Semilinear idempotent distributive l-monoids

We prove a representation theorem for totally ordered idempotent monoids via a nested sum construction. Using this representation theorem we obtain a characterization of the subdirectly irreducible members of the variety of semilinear idempotent distributive l-monoids and a proof that its lattice of subvarieties is countably infinite. For the variety of commutative idempotent distributive l-monoids we give an explicit description of its lattice of subvarieties and show that each of its subvarieties is finitely axiomatized. Finally we give a characterization of which spans of totally ordered idempotent monoids have an amalgam in the class of totally ordered monoids, showing in particular that the class of totally ordered commutative idempotent monoids has the strong amalgamation property and that various classes of distributive l-monoids do not have the amalgamation property. We also show that exactly seven non-trivial finitely generated subvarieties of the variety of semilinear idempotent distributive l-monoids have the amalgamation property; we are able to determine for all but three of its subvarieties whether they have the amalgamation property or not.


Introduction
Distributive ℓ-monoids are monoids with a distributive lattice-order such that multiplication distributes over both binary meets and binary joins.They occur naturally as inverse-free reducts of lattice-ordered groups (ℓ-groups) and indeed have the same equational theory as the latter [9].More generally, distributive ℓ-monoids occur as residual-free reducts of fully distributive residuated lattices (see e.g., [6]) Email address: simon.santschi@unibe.ch(Simon Santschi) 1 Supported by Swiss National Science Foundation grant 200021 184693.and a systematic study of distributive ℓ-monoids can further the investigation of these reducts and the corresponding implication-free fragments of substructural logics [19] (see e.g, [20]).For example, it can help answer the question whether two classes of fully distributive residuated lattices can be distinguished by residualfree equations or quasi-equations.
Despite providing one of the most natural generalizations of ℓ-groups, the structure theory of distributive ℓ-monoids is not yet at a very sophisticated state, possibly because the tools and techniques of ℓ-group theory do not extend well in the absence of inverses.In particular, there does not yet exist a good characterization of congruences of distributive ℓ-monoids (but see [30,4,9], where congruences are constructed from prime lattice ideals of distributive ℓ-monoids) and not much is known about subdirectly irreducible members of this class to date (see [40], covering the finite commutative case).On the other hand, there exists a natural analogue of the representation theorem of Holland [24] of ℓ-groups as subalgebras of ℓ-groups of automorphisms of chains.That is, in [2] Anderson and Edwards show that every distributive ℓ-monoid embeds into the distributive ℓ-monoid of endomorphisms of a chain.Note, however, that in [39] Repnitskii shows that the variety generated by the inverse-free reducts of abelian ℓ-groups is not finitely axiomatizable, gives a recursive axiomatization for this variety, and proves that in contrast to the case of all distributive ℓ-monoids, there are equations that hold in this variety but not in all commutative distributive ℓ-monoids.Extending the result of Repnitskii, it is proved in [9] that there are equations that hold in all inverse-free reducts of totally ordered ℓ-groups that do not hold in all totally ordered distributive ℓ-monoids.
In this paper, we consider the variety of semilinear idempotent distributive ℓmonoids, i.e., distributive ℓ-monoids that are subdirect products of totally ordered idempotent monoids.At the center of this paper lies the nested sum construction.The nested sum construction was considered by Aglianò and Montagna in [1] under the name 'ordinal sum' for totally ordered BL-algebras.It was generalized by Galatos in [18].The nested sum construction has been used to obtain representation results in a range of settings, including totally ordered BL-algebras [1], totally ordered n-contractive MTL-algebras [25], semilinear commutative idempotent residuated lattices [35,36], and ⋆ -involutive idempotent residuated chains [16].
In the current paper, we extend on one hand the theory of nested sum representations to totally ordered idempotent monoids, and on the other hand we use the nested sum construction to overcome some of the current deficits of the theory of distributive ℓ-monoids.In particular, we give an explicit characterization of subdirect irreducibility in the variety of semilinear idempotent distributive ℓ-monoids in terms of the nested sum representation which also leads to a better understanding and description of the subvariety lattices.The nested sum representation is not only useful to further the structure theory, but also helps us better understand homomorphisms between totally ordered idempotent monoids.In particular, it is useful for establishing the amalgamation property for classes of idempotent distributive ℓmonoids, a fundamental algebraic property (see e.g., [28]) that has been studied in great depth for many varieties of residuated lattices (see [34] for an overview and further references).
The paper is structured as follows.In Section 2, we introduce the necessary definitions and preliminary results about distributive ℓ-monoids that will be used in the later sections.In Section 3, we lay the ground work for the paper.We first introduce the nested sum construction for the finite case and prove a representation theorem (Theorem 3.7) for finite totally ordered idempotent monoids in terms of nested sums, followed by some results obtained using this representation.Then we introduce the nested sum construction in the general case and prove a representation theorem (Theorem 3.15) for arbitrary totally ordered idempotent monoids.Using the general representation theorem we characterize embeddings between totally ordered idempotent monoids with respect to their nested sum decomposition (Lemma 3.17).
In Section 4, we provide a characterization of the subdirectly irreducible members of the variety of semilinear idempotent distributive ℓ-monoids in terms of nested sums first for the finite case (Theorem 4.3) and then for the general case (Theorem 4.6).The characterization in the finite case also yields a counting result for finite subdirectly irreducibles (Corollary 4.4).
In Section 5, we use the characterization of finite subdirectly irreducibles to investigate the lattice of subvarieties of the variety of idempotent distributive ℓmonoids.Similarly to [36] we use the nested sum representation to obtain a description of the lattice of subvarieties of the variety of semilinear idempotent distributive ℓ-monoids, resulting in a proof that the lattice of subvarieties is countably infinite (Theorem 5.1).
In Section 6, we first specialize the results of Section 4 and Section 5 to the commutative case and then obtain an explicit description of the lattice of subvarieties of the variety of commutative idempotent distributive ℓ-monoids (Theorem 6.6).Using the explicit description of its lattice of subvarieties we establish that every proper subvariety of the variety of commutative idempotent distributive ℓ-monoids can be relatively axiomatized by a single equation (Theorem 6.10).We conclude the section with a brief discussion of some consequences of the results in the context of Sugihara monoids and commutative fully distributive residuated lattices.
In Section 7, we shift our focus to the amalgamation property for classes of semilinear idempotent distributive ℓ-monoids.First, we use the nested sum repre-sentation to give a complete characterization of the spans of totally ordered idempotent monoids that have an amalgam in the class of totally ordered monoids (Theorem 7.10) in terms of not 'restricting' to one of two forbidden spans.In particular, we show that the class of totally ordered commutative idempotent monoids has the strong amalgamation property (Corollary 7.5) and the forbidden spans enable us to show that several classes of distributive ℓ-monoids do not have the amalgamation property.Finally, we show that exactly seven non-trivial finitely generated subvarieties of the variety of semilinear idempotent distributive ℓ-monoids have the amalgamation property (Theorem 7.27), and establish the failure of this property for all but three non-finitely generated subvarieties (Corollary 7.31).

