Finitary Monads on the Category of Posets

Finitary monads on $\mathsf{Pos}$ are characterized as the precisely the free-algebra monads of varieties of algebras. These are classes of ordered algebras specified by inequations in context. Analagously, finitary enriched monads on $\mathsf{Pos}$ are characterized: here we work with varieties of coherent algebras which means that their operations are monotone.


Introduction
Equational specification usually applies classes of (often many-sorted) finitary algebras specified by equations. That is, varieties of algebras over the category Set S of S-sorted sets. This is well known to be equivalent to applying finitary monads over Set S , i.e. monads preserving filtered colimits: every variety V yields a free-algebra monad T V on Set S which is finitary and whose Eilenberg-Moore category is isomorphic to V. Conversely, every finitary monad T on Set S defines a canonical S-sorted variety V whose free-algebra monad is isomorphic to T.
There are cases in which algebraic specifications use partially ordered sets rather than sets without a structure. The goal of our paper is to present for the category Pos of partially ordered sets an analogous characterization of finitary monads: we define varieties of ordered algebras which allow us to represent (a) all finitary monads on Pos and (b) all enriched finitary monads on Pos as the free-algebra monads of varieties. 'Enriched' refers to Pos as a cartesian closed category: a monad is enriched if its underlying functor T is locally monotone (f ≤ g in Pos(A, B) implies T f ≤ T g in Pos(T A, T B)). Case (b) works with algebras on posets such that the operations are monotone (and as morphisms we take monotone homomorphisms). Whereas for (a) we have to work with algebras on posets whose operations are not necessarily monotone (but whose morphisms are). To distinguish these cases, we shall call an algebra coherent if its operations are all monotone.
A basic step, in which we follow the excellent presentation of finitary monads on enriched categories due to Kelly and Power [11], is to work with operation symbols whose arity is a finite poset rather than a natural number; we briefly recall the approach of op. cit. in Section 2. Just as natural numbers n = {0, 1, . . . , n − 1} represent all finite sets up to isomorphism, we choose a representative set Pos f of finite posets up to isomorphism. Members of Pos f are called contexts. A signature is then a set Σ of operation symbols of arities from Pos f . More precisely, Σ is a collection of sets (Σ Γ ) Γ∈Pos f . Thus, a Σ-algebra is a poset A together with an operation σ A , for every σ ∈ Σ Γ , which assigns to every monotone map u : Γ → A an element σ A (u) of A.
For example, let ¾ be the two-chain in Pos f given by x < y. Then an operation symbol σ of arity ¾ is interpreted in an algebra A as a partial function σ A : A × A → A whose definition domain consists of all comparable pairs in A.
Given a signature Σ we form, for every context Γ ∈ Pos f , the set T (Γ) of terms in context Γ. It is defined as usual in universal algebra by ignoring the order structure of contexts. Then, for every Σ-algebra A, whenever a monotone function f : Γ → A is given (i.e. whenever the variables of context Γ are interpreted in A) we define an evaluation of terms in context Γ. This is a partial map f # assigning a value to a term t provided that values of the subterms of t are defined and respect the order of Γ. This leads to the concept of inequation in context Γ: it is a pair (s, t) of terms in that context. An algebra A satisfies this inequation if for every monotone interpretation f : Γ → A we have that both f # (t) and f # (s) are defined and f # (s) ≤ f # (t) holds in A. We use the following notation for inequations in context: By a variety we understand a category V of Σ-algebras presented by a set E of Σinequations in context. Thus the objects of V are all algebras satisfying each Γ ⊢ s ≤ t in E, and morphisms are monotone homomorphisms. We prove that every variety V is monadic over Pos, that is, for the monad T V of free V-algebras V is isomorphic to the category Pos T V of algebras for T V . Moreover, T V is a finitary monad and, in case V consists of coherent algebras, T V is enriched.
Conversely, with every finitary monad T on Pos we associate a canonical variety whose free-algebra monad is isomorphic to T. This process from monads to varieties is inverse to the above assignment V → T V . Moreover, if T is enriched, the canonical variety consists of coherent algebras. This leads to a bijection between finitary enriched monads and varieties of coherent algebras.
Is it really necessary to work with signatures of operations with partially ordered arities and terms in context? There is a 'natural' concept of a variety of ordered (coherent) algebras for classical signatures Σ = (Σ n ) n∈N . Here terms are elements of free Σ-algebras on finite sets (of variables) and a variety is given by a set of inequations s ≤ t where s and t are terms. Such varieties were studied e.g. by Bloom and Wright [6,7]. Kurz and Velebil [12] characterized these classical varieties as precisely the exact categories (in an enriched sense) with a 'suitable' generator. In a recent paper, the first author, Dostál, and Velebil [2] proved that for every such variety V the free-algebra monad T V is enriched and strongly finitary in the sense of Kelly and Lack [10]. This means that the functor T V is the left Kan extension of its restriction along the full embedding E : Pos fd ֒→ Pos of finite discrete posets: Conversely, every strongly finitary monad on Pos is isomorphic to the free-algebra monad of a variety in this classical sense. This answers our question above affirmatively: contexts are necessary if all (possibly enriched) finitary monads are to be characterized via inequations.
Example 1.1. We have mentioned above a binary operation σ(x, y) in context x < y. For the corresponding variety Alg Σ (with no specified inequations) the free-algebra monad is described in Example 4.3. This monad is not strongly finitary [2,Ex. 3.15], thus no variety with a classical signature has this monad as the free-algebra monad.
Related work As we have already mentioned, the idea of using signatures in context stems from the work of Kelly and Power [11]. They presented enriched monads by operations and equations. A signature in their sense is more general than what we use: it is a collection of posets (Σ Γ ) Γ∈Pos f , and a Σ-algebra A is then a poset together with a monotone functions from Σ Γ to the poset of monotone functions from Pos(Γ, A) to A for every context Γ. Whereas we deal with the monadic view on varieties of ordered algebras in the present paper, the view using algebraic theories has been investigated by Power with coauthors, e.g. [20][21][22][23], see Section 5. In particular, the paper [20] works with enriched categories over a monoidal closed category V for which a V -enriched base category C has been chosen. Then enriched algebraic C -theories are shown to correspond to Venriched monads on C . This is particularly relevant for the current paper: by choosing V = Set and C = Pos we treat non-enriched finitary monads on Pos, whereas the choice V = C = Pos covers the enriched case.
Acknowledgement The authors are grateful to Jiří Rosický for fruitful discussion.

