFSI has the 1AP and the EP, by Proposition 4.1, and hence Q has the Q-CEP, by Theorem 2.12.Moreover, Q has the AP, by Theorem 3.3.So Q has the TIP, by Proposition 4.1.

Theorem 4 . 5 ([ 6 ,
Theorem 22]).Let V be an arithmetical variety such that V FSI is a universal class.Then V has surjective epimorphisms if and only if V FSI has surjective epimorphisms.Combining Theorems 4.4 and 4.5 with Corollary 3.4 yields the following result.

Lemma 4 . 7 .
Let K be a class of similar algebras closed under subalgebras.Then K has surjective epimorphisms if and only if no member of K has a proper K-epic subalgebra.Proof.Let A, B ∈ K and suppose that ϕ : A → B is a non-surjective Kepimorphism.Then ϕ[A] is a proper subalgebra of B. Observe that if C ∈ K and ψ 1 , ψ 2 : B → C are homomorphisms that coincide on their restriction to ϕ[A], then ψ 1 ϕ = ψ 2 ϕ and hence ψ 1 = ψ 2 since ϕ is an epimorphism.It follows that ϕ[A] is a proper K-epic subalgebra of B.
Notable subvarieties of BL include (i) the variety MV of MV-algebras consisting of BL-algebras satisfying a ∨ b = (a → b) → b for all a, b ∈ A; (ii) the variety G of Gödel algebras consisting of BL-algebras satisfying a∧b = a•b for all a, b ∈ A; (iii) the variety P of product algebras consisting of BL-algebras satisfying a ∧ (a → 0) = 0 and (a → 0) ∨ ((a → (a • b)) → b) = 1 for all a, b ∈ A. The varieties MV, G, and P are generated by algebras L, G, and P of the form [0, 1], min, max, ⋆, → ⋆ , 0, 1 , where x ⋆ y is max(0, x + y − 1), min(x, y), and xy, respectively.Every member of BL FSI (i.e., every totally ordered BL-algebra) can be constructed as a certain ordinal sum of members of MV FSI and P FSI [1, Theorem 3.7].Let us briefly recall some further relevant facts about MV, G, and P. First, given any totally ordered Abelian group L = L, +, −, 0, ≤ and u ∈ L with u ≥ 0, defining a • b := max(a + b − u, 0) and a → b := min(u − a + b, u) yields a totally ordered MV-algebra Γ(L, u) := [0, u], min, max, •, →, 0, u .Each proper non-trivial subvariety of MV is generated by a non-empty finite set of algebras of the form S n := Γ(Z, n) or S ω n := Γ(Z × lex Z, n, 0 ), where Z is the ordered group of integers, Z × lex Z is the lexicographic product of two copies of Z, and n ∈ N >0 [28, Theorem 4.11].Notably, S 1 generates the variety BA of Boolean algebras, and S ω 1 , known as the Chang algebra, generates a variety denoted by C. Each proper non-trivial subvariety G n of G is generated by the algebra G n := {0, 1 n , . . ., n−1 n , 1}, min, max, min, →, 0, 1 , where n ∈ N >0 and a → b = 1 if a ≤ b, otherwise a → b = b.Finally, the only proper non-trivial subvariety of P is BA (which coincides with G 2 ).
RFSI = Q FSI is closed under subalgebras, then the EP and Q-CEP for Q and Q RFSI = Q FSI all coincide (Theorem 2.12).Now let K and K ′ be two classes of algebras of the same signature.Anamalgam in K ′ of a doubly injective span A, B, C, ϕ B , ϕ C in K is a triple D, ψ B , ψ C where D ∈ K ′ and ψ B , ψ C are embeddings of B and C into D, respectively, such that ψ B ϕ B = ψ C ϕ C (see Figure 1(ii)).The class K has the amalgamation property (for short, AP) if every doubly injective span in K has an amalgam in K.We also say that K has the one-sided amalgamation property (for short, 1AP) if for any doubly injective span A, B, C, ϕ B , ϕ C in K, there exist a D ∈ K, a homomorphism ψ B : B → D, and an embeddingψ C : C → D such that ψ B ϕ B = ψ C ϕ C (see Figure 1(iii)).Itis easy to see using [20, Lemma 2] that a variety V has the 1AP if and only if it has the AP, but this is not always the case for other classes, in particular, V FSI .In Section 3, we prove that when Q has the Q-CEP and Q RFSI is closed under subalgebras, Q has the 1AP (equivalently, the AP) if and only if Q RFSI has the 1AP (Theorem 3.3).
FSI .Hence, by Corollary 2.4, it can be decided if V has the CEP by checking if each member of V * FSI has the CEP.Since V is clearly residually small, if V does not have the CEP, it cannot have the AP.Otherwise, V has the CEP and, by Corollary 3.4, it can be decided if V has the AP by checking -by considering the finitely many finite algebras in V * FSI -if V FSI has the 1AP.Finally, if V is also arithmetical, to check if V has SE, it suffices to check -again, considering the algebras in V * FSI -if V FSI has SE [6, Theorem 22], and V then has the SAP if and only if it has SE and the AP [22].Hence we have established the following result. *