Preliminaries
Conventions.We assume that the reader is familiar with the basic notions of universal algebra which can be found for example in [5].We mostly follow the notation of [5].In particular for an algebra A we denote its universe by A, for S ⊆ A we denote the subalgebra of A generated by S by Sg(S), the congruence lattice of A by Con(A), the trivial congruence on A by ∆ A , i.e., ∆ A = {⟨a, a⟩ | a ∈ A}, and for a, b ∈ A, the principal congruence generated by {⟨a, b⟩} by Θ A (a, b).We write Θ(a, b) if the algebra is clear.For a class of algebras K we denote by H(K), S(K), P (K), P U (K), P S (K), and I(K) the closure of K under homomorphic images, subalgebras, products, ultraproducts, subdirect products, and isomorphic images, respectively.Moreover, we will denote by V (K) and Q(K) the variety and quasivariety generated by K, respectively.For a variety V and a set X, we denote by F V (X) the V-free algebra over X and for n ∈ N we write F V (n) for F V ({1, . . ., n}).
A distributive ℓ-monoid is an algebra M = ⟨M, ∧, ∨, •, e⟩ such that As in (3) we will sometimes write ab for a • b, we will drop unnecessary brackets if no confusion arises, and we will assume that • binds stronger than ∧ and ∨.
The class DLM of distributive ℓ-monoids forms a variety (equational class).We call a distributive ℓ-monoid idempotent if its monoid reduct is idempotent, i.e., satisfies the equation x 2 ≈ x, and commutative if its monoid reduct is commutative.
We denote the variety of idempotent distributive ℓ-monoids by IdDLM and the variety of commutative idempotent distributive ℓ-monoids by CIdDLM.
Example 2.1.An ℓ-group is an algebra L = ⟨L, ∧, ∨, •, −1 , e⟩ such that ⟨L, ∧, ∨⟩ is a lattice, ⟨L, •, −1 , e⟩ is a group, and a ≤ b implies cad ≤ cbd for all a, b, c, d ∈ L, where ≤ is the lattice order of L. It is well-known that the lattice reduct of an ℓ-group is distributive and that products distribute over meets and joins.Hence, the inverse-free reducts of ℓ-groups are distributive ℓ-monoids.
In fact every distributive ℓ-monoid can be seen as a subalgebra of an ℓ-monoid of endomorphisms on a chain.Similarly to ℓ-groups (see [24]) there is a Hollandstyle representation theorem for distributive ℓ-monoids.
We call a distributive ℓ-monoid M = ⟨M, ∧, ∨, •, e⟩ an ordered monoid if its lattice order is a total order.As there is a one-to-one correspondence, we will also consider ordered monoids as relational structures M = ⟨M, •, e, ≤⟩, where ⟨M, •, e⟩ is a monoid, ≤ is a total order on M , and for all a, b, c, d ∈ M , a ≤ b implies cad ≤ cbd.We note that a map between ordered monoids is a homomorphism if and only if it is a monoid homomorphism and it is order-preserving.We denote the class of ordered monoids by OM and the class of idempotent ordered monoids by IdOM.We will also denote the class of commutative idempotent ordered monoids by CIdOM.
We call a distributive ℓ-monoid semilinear if it is contained in the variety SemDLM generated by the class OM.We denote the variety of semilinear idempotent distributive ℓ-monoids by SemIdDLM.Note that in the literature semilinear distributive ℓ-monoids are also called representable (see e.g., [9]) following the nomenclature for ℓ-groups.
Recall that an algebra A is called congruence-distributive if its congruence lattice Con(A) is a distributive lattice and a class of algebras is called congruencedistributive if all of its members are congruence-distributive.It is well-known that lattices are congruence-distributive and hence every algebra with a lattice reduct is also congruence-distributive.In particular, every distributive ℓ-monoid is congruence-distributive.We also recall the following useful result by Jónsson about subdirectly irreducible algebras in congruence-distributive varieties: Theorem 2.4 (Jónsson's Lemma [26,Theorem 3.3.]).Let K be a class of algebras such that V (K) is congruence-distributive.Then every subdirectly irreducible algebra in V (K) is contained in HSP U (K) and V (K) = IP S HSP U (K).
In this paper we will often use a special case of Theorem 2.4.
Corollary 2.5 ([26, Corollary 3.4.]).If K is a finite set of finite algebras such that V (K) is congruence-distributive, then the subdirectly irreducible algebras of V (K) are in HS(K), and V (K) = IP S (HS(K)).
The class OM of ordered monoids is a positive universal class, i.e., closed under HSP U .So, by Theorem 2.4, we get the following result.
Proposition 2.6.Every subdirectly irreducible algebra in SemDLM is totally ordered and a distributive ℓ-monoid is semilinear if and only if it is isomorphic to a subdirect product of ordered monoids.
It follows, in particular, that SemIdDLM is the variety generated by IdOM.We have the following equational characterization of semilinearity, where for terms s and t we denote by s ≤ t the equation s ∧ t ≈ s.Proposition 2.7 ([9, Proposition 5. 4.], see also [3,Corollary 6.10]).A distributive ℓ-monoid is semilinear if and only if it satisfies the equation z An important consequence of Proposition 2.7 is the following result, which was first proved by Merlier in [30].
Hence, in particular CIdDLM is the subvariety of SemIdDLM consisting of its commutative members and is generated by CIdOM.
For a distributive ℓ-monoid M = ⟨M, ∧, ∨, •, e⟩, we define its order dual M ∂ by M ∂ = ⟨M, ∨, ∧, •, e⟩.Taking the order dual of a distributive ℓ-monoid corresponds to reversing its lattice order, so in particular for an ordered monoid M = ⟨M, •, e, ≤⟩ we have M ∂ = ⟨M, •, e, ≥⟩.For a term t we define the dual t ∂ of t recursively by and we extend the notion of a dual to equations by setting (t ≈ s) For a variety V of distributive ℓ-monoids we denote by V ∂ the class consisting of the order duals of the algebras in V.It is clear that V ∂ is also a variety axiomatized by the duals of the axioms of V. We call a variety V self-dual if V = V ∂ .Lemma 2.9.Let V be a variety of distributive ℓ-monoids axiomatized by a set Σ of equations.Then V is self-dual if and only if V |= ε ∂ for every ε ∈ Σ.
So we obtain the following corollary, where we note that the dual of the equation from Proposition 2.7 is a substitution instance of the original equation.Most of the varieties we consider in this paper are self-dual, so in the following we will use this fact to shorten some of the proofs by referring to duality.
A variety V is called locally finite if every finitely generated algebra contained in V is finite.For a variety V of distributive ℓ-monoids we denote by V m the variety generated by the monoid reducts of V. Note that V m is a variety of monoids.Lemma 2.11.A subvariety V of DLM is locally finite if and only if the variety V m generated by the monoid reducts of the members of V is locally finite.
Proof.First note that a variety is locally finite if and only if every finitely generated free algebra is finite.
For the right-to-left direction assume that V m is locally finite.Then for every n ∈ N the free algebra F Vm (n) is finite and, using the distributivity of the lattice reduct and the distributivity of products over meets and joins, it is straightforward to see that every member of the free algebra F V (n) corresponds to a meet of joins of members of F Vm (n).Thus, since F Vm (n) is finite, modulo the lattice axioms there are only finitely many different such terms.Hence also F V (n) has to be finite for every n ∈ N, i.e., V is locally finite.
For the left-to-right direction note first that, by the definition of V m , for any monoid terms s, t we have V |= s ≈ t if and only if V m |= s ≈ t, so F Vm (n) can be considered as a subset of F V (n).Now suppose that V is locally finite.Then for every n ∈ N the free algebra F V (n) is finite and thus also F Vm (n) is finite.Hence V m is locally finite.Theorem 2.12 ( [22], [29,Theorem 2]).The variety of idempotent monoids is locally finite.
Hence, by Lemma 2.11, we get some immediate corollaries.
Since every variety of idempotent distributive ℓ-monoids is locally finite, recursive axiomatizability yields the decidability of their quasi-equational theories.
Corollary 2.14.Every recursively axiomatized subvariety of IdDLM has a decidable quasi-equational theory.In particular the variety SemIdDLM has a decidable quasi-equational theory.
Corollary 2.15.Every subvariety of SemIdDLM is generated by its finite subdirectly irreducible (totally ordered) members as a quasivariety.
Thus to describe subvarieties of SemIdDLM, it is enough to consider the finite subdirectly irreducible members of a given subvariety.

Nested sums
For idempotent distributive ℓ-monoids the product is much more restricted than in the general non-idempotent case.Lemma 3.1 (cf.[33,42]).For every idempotent distributive ℓ-monoid M and a, b ∈ M we have: (iv) If M is totally ordered, then ab ∈ {a, b}.
Note that it follows from Lemma 3.1 that for every idempotent ordered monoid M and subset S ⊆ M , the set S ∪ {e} is the universe of the subalgebra Sg(S) of M and ab ∈ {a, b} for all a, b ∈ S ∪ {e}.In the sequel we will use this fact without explicitly mentioning it.Lemma 3.2.Let M be a non-trivial idempotent ordered monoid with top element ⊤ and bottom element ⊥.Then exactly one of the following holds: (1) For all a ∈ M we have ⊥ • a = a • ⊥ = ⊥.
(3) For all a ∈ M \{⊤, ⊥} we have So, since M is totally ordered it suffices to consider the case e ≤ a < ⊤.Then, by Lemma 3.1, a  As discussed in Section 1, nested sums were introduced by Galatos in [18] and since used to provide a plethora of structure theorems for classes of residuated lattices (see [1,25,35,36]).They have been deployed especially heavily in the context of BL-algebras, where they are usually called ordinal sums [1].Following the discussion in Fussner and Galatos [16], we adopt the terminology nested sum instead of ordinal sum.
Let M and N be idempotent ordered monoids, where we relabel the elements such that M ∩ N = {e}.We define the nested sum of M and N by Intuitively this means that we replace the identity e in M with N and extend the order and product in such a way that the elements of N behave with respect to elements of M like e.
Example 3.4.Consider the algebras C 2 and G 3 of Example 3.3, where we rename the element ⊥ in C 2 as 1.The order of the nested sum G 3 ⊞C 2 is drawn in Figure 1 and it has the following multiplication table: It is no coincidence that we consider the algebras of Example 3.3.We will see later that every finite idempotent ordered monoid is isomorphic to a nested sum of these four algebras.Lemma 3.5 (cf.[35,Proposition 4.8]).Let M and N be idempotent ordered monoids.Then M ⊞ N is an idempotent ordered monoid.Moreover, M and N embed into M ⊞ N via the inclusion maps.Lemma 3.6 (cf.[36,Proposition 3.3]).Let M 1 , M 2 , and M 3 be idempotent ordered monoids.Then Since the nested sum is associative up to isomorphism, it makes sense to use the notation To shorten the notation we will denote this nested sum also by ⊞ n i=1 M i and we set ⊞ 0 i=1 M i = 0, where 0 is a fixed trivial algebra.So we assume that for all nested sums there is a fixed relabeling of the elements.We note that M ⊞ 0 ∼ = 0 ⊞ M ∼ = M for all M, i.e., 0 can be seen as the neutral element of the nested sum operation.Theorem 3.7.Each finite idempotent ordered monoid M is isomorphic to a unique Proof.We prove the claim by induction on the cardinality of M.
and the claim is already proved for all k < m.By Lemma 3.2, there are four cases: In each case we have |L| < m, so the claim follows by the induction hypothesis.
For uniqueness note that in every step the first term of the nested sum is unique, yielding the uniqueness for every M i .
From the uniqueness of the above nested sum decomposition we obtain the following counting result: Corollary 3.8.The number I(n) of idempotent ordered monoids with n ∈ N \{0} elements (up to isomorphism) is recursively defined by I(1) = 1, I(2) = 2, and Moreover, Proof.It is clear that the trivial algebra is the only 1-element algebra and that C 2 and C ∂ 2 are the only 2-element idempotent ordered monoids up to isomorphism.Now every idempotent ordered monoid with n > 2 elements, by Theorem 3.7, is isomorphic to a nested sum L ⊞ M where either A standard calculation then yields the closed formula for I(n).
Remark 3.9.The closed formula for I(n) is the same formula as in [21] for the number of idempotent residuated chains with n elements, but in [21] the number of n-element idempotent residuated chains should be I(n − 1), since every finite idempotent residuated chain is of the form C 2 ⊞ M, where M is an idempotent ordered monoid, i.e., the number of idempotent residuated chains with n + 1 elements is equal to the number of idempotent ordered monoids with n elements.So an alternative proof of Corollary 3.8 can be obtained by using the result of [21] about idempotent residuated chain.Theorem 3.10.Let M 1 , . . ., M n , N 1 , . . ., N n be idempotent ordered monoids and for i = 1, . . ., n let φ i : Corollary 3.11.Let M 1 , . . ., M n , N 1 , . . ., N n be idempotent ordered monoids and for i = 1, . . ., n let φ i : M i → N i be an ℓ-monoid embedding.Then the map Analogous to the case of similar constructions the nested sum can be generalized to infinite nested sums.Let C = ⟨C, ⊑⟩ be a chain and let {M c } c∈C be a family of idempotent ordered monoids M c = ⟨M c , • c , e, ≤ c ⟩, where we relabel the elements such that M c ∩ M d = {e} for all c ̸ = d.We define the nested sum We set ⊞c∈∅ M c = 0. Note that for a non-empty subchain D of C, if we restrict the operations of ⊞c∈C M d to d∈D M d , we obtain the nested sum ⊞d∈D M d .
The finite nested sums that we considered above correspond to the cases where C is a finite chain.The next lemma generalizes Lemma 3.5.
Lemma 3.12.Let C = ⟨C, ⊑⟩ be a chain and let {M c } c∈C be a family of idempotent ordered monoids.Then the nested sum ⊞c∈C M c is an idempotent ordered monoid.Moreover, M c embeds into ⊞c∈C M c via the inclusion map for every Proof.Note that all the properties we need to check involve at most three different algebras in the nested sum.Thus the claim follows from the fact that for idempotent ordered monoids Let M be an idempotent monoid.We consider the Green equivalence relation D M defined by letting for a, b ∈ M a D M b : ⇐⇒ aba = a and bab = b.
In [13] Dubreil-Jacotin proved that an idempotent monoid M is totally orderable if and only if for all a, b ∈ M we have ab ∈ {a, b} and every equivalence class of D M contains at most two elements (see also [31,32] or [41] for a characterization of totally orderable idempotent semigroups).Lemma 3.13 ([13]).Let M be an idempotent ordered monoid.For two distinct elements a, b ∈ M the following are equivalent: On the other hand if we have an order-embedding f : C → D such that for every c ∈ C, M c is a subalgebra of N f (c) , then we can define the map (1) The map f φ : C → D is an order-embedding such that for every c ∈ C, M c is a subalgebra of N fφ(c) .
( (2) By Corollary 3.11 and since f is an order-embedding, if . So, since checking whether φ is an ℓ-monoid embedding involves at most two terms of the nested sum, φ f is an ℓ-monoid embedding.
(3) is immediate from the definition.
Then M embeds into N if and only if there exists an order-embedding f : {1, . . ., m} → {1, . . ., n} such that for every i ∈ {1, . . ., m} we have that M i is a subalgebra of N f (i) .

Subdirectly irreducible algebras in SemIdDLM
In this section we use the nested sum decomposition to give a characterization of the subdirectly irreducible members of SemIdDLM.Note that in contrast to ℓ-groups and residuated lattices not very much is known about congruences and subdirectly irreducible elements in varieties of distributive ℓ-monoids.For the semilinear and idempotent case the nested sum representation helps us overcome this problem.
The next lemma shows that the nested sum decomposition of a finite subdirectly irreducible idempotent ordered monoid does not contain any consecutive occurrences of C 2 or any consecutive occurrences of C ∂ 2 .
Lemma 4.1.Every finite subdirectly irreducible idempotent ordered monoid is Proof.Suppose that M is a finite idempotent ordered monoid.Then by Theorem 3.7, there is a unique nested sum decomposition Let a be the non-identity element in M j and b be the non-identity element in and  (1) M is subdirectly irreducible. ( It follows from Theorem 4.3, that if the algebras C 2 and C ∂ 2 do not occur in the nested sum decomposition of a non-trivial finite idempotent ordered monoid, then it is subdirectly irreducible.As the next example shows this is no longer true in the infinite case. Example 4.5.Consider the chain ω of natural numbers and set for n ∈ ω, M n = G 3 and M = ⊞n∈ω M n .Then, if we define for n ∈ ω the congruence θ n = ∆ M ∪ ( m≥n M m ) 2 , we get n∈ω θ n = ∆ M .Hence M is not subdirectly irreducible.Thus, there are infinite nested sums that are not subdirectly irreducible even though they do not have any occurrence of C 2 or C ∂ 2 in their decomposition.
The next theorem is the correct generalization of Theorem 4.3.
Theorem 4.6.Let M = ⊞c∈C M c be an idempotent ordered monoid such that for all c ∈ C, Then the following are equivalent: (1) M is subdirectly irreducible.
(2) C contains a maximal element and for all (3) Con(M) is a chain containing an atom.
(1) ⇒ (2): Suppose that M is subdirectly irreducible.If C does not contain a maximal element, then we can define for every c ∈ C the congruence θ c = ∆ M ∪ ( c⊑d M d ) 2 such that ∆ M = c∈C θ c , contradicting the fact that M is subdirectly irreducible.So C must contain a maximal element m and we can define the congruence µ = M 2 m ∪ ∆ M .Now suppose for a contradiction that there exist (2) ⇒ (3): Suppose that C contains a maximal element m and for all is a chain containing an atom.

The lattice of subvarieties of SemIdDLM
In this section we use the nested sum representation and the characterization of the finite subdirectly irreducibles to investigate the lattice of subvarieties of SemIdDLM.
Recall that a poset ⟨P, ≤⟩ is called well-founded if it neither contains an infinite antichain nor an infinite descending chain.For two algebras A and B we define the relation ≤ HS (≤ IS ) by A ≤ HS B if and only if A ∈ HS(B) (A ≤ IS B if and only if A ∈ IS(B)).Moreover, for a variety V of finite type we denote by V * a set which contains exactly one algebra from each of the isomorphism classes of the finite subdirectly irreducible members of V. Then ⟨V * , ≤ HS ⟩ and ⟨V * , ≤ IS ⟩ are posets.The following result by Davey shows how to describe the lattice of subvarieties of V using ⟨V * , ≤ HS ⟩ if V is congruence-distributive and locally finite, where we recall that a complete lattice D = ⟨D, ∧, ∨⟩ is called completely distributive if for all index sets I, J and {a i,j } i∈I,j∈J ⊆ D we have Theorem 5.1 ([11, Theorem 3.3]).Let V be a congruence-distributive, locally finite variety of finite type.Then the lattice of subvarieties of V is completely distributive and is isomorphic to the lattice of order ideals of the poset ⟨V * , ≤ HS ⟩ via the map I → V (I) mapping an order ideal of ⟨V * , ≤ HS ⟩ to the subvariety of V that it generates.

Corollary 5.2 ([36, Corollary 4.3]
).Let V be a congruence-distributive, locally finite variety of finite type.Then the lattice of subvarieties of V is countable if and only if ⟨V * , ≤ HS ⟩ is well-founded.
As SemIdDLM is a congruence-distributive, locally finite variety of finite type we want to use Theorem 5.1 and Corollary 5.2.By Theorem 3.7, we can assume that SemIdDLM * consists of the unique nested sum decompositions of the finite subdirectly irreducible algebras.Lemma 5.3.Every homomorphic image of an idempotent ordered monoid M is isomorphic to a subalgebra of M.
Proof.We prove that for every congruence θ on M, the quotient M/θ is isomorphic to a subalgebra of M. For a congruence θ on M let φ : M → M be a map such that φ(e) = e and that maps every element to a fixed representative of its congruence class.So in particular for the congruence class of e the fixed representative is e.Since the congruence classes of θ are convex it is clear that φ is order-preserving.Let ⟨P, ≤⟩ be a poset.We define the order ≤ * on the set σ(P ) of finite sequences of P by ⟨p 1 , . . ., p n ⟩ ≤ * ⟨q 1 , . . ., q m ⟩ : ⇐⇒ there exists an order-embedding f : {1, . . ., n} → {1, . . ., m} such that p i ≤ q f (i) for all i ∈ {1, . . ., n}.Lemma 5.5 (Higman's Lemma [23,Theorem 4.3]).If ⟨P, ≤⟩ is a well-founded poset, then ⟨σ(P ), ≤ * ⟩ is a well-founded poset.
So, since restrictions of well-founded partial orders are well-founded, Lemma 5.5 yields that ⟨SemIdDLM * , ≤ IS ⟩ is well-founded.
Proof.Since ⟨SemIdDLM * , ≤ IS ⟩ is well-founded, it follows from Corollary 5.4 and Corollary 5.2 that the lattice of subvarieties of SemIdDLM is countable.That it is infinite follows from Corollary 2.5 and the fact that, by Corollary 4.4, SemIdDLM contains infinitely many finite subdirectly irreducibles.
Remark 5.7.By Theorem 5.1, the lattice of subvarieties of SemIdDLM is isomorphic to the lattice of order ideals of ⟨SemIdDLM * , ≤ IS ⟩ via the map I → V (I).Moreover, by Theorem 4.3, we have a description of the finite subdirectly irreducible elements of SemIdDLM in terms of the nested sum decomposition and, by the equivalence (1), we have a description of the order ≤ IS on SemIdDLM * in terms of the nested sum decomposition.Hence, we get a more or less explicit description of the lattice of subvarieties of SemIdDLM in terms of the nested sum decomposition of its finite subdirectly irreducible members.

The commutative case
In this section we apply the results of the previous two sections to the commutative case to get an explicit description of the lattice of subvarieties of CIdDLM.
Using this description we obtain a finite axiomatization for every subvariety of CIdDLM. 2rom the nested sum decomposition and the characterization of subdirectly irreducibles we immediately get the following result for the commutative case: Corollary 6.1.Let M be a finite commutative idempotent ordered monoid.Then . Moreover, M is subdirectly irreducible if and only if for all i ∈ {1, . . ., n − 1} we have Furthermore, we get the following counting result: Corollary 6.2.There are up to isomorphism 2 n−1 commutative idempotent ordered monoids of size n ≥ 1.
Proof.Note that every commutative idempotent ordered monoid of size n ≥ 1 has a unique nested sum representation So it is clear that there are 2 n−1 idempotent ordered monoids of size n ≥ 1. Remark 6.3.Again Corollary 6.2 also follows from the corresponding counting result about commutative idempotent residuated chains in [21].In [12] a counting result is obtained for finite totally ordered commutative idempotent semigroups.
For n > 2 we define inductively the algebras C n and C ∂ n by and we set C 1 = C ∂ 1 = 0. Note that these are exactly the commutative idempotent ordered monoids that do not contain consecutive occurrences of C 2 or C ∂ 2 .Hence, Corollary 6.1 yields an explicit characterization of the subdirectly irreducible idempotent ordered monoids in the commutative case: Corollary 6.4.For every n > 1 the algebras C n and C ∂ n are up to isomorphism the only subdirectly irreducible commutative idempotent ordered monoids with n elements.Remark 6.5.For n ≥ 2 the algebras C n are exactly the distributive ℓ-monoid reducts of the finite Sugihara chains of length n (for the definition of a Sugihara monoid see e.g., [38]).Indeed, the same is true for n ≥ 2 and the unique extension of C n with an implication.Let us recall that a class of algebras K is called a prevariety if it is closed under I, S, and P or equivalently K = ISP (K).Corollary 6.7.Every non-trivial proper subvariety of CIdDLM can be generated as a prevariety by two or fewer finite subdirectly irreducible algebras.Corollary 6.8.Every infinite subdirectly irreducible member of CIdDLM generates CIdDLM as a quasi-variety.
Let us for every natural number n ≥ 2 rename the elements of C n such that C n = ⟨{e, 1, . . ., n − 1}, •, e, ≤⟩ with V (C 1 ) CIdDLM and m • k = max N (m, k).We define the equations σ n for n ≥ 2 inductively as follows: we set σ 2 := (x 1 ≤ e) and if σ n = (s n ≤ t n ) is already defined, we set So for n ≥ 3 we have Lemma 6.9.
(i) Let n ≥ 2 be even.Then we have C ∂ n ̸ |= σ n .Moreover, for every M ∈ CIdOM with M ̸ |= σ n and elements a 1 , . . ., a n−1 ∈ M witnessing the failure of σ n via the valuation x k → a k , the subalgebra Sg(a 1 , . . ., a n−1 ) is isomorphic to C ∂ n via the map a k → k.
(ii) Let n ≥ 2 be odd.Then we have C n ̸ |= σ n .Moreover, for every M ∈ CIdOM with M ̸ |= σ n and elements a 1 , . . ., a n−1 ∈ M witnessing the failure of σ n via the valuation x k → a k , the subalgebra Sg(a 1 , . . ., a n−1 ) is isomorphic to C n via the map a k → k.
Proof.For the first part of (i) note that for n = 2 we clearly have 1 > e in C ∂ 2 , and for n > 2 even and 1, . . ., n − 1 ∈ C ∂ n we get, by Equation (3), that Similarly we get for the first part of (ii) that for n ≥ 2 odd, C n ̸ |= σ n .Now we prove the second parts of (i) and (ii) together by induction on n ≥ 2. For n = 2 we have σ 2 = (x 1 ≤ e) and if M ∈ CIdOM and a 1 ∈ M such that a 1 ≰ e, then, since M is totally ordered, we get a 1 > e and the subalgebra Sg(a Now suppose that n > 2 is odd and (i) holds for n − 1.Then for σ n−1 = (s n−1 ≤ t n−1 ), σ n has the following form Let M ∈ CIdOM and a 1 , . . ., a n−1 ∈ M witnessing the failure of σ n .Then, since M is totally ordered we have . ., a n−1 ) So the elements a 1 , . . ., a n−2 witness the failure of σ n−1 via the map x k → a k and the induction hypothesis yields that Sg(a 1 , . . ., a n−2 ) is isomorphic to C ∂ n−1 via the map a k → k.Thus, by Equation (3), we get that yielding e > a n−1 and a n−2 a n−1 = a n−1 .Thus, since a n−2 • a n−3 = a n−2 , we get a n−1 < a n−3 and, by Equation (2), Sg(a 1 , . . ., a n−1 ) is isomorphic to C n via the map a k → k.
The case where n is even and (ii) holds for n − 1 is very similar.
For n ≥ 2 and two disjoint sets {x 1 , . . ., x n−1 } and {y 1 , . . ., Note that s ∂ n and t ∂ n are strictly speaking not the duals of s n and t n , since we change the variables.Theorem 6.10.
(1) If n ≥ 2 is even, then the equations σ n and σ ∂ n axiomatize the subvarieties V (C n ) and V (C ∂ n ), respectively, relative to CIdDLM.
(2) If n ≥ 2 is odd, then the equations σ n and σ ∂ n axiomatize the subvarieties V (C ∂ n ) and V (C n ), respectively, relative to CIdDLM.
(3) For n ≥ 2 the equation γ n+1 axiomatizes the subvariety Proof.First note that it follows from Theorem 6.6 that if an equation holds in Thus ( 1) and ( 2) follow immediately from Lemma 6.9 and the fact that C ∂ n is the order dual of C n .For (3) note that if two varieties V 1 and V 2 are axiomatized by sets of equations Σ 1 and ) is axiomatized by the set {σ n+1 , σ ∂ n+1 }.But also for every idempotent ordered monoid M and a, b, c, d ∈ M we have that a ≤ b and c ≤ d implies ac ≤ bd.So, by definition of γ n+1 , we get that On the other hand if we substitute in γ n+1 the variables y 1 , . . ., y n by e, we obtain σ n+1 and if we substitute the variables x 1 , . . ., x n by e, we obtain Remark 6.11.It follows from Theorem 6.6 and the fact that for every n ∈ N \{0} we have V (C n ) = ISP (C n ) that every finite member of CIdDLM embeds into a power of C n for some n ∈ N \{0}.Moreover, clearly every finite distributive ℓ-monoid with an absorbing bottom element is the residual-free reduct of a fully distributive residuated lattice 3 .Thus the class of Sugihara monoids and the class of commutative idempotent fully distributive residuated lattices satisfy the same universal first-order sentences in the language {∧, ∨, •, e}.This slightly generalizes Corollary 25 of [38].Furthermore it follows from Theorem 6.10 that nonisomorphic finite Sugihara chains can be distinguished by equations in the language {∧, ∨, •, e}.

Amalgamation
In this section we use the nested sum representation to study the amalgamation property for idempotent ordered monoids.
Let K be a class of algebras of the same type.A span in K is a pair ⟨i 1 : A → B, i 2 : A → C⟩ of embeddings between algebras A, B, C ∈ K.The class K has the amalgamation property if for every span ⟨i i.e., the following diagram commutes: The triple ⟨D, j 1 , j 2 ⟩ is called an amalgam of the span.We say that K has the strong amalgamation property if for every span ⟨i 1 : A → B, i 2 : A → C⟩ there exists an amalgam ⟨D, which we call a strong amalgam of the span.
Proposition 7.1.The variety DLM does not have the amalgamation property.
Proof.The set of invertible elements of a distributive ℓ-monoid gives rise to a subalgebra which is the inverse-free reduct of an ℓ-group.Suppose for a contradiction that DLM has the amalgamation property.Then every span of inverse-free reducts of ℓ-groups has an amalgam and, by taking the subalgebra of invertible elements of the amalgam, we can assume that this amalgam is the inverse-free reduct of an ℓ-group, since monoid homomorphisms map invertible elements to invertible elements.Hence, it follows that the variety of ℓ-groups has the amalgamation property, contradicting [37, Theorem 3.1].
It is well-known that the variety of distributive lattices does not have the strong amalgamation property.We will now show that this failure of the strong amalgamation property is inherited by every non-trivial subvariety of DLM.
Proof.Let V be a non-trivial subvariety of DLM.Then there is an M ∈ V that is non-trivial, i.e., there is an a 0 ∈ M with a 0 ̸ = e.But then either a 0 ∧ e < e or a 0 ∨ e > e.If a = a 0 ∧ e < e, then we consider the subalgebra A of M generated by a.We have A = {a n | n ∈ N} and a n+1 ≤ a n , so the map φ : A → C 2 defined by φ(e) = e and φ(a n ) = ⊥ for n > 0 is a homomorphism.Hence C 2 ∈ V. Otherwise if a 0 ∨ e > e we consider a = a 0 ∨ e and, arguing dually, we get The second part follows from the fact that C 2 = ⟨{⊥, e}, ∧, ∨, ∧, e⟩, where ⟨{⊥, e}, ∧, ∨⟩ is the two element distributive lattice, and V (C 2 ) = ISP (C 2 ).Dually the claim for V (C ∂ 2 ) follows.
Proposition 7.3.No non-trivial subvariety of DLM has the strong amalgamation property.
Proof.Let V be a non-trivial subvariety of DLM.By Lemma 7.2, we may assume, without loss of generality, that It is easy to see that φ and ψ are ℓ-monoid embeddings.Moreover, it is well-known (see e.g., [15]) that the span ⟨φ, ψ⟩ considered as a span in the variety of distributive lattices does not have a strong amalgam, so it does not have a strong amalgam in V, since the lattice reduct of a strong amalgam in V would be a strong amalgam in the variety of distributive lattices.
Let S = ⟨i 1 : L → M, i 2 : L → N⟩ and S ′ = ⟨j 1 : L ′ → M ′ , J 2 : L ′ → N ′ ⟩ be spans of idempotent ordered monoids.We say that S restricts to S ′ if the algebras L ′ , M ′ , and N ′ are subalgebras of L, M, and N, respectively, and i 1 ↾ L ′ = j 1 and i 2 ↾ L ′ = j 2 .Moreover, we extend this notion in the obvious way to the case where the algebras L ′ , M ′ , and N ′ embed into L, M, and N, respectively, by identifying L ′ , M ′ , and N ′ with the images of the respective embeddings.
Recall from Section 5 that C 2 embeds into G 3 and D 3 via the obvious inclusion maps.Moreover, clearly these are the only possible embeddings from C 2 into G 3 and D 3 , respectively.We denote the resulting span by S 1 := ⟨C 2 → G 3 , C 2 → D 3 ⟩ and we define the span From here on we will denote the top element of G 3 by ⊤ and the top element of D 3 by ⊤ * .We call a span ⟨i 1 : L → M, i 2 : L → N⟩ of idempotent ordered monoids a compatible span if it does not restrict to any of the spans S 1 or S 2 up to permuting the embeddings in the span.We define for every a ∈ A the algebra P a as the largest algebra of the set {M a , N a }.Note that the algebras M a and N a cannot be two different algebras of the same cardinality, since clearly {M a , N a } = {C 2 , C ∂ 2 } is not possible because L a is a non-trivial subalgebra of M a and N a , and {M a , N a } = {G 3 , D 3 } would contradict the assumption that the span is compatible.Hence P a is well-defined for each a ∈ A and we get, in particular, that both M a and N a embed into P a and the embeddings agree on the images of L a under i 1 and i 2 .Moreover, we define the algebras P d for d ∈ D \ A by Then, by construction, for every b ∈ B, M b embeds into P b and for every c ∈ C, N c embeds into P c .Define P = ⊞d∈D P d .Then, by construction, Lemma 3.17 yields that the maps j 1 : M → P and j 2 : N → P corresponding to the inclusions B → D and C → D are embeddings and together with the algebra P they form an amalgam for the span.
If L, M, and N are commutative, then the algebras G 3 and D 3 cannot occur in their e-sum decomposition, i.e., the span is always compatible.Moreover, in the construction of P we have P a = N a = M a for every a ∈ A, so P is a strong amalgam.
Corollary 7.5.The class CIdOM of commutative idempotent ordered monoids has the strong amalgamation property.
The next example shows that the amalgam constructed in the proof of Proposition 7.4 is not necessarily a strong amalgam and even if we restrict to compatible spans a strong amalgam does not always exist.
Example 7.6.Consider the span ⟨C 2 → G 3 , C 2 → G 3 ⟩.Clearly it is compatible, but it does not have a strong amalgam in IdOM.To see this let ⟨M, j 1 : G 3 → M, j 2 : G 3 → M⟩ be an amalgam of the span in IdOM.Then, by Lemma 3.17, j 1 (G 3 ) and j 2 (G 3 ) are components of the nested sum decomposition of M isomorphic to G 3 .But we have j 1 (⊥) ∈ j 1 (G 3 ) ∩ j 2 (G 3 ), so we get j 1 (G 3 ) = j 2 (G 3 ), i.e., the amalgam is not strong.
Note that every span ⟨i 1 : 0 → L, i 2 : 0 → M⟩ is compatible.So we get the following result.
Corollary 7.7.The class IdOM of idempotent ordered monoids has the joint embedding property, i.e., each pair L, M of idempotent ordered monoids embeds into a common idempotent ordered monoid N.
The next proposition shows that being compatible is not only a sufficient condition for spans of idempotent ordered monoids to have an amalgam in the class of ordered monoids but also a necessary condition.
Proposition 7.8.The spans S 1 and S 2 do not have an amalgam in the class OM of ordered monoids.
Proof.Suppose for a contradiction that there exists an amalgam E in the class of ordered monoids for the span S 1 with embeddings j 1 : G 3 → E and j 2 : So, since j 1 is order-preserving, j 1 (⊤) = j 1 (⊥), contradicting the injectivity of j 1 .Similarly j 2 (⊤ * ) ≤ j 1 (⊤), yields j 2 (⊤ * ) = j 2 (⊥) which is again a contradiction.Hence S 1 does not have an amalgam in OM.For the span S 2 the claim follows by duality.
Corollary 7.9.The classes OM and IdOM do not have the amalgamation property.From Proposition 7.8 and Proposition 7.4 we get the following result.
Theorem 7.10.For a span S = ⟨i 1 : L → M, i 2 : L → N⟩ of idempotent ordered monoids the following are equivalent: (1) S has an amalgam in OM.
(2) S has an amalgam in IdOM.
So to check whether a span of idempotent ordered monoids has an amalgam in the class of ordered monoids it suffices to check if it restricts (up to permuting the embeddings in the span) to one of the spans S 1 and S 2 .The next proposition shows that the spans S 1 and S 2 do also not have an amalgam in the bigger classes SemDLM and IdDLM.
Proposition 7.11.The spans S 1 and S 2 do not have an amalgam in the varieties SemDLM and IdDLM.
Proof.For SemDLM suppose for a contradiction that the span S 1 has an amalgam E ∈ SemDLM with inclusion maps G 3 → E and D 3 → E.Then, since E is semilinear, we have without loss of generality E = i∈I E i for a family of ordered monoids {E i } i∈I .So there is an i ∈ I and there are homomorphisms j 1 : G 3 → E i and j 2 : D 3 → E i such that j 1 ↾ C 2 = j 2 ↾ C 2 and j 1 (e) ̸ = j 1 (⊥), so also j 2 (e) ̸ = j 2 (⊥).But, since G 3 and D 3 are simple, j 1 and j 2 are embeddings.Hence E i is an amalgam for the span which is an ordered monoid, contradicting Proposition 7.8.For the span S 2 the claim follows by duality.
For IdDLM suppose again for a contradiction that the span S 1 has an amalgam E ∈ IdDLM with embeddings j 1 : G 3 → E and j 2 : D 3 → E and E ∈ IdDLM.Then j 1 (e) ≤ j 1 (⊤), j 2 (⊤ * ), so by Lemma 3.1 (ii).Thus we get But we also have Hence j 1 (⊤) ∨ j 2 (⊤ * ) = j 1 (⊤), i.e., j 2 (⊤ * ) ≤ j 1 (⊤), and as in the proof of Proposition 7.8 we get a contradiction.For the span S 2 the claim follows by duality.Proof.Let V be a finitely generated subvariety of DLM that contains one of the algebras.Then, by Proposition 7.17, V does not have the congruence extension property.Moreover, since V is finitely generated, it follows from Corollary 2.5 that V * contains only finitely many subdirectly irreducible algebras, yielding that it is residually small.Thus the claim follows from Lemma 7.19.Let V be a variety.We denote by V FSI the class of finitely subdirectly irreducible members of V, i.e., V FSI contains all algebras A ∈ V such that ∆ A is meet-irreducible in Con(A).Moreover, we say that a class of algebras K has the one-sided amalgamation property if for every span ⟨i 1 : A → B, i 2 : A → C⟩ in V there exists an algebra D ∈ K, a homomorphism j 1 : B → D, and an embedding j 2 : C → D such that j 1 • i 1 = j 2 • i 2 .The triple ⟨D, j 1 , j 2 ⟩ is called a one-sided amalgam of the span.We note that the amalgamation property implies the one-sided amalgamation property.Lemma 7.22 (Relativized Jónsson's Lemma [10, Lemma 1.5.]).Let K be a class of algebras.Then Q(K) FSI ⊆ ISP U (K).In particular, if K is a finite set of finite algebras and at least one member of K has the trivial algebra as a subalgebra, then Q(K) FSI ⊆ IS(K).
Theorem 7.23 ([17, Corollary 3.5.]).Let V be a congruence-distributive variety with the congruence extension property such that V FSI is closed under subalgebras.Then V has the amalgamation property if and only if V FSI has the one-sided amalgamation property.Lemma 7.24.For n ∈ {2, 3} we have Proof.For the left-to-right inclusions note that all the varieties are generated by the respective algebras as quasi-varieties, by Corollary 2.5 and Lemma 5.3.Thus the left-to-right inclusion follows from Lemma 7.22.
For the right-to-left inclusions note that for n ∈ {2, 3} every member of IS(C n , C ∂ n ) is either trivial or subdirectly irreducible.The same is also true for V (G 3 ), V (D 3 ) and V (G 3 , D 3 ).
For U note that U FSI = IS(C 2 ) consists up to isomorphism of the algebras C 2 and 0. Thus every span in U FSI has C 2 with the respective inclusion maps as a one-sided-amalgam.So U FSI has the amalgamation property.
Lemma 7.29.If V is a subvariety of SemIdDLM that is not finitely generated, then CIdDLM ⊆ V.
Proof.Suppose that V is a subvariety of SemIdDLM that is not finitely generated.Then, since V is locally finite, there exists an indexed family of finite subdirectly irreducible algebras {M i } i∈N with |M i | < |M i+1 | for each i ∈ N, i.e., the family contains finite subdirectly irreducible algebras which strictly increase in size.By Proposition 2.6, each subdirectly irreducible member of V is totally ordered.So let c(M i ) be the number of components in the nested sum decomposition of M i .Note that, since the biggest possible components are G 3 and D 3 , we have Therefore, without loss of generality we may also assume that also c(M i ) < c(M i+1 ) for each i ∈ N. Thus, by Theorem 4.3 and the fact that C 2 , C ∂ 2 ≤ IS G 3 , D 3 , for every n ≥ 1 there exists an i ∈ N such that C n ≤ IS M i .Hence, CIdDLM ⊆ V, by Theorem 6.6.Proposition 7.30.Let V be a subvariety of SemIdDLM.If V is not finitely generated and each span of totally ordered members has an amalgam in V, then Proof.Since V is not finitely generated, by Lemma 7.29, we have CIdDLM ⊆ V.If V ̸ = CIdDLM, then not all members of V are commutative.Hence either G 3 ∈ V or D 3 ∈ V, but not both, by Proposition 7.11.Assume, without loss of generality, that G 3 ∈ V. Then every finite subdirectly irreducible member of V is isomorphic to a finite nested sum of the algebras C 2 , C ∂ 2 , G 3 , so clearly V ⊆ V ({G 2n+1 | n ∈ N}), by Lemma 3.17.For the reverse inclusion we prove by induction that for every n ∈ N we have G 2n+1 ∈ V. Before we start the induction note that, since CIdDLM ⊆ V, we have C 3 ∈ V and, by assumption, the span ⟨i 1 : C ∂ 2 → C 3 , i 2 : C ∂ 2 → G 3 ⟩ has an amalgam ⟨A, j 1 : C 3 → A, j 2 : G 3 → A⟩ in V.Moreover, using the notation C 3 = {2 < e < 1} we get j 1 (2)•j 2 (⊥) = j 1 (2)•j 1 (1)•j 2 (⊥) = j 1 (2)•j 2 (⊤)•j 2 (⊥) = j 1 (2)•j 2 (⊤) = j 1 (2).
Note that the class CIdOM of commutative idempotent ordered monoids contains all finitely subdirectly irreducible members of CIdDLM, is closed under subalgebras, and has the strong amalgamation property.Similarly the classes containing the totally ordered members of the varieties V ({G 2n+1 | n ∈ N}) and V ({D 2n+1 | n ∈ N}), respectively, contain all finitely subdirectly irreducible members, are closed under subalgebras, and have the amalgamation property, by using the construction in the proof of Proposition 7.4.But the varieties CIdDLM, V ({G 2n+1 | n ∈ N}), and V ({D 2n+1 | n ∈ N}) do not have the congruence extension property, so there is no obvious way of extending the amalgamation property from their totally ordered members to the whole variety.Hence, it remains open whether these three varieties have the amalgamation property or not.

(
ii) If e ≤ ab, then ab = a ∨ b. (iii) If ab ≤ e, then ab = a ∧ b.
2 and C ∂ 2 also indicates that the algebras are commutative and have two elements.The notation for G 3 indicates that it has three elements and that the non-identity elements are left-absorbing (French: 'à gauche') and similarly D 3 has three elements and the non-identity elements are right-absorbing (French: 'à droite').Also, the algebras G 3 and D 3 are connected.Indeed, we have G 3 ∼ = ⟨D 3 , * , e, ≤⟩, where a * b := b • a.Moreover, C 2 and C ∂ 2 are subalgebras of G 3 and D 3 and it is easy to see that C 2 , C ∂ 2 , G 3 , and D 3 are simple.Indeed, already their monoid reducts are simple.We also note that each of these four algebras corresponds to a different case of Lemma 3.2.They are the minimal examples of each case.
where • is the extension of the monoid operations on M and N with a • b = b • a = a for all a ∈ M \{e} and b ∈ N , and ≤ is the least extension of the orders of M and N that satisfies for all a ∈ M \{e} and b where L is the subalgebra of M with the subuniverse L = M \{⊤}.• If M has bottom and top elements ⊥ and ⊤ such that ⊥ • ⊤ = ⊥ and ⊤ • ⊥ = ⊤, then we have M ∼ = G 3 ⊞L, where L is the subalgebra of M with the subuniverse L = M \{⊥, ⊤}.• If M has bottom and top elements ⊥ and ⊤ such that ⊤ • ⊥ = ⊥ and ⊥ • ⊤ = ⊤, then we have M ∼ = D 3 ⊞L, where L is the subalgebra of M with the subuniverse L = M \{⊥, ⊤}.

Proof. It suffices to consider the case n = 2 .
Note that the map φ is well defined, since φ 1 (e) = φ 2 (e) = e and M 1 ∩ M 2 = {e}.First we show that φ is a monoid homomorphism.By definition we have φ(e) = e.Moreover, φ 1 and φ 2 are monoid homomorphisms, so it suffices to take a ∈ M 1 \{e} and b ∈ M 2 , and therefore ab = ba = a.Since φ −1 1 ({e}) = {e}, we have φ(a) ∈ N 1 \{e} and clearly φ(b) ∈ M 2 .So we get φ(ab) = φ(a) = φ(a)φ(b), by the definition of the nested sum.Hence φ is a monoid homomorphism.To see that φ is order-preserving it again suffices to consider a ∈ M 1 \{e} and b ∈ M 2 , since both φ 1 and φ 2 are order-preserving.If a ≤ b, then a ≤ M 1 e.So, since φ 1 is orderpreserving and φ −1 1 ({e}) = {e}, we have φ(a) ≤ N 1 e, φ(a) ̸ = e, and φ(b) ∈ N 2 , yielding φ(a) ≤ φ(b).Similarly if b ≤ a, we get φ(b) ≤ φ(a).Hence φ is a homomorphism.That we have ker where • the product • extends • c for all c ∈ C such that for all c, d ∈ C with c ⊏ d and a ∈ M c \{e}, b ∈ M d we have a • b = b • a = a, • and the order ≤ is the least extension of the orders ≤ c such that for all c, d ∈ C with c ⊏ d and a ∈ M c \{e}, b ∈ M d we have a ≤ b if a ≤ c e and b ≤ a if e ≤ c a.
Moreover, every equivalence class of D M has at most two elements.Proof.Let a, b ∈ M be two distinct elements.(1) ⇒ (3): Suppose contrapositively that ab = ba, then we get aba = baa = ba = bba = bab.Hence ⟨a, b⟩ / ∈ D M .(3) ⇒ (2): Suppose ab ̸ = ba.Then clearly a ̸ = e and b ̸ = e.So the subalgebra Sg(a, b) of M has universe {a, b, e}.Moreover, by Lemma 3.1, since ab ̸ = ba, we get a < e < b or b < e < a.So Sg(a, b) is isomorphic to G 3 or D 3 .(2) ⇒ (1): Suppose that the subalgebra Sg(a, b) of M is isomorphic to G 3 or D 3 .Then clearly aba = a and bab = b, so a D M b.That every equivalence class of D M has at most two elements is immediate, by Lemma 3.1 and the fact that for any three distinct elements of M either at least two are positive or at least two are negative, and thus do not generate G 3 or D 3 .Lemma 3.14.Let M be an idempotent ordered monoid and let a, b, c ∈ M be three distinct elements with a D M b.Then ac = ca and bc = cb, and ac = a if and only if bc = b.Proof.First note that, by Lemma 3.13, every equivalence class of D M contains at most two elements, so ⟨a, c⟩, ⟨b, c⟩ / ∈ D M and we get ac = ca and bc = cb.Suppose now that ab = a and ba = b.Then, if ac = a, we get bc = bac = ba = b.Conversely, if bc = b, then we get ac = abc = ab = a.For the case ab = b and ba = a the proof is symmetric, noting that ac = ca and bc = cb.The next theorem generalizes Theorem 3.7.Theorem 3.15.Let M be an idempotent ordered monoid.Then we have M ∼ = ⊞c∈C M c for some chain C = ⟨C, ⊑⟩ and M c ∈ {C 2 , C ∂ 2 , G 3 , D 3 }.Moreover, this nested sum is unique over the algebras {C 2 , C ∂ 2 , G 3 , D 3 }.Proof.Let C = {[a] | a ∈ M \{e}}, where [a] = {b ∈ M | a D M b} for a ∈ M , and define the relation ⊑ on C by [a] ⊑ [b] ⇐⇒ ab = ba = a or a D M b.By Lemma 3.1 (iv) and Lemma 3.14, ⊑ is well-defined and C = ⟨C, ⊑⟩ is a chain.Define for a ∈ M \{e}, the algebra M [a] to be the subalgebra Sg([a]) of M with universe [a] ∪ {e}.By Lemma 3.13, we get for every c ∈ C that M c is isomorphic to one of the algebras in {C 2 , C ∂ 2 , G 3 , D 3 }.Moreover, by construction, we get M = ⊞c∈C M c .The nested sum is unique over the algebras {C 2 , C ∂ 2 , G 3 , D 3 }, since, by Lemma 3.13, for every nested sum N = ⊞d∈D N d over the algebras {C 2 , C ∂ 2 , G 3 , D 3 } with M = N we have for all a, b ∈ N \{e} that a D M b if and only if a, b ∈ N d for some d ∈ D. This also implies that D and C are isomorphic.Remark 3.16.Similar constructions also using the Green equivalence relation D were considered in [8, 7] for idempotent residuated chains and conic idempotent residuated lattices.Let C, D be chains and M = ⊞c∈C M c , N = ⊞d∈D N d idempotent ordered monoids with M c , N d ∈ {C 2 , C ∂ 2 , G 3 , D 3 }.If φ : M → N is an embedding, then for every c ∈ C, φ(M c ) is a subalgebra of N and, by Lemma 3.13 and Theorem 3.15, there exists a unique d ∈ C such that φ(M c ) ⊆ N d .Thus we get an injective map f φ : C → D, where f φ (c) is the unique element of D such that φ(M c ) ⊆ N fφ(c) .

)
We have φ fφ = φ and f φ f = f .Proof.(1) By definition of the map f φ , we already know that f φ is injective and that M c is a subalgebra of N fφ(c) for each c ∈ C.So it remains to show that f φ is order-preserving.Let c 1 , c 2 ∈ C such that c 1 ⊏ c 2 and suppose for a contradiction that f (c 2 ) ⊑ f (c 1 ).Then for a ∈ M c 1 \{e} and b ∈ M c 2 \{e} we have φ(a) = φ(a • b) = φ(a) • φ(b) = φ(b).Hence a = b a contradiction.Hence f φ is orderpreserving.
3 and, by Theorem 3.10, the homomorphism φ :N 1 ⊞N 2 ⊞N 3 → N 1 ⊞M j ⊞N 3 induced by the homomorphisms φ 1 = id N 1 , φ 2 : N 2 → M j with φ 2 (e) =e, φ 2 (a) = φ 2 (b) = a, and φ 3 = id N 3 has kernel ker(φ) = ker(φ 1 ) ∪ ker(φ 2 ) ∪ ker(φ 3 ) = ∆ M ∪ {⟨a, b⟩, ⟨b, a⟩}.Moreover, by Theorem 3.10, the map ψ : M → ⊞ n−1 i=1 M i , induced by the homomorphisms and a • b = a or b • a = a.Then Θ(a, b) = Θ(a, e).Proof.We show the claim for a • b = a (the case b • a = a is symmetric).Since a • b = a and e • b = b we have Θ(a, b) ⊆ Θ(a, e).Conversely, if a ≤ e ≤ b or b ≤ e ≤ a, then we are done, since lattice congruence classes are convex.Otherwise using the assumptions that a ̸ = b and a • b = a, we get that either e < b < a or a < b < e.Without loss of generality we assume that e < b < a.So for a ∈ M i and b ∈ M j we get i < j, since both a and b are strictly positive.If M i ∈ {G 3 , D 3 } or M j ∈ {G 3 , D 3 }, then there exists c ∈ M i ∪ M j with c < e < b < a such that ac = a and bc = c, or ca = a and cb = c, yielding ⟨a, c⟩ ∈ Θ(a, b) and, by convexity, ⟨a, e⟩ ∈ Θ(a, b).Otherwise M i = M j = C ∂ 2 .So, by assumption, i + 1 ̸ = j and M i+1 ̸ = C ∂ 2 , i.e., M i+1 ∈ {C 2 , G 3 , D 3 }.But then, there exists a c ∈ M i+1 such that c < e < b < a, ac = a, and bc = c, yielding ⟨a, c⟩ ∈ Θ(a, b).Thus, by convexity, we get ⟨a, e⟩ ∈ Θ(a, b).Theorem 4.3.Let M be a non-trivial finite idempotent distributive ℓ-monoid.Then the following are equivalent:

Figure 3 :
Figure 3: The poset ⟨CIdDLM * , ≤IS⟩ and C ∂ m ≤ IS C n .Thus the poset ⟨CIdDLM * , ≤ IS ⟩ has the form represented by Figure 3.If for X ⊆ CIdDLM * we denote by ↓X the downwards closure of X under ≤ IS , then it is clear from Figure 3 that every proper order ideal of ⟨CIdDLM * , ≤ IS ⟩ is equal to either ↓{C n }, ↓{C ∂ n } or ↓{C n , C ∂ n } for some n ∈ N. Thus Corollary 5.4 together with Theorem 5.1 yields the following characterization of the lattice of subvarieties of CIdDLM: Theorem 6.6.The lattice of subvarieties of CIdDLM is the lattice represented by Figure 4.

Proposition 7 . 4 .
Let L, M, N be idempotent ordered monoids.Then every compatible span ⟨i 1 : L → M, i 2 : L → N⟩ has an amalgam in the class IdOM of idempotent ordered monoids.In particular, every span of commutative idempotent ordered monoids has a strong amalgam.Proof.Let A, B, C be chains, L = ⊞a∈A L a , M = ⊞b∈B M b , N = ⊞c∈C N c such that L a , M b , N c ∈ {C 2 , C ∂ 2 , G 3 , D 3 },and let ⟨i 1 : L → M, i 2 : L → N⟩ be a compatible span.Then, by Lemma 3.17, there are corresponding orderembeddings f : A → B and g : A → C. Without loss of generality we may assume that A = B ∩ C and f, g are the inclusion maps.By [14, Section 9.2], there exists an extension D of the chains B and C with D = B ∪ C.

Corollary 7 . 12 .Proposition 7 . 20 .
No subvariety of SemDLM or IdDLM that contains the algebras G 3 and D 3 has the amalgamation property.In particular, the varieties SemDLM, SemIdDLM, and IdDLM do not have the amalgamation property.The next example shows that the spans S 1 and S 2 have amalgams in the variety DLM.No finitely generated subvariety of DLM that contains any of the algebras.C 4 , C ∂ 4 , M ⊞ N, or N ⊞ M for M ∈ {C 2 , C ∂ 2 }, N ∈ {G 3 , D 3 } has the amalgamation property.

Corollary 7 . 21 .
No proper subvariety of CIdDLM that contains C 4 or C ∂ 4 has the amalgamation property.
e) = f (⊥) = e and id C ∂ 3 form a one-sided amalgam, and for the span ⟨i 2 , i 1 ⟩ the maps id C 2 and g :C ∂ 2 → C 2 , g(e) = g(⊤)= e form a one-sided amalgam.Thus V FSI has the one-sided amalgamation property.For W again note that W FSI = IS(C 3 ) consists up to isomorphism of the algebras 0, C 2 , C ∂ 2 , andC 3 .Let ⟨i 1 : A → B, i 2 : A → C⟩ be a span in W FSI , i.e., without loss of generality A, B, C ∈ {0, C 2 , C ∂ 2 , C 3 }.Then the inclusion maps j 1 : B → C 3 and j 2 : C → C 3 form an amalgam, since for every A ∈ {0, C 2 , C ∂ 2 , C 3 } there exists a unique embedding into C 3 .Hence W FSI has the amalgamation property.For V (C ∂ 2 ) and V (C ∂ 3 ) the claim follows by duality and for V (D 3 ) and V (G 3 ) the proofs are very similar to the proof for V (C 3 ), also using the uniqueness of embeddings.Combining Propositions 7.20, 7.25, and 7.26 we get the following characterization for amalgamation in non-trivial finitely generated subvarieties of SemIdDLM.Theorem 7.27.The non-trivial finitely generated subvarieties of SemIdDLM that have the amalgamation property areV (C 2 ), V (C ∂ 2 ), V (C 2 , C ∂ 2 ), V (C 3 ), V (C ∂ 3 ), V (G 3 ),and V (D 3 ).Corollary 7.28.The non-trivial proper subvarieties of CIdDLM with the amalgamation property are V (C 2 ), V (C ∂ 2 ), V (C 2 , C ∂ 2 ), V (C 3 ), and V (C ∂ 3 ).Let us define for n ∈ N the algebras G 2n+1 := The algebras C 2 , C ∂ 2 , G 3 , and D 3 are idempotent ordered monoids.The notation of C ∂ 2 makes sense, since it is isomorphic to the order dual of C 2 .The notation for C