Equational Presentations of Monads
We now recall the approach to equational presentations of finitary monads introduced by Kelly and Power [11]; our aim here is to bring the rest of the paper into this perspective. However, we note that the signatures used here are more general than those of the subsequent sections, and (unlike later) some enriched category theory is used. The reader can decide to skip this section without losing the connection.
For a locally finitely presentable category C enriched over a symmetric monoidal closed category V Kelly and Power consider (enriched) monads on C that are finitary, i.e. the ordinary underlying endofunctor preserves filtered colimits. Below we specialize their approach to C = Pos considered as an ordinary category (V = Set) or as a category enriched over itself (V = Pos) as a cartesian closed category. In the first case, the homobject Pos(A, B) is the set of all monotone functions from A to B; in the latter case, this is the poset of those functions, ordered pointwise. As in Section 1, a representative set Pos f of finite posets (called contexts) is chosen which is to be viewed as a full subcategory of Pos. We denote by |Pos f | the corresponding discrete category. In the introduction we considered the special case of signatures where each poset Σ Γ is discrete, i.e. we just have a set of operation symbols in context Γ; for emphasis, we will call such signatures discrete.
Remark 2.2. Recall [8, Def. 6.5.1] the concept of a tensor for objects V ∈ V and C ∈ C : it is an object V ⊗ C of C together a natural isomorphism In the case where C = Pos and V = Set we get the copower and for C = V = Pos we just get the product in Pos: (2) The category of finitary enriched monads on Pos is denoted by FinMnd(Pos). We have a forgetful functor U : FinMnd(Pos) → Fin(Pos).
By precomposing endofunctors with the non-full embedding J : |Pos f | → Pos we obtain a forgetful functor from Fin(Pos) to Sig. It has a left adjoint assigning to every signature Σ the polynomial functor P Σ given on objects by and similarly on morphisms. As previously explained, the hom-object Pos(Γ, X) can have one of the two meanings: for V = Set this is regarded as a set and for V = Pos as a poset. Henceforth, we will use that notation for hom-objects only in the latter case and write Pos 0 (Γ, X) for the set of monotone maps.
Observation 2.4. The usual category of algebras for the functor P Σ , whose objects are posets A with a monotone map α : P Σ A → A, has the following form for our two enrichements: (1) Let V = Set. Then α as above is a monotone map Γ∈Pos f u∈Pos 0 (Γ,A) Σ Γ → A, and as such has components assigning to every monotone function u : Γ → A (that is, a monotone interpretation of the variables in Γ) a monotone function Σ Γ → A. We denote this function by σ → σ A (u).
In other words, the poset A is equipped with operations σ A : Pos 0 (Γ, A) → A (which need not be monotone since Pos 0 (Γ, A) is just a set) satisfying σ A (u) ≤ τ A (u) for all pairs σ ≤ τ in Σ Γ and u in Pos(Γ, A). If Σ is discrete, this is precisely a Σ-algebra (see the introduction).
(2) Now let V = Pos. Then α : P Σ A → A is a monotone map Γ∈Pos f Pos(Γ, A) × Σ Γ → A, and thus has as components monotone functions (u, σ) → σ A (u). That is, in addition to the condition that σ A (u) ≤ τ A (u) for all pairs σ ≤ τ in Σ Γ and u in Pos(Γ, A) as above, we also see that each σ A is monotone. Thus, if Σ is discrete, this is precisely a coherent algebra (again, see the introduction).
Observe also that 'homomorphism' has the usual meaning: a monotone function preserving the given operations. In fact, given algebras α : P Σ A → A and β : P Σ B → B a homomorphism is a monotone function f : A → B such that f · α = β · P Σ f . This is equivalent to f (σ A (u)) = σ B (f · u) for all u ∈ Pos(Γ, A) and all σ ∈ Σ Γ . Remark 2.5. (1) As shown by Trnková et al. [24] (see also Kelly [9]) every ordinary finitary endofunctor H on Pos generates a free monad whose underlying functor H is a colimit of the ω-chain H = colim n<ω W n of functors, where W 0 = Id and W n+1 = HW n + Id Connecting morphisms are w 0 : Id → H + Id, the coproduct injection, and w n+1 = Hw n + Id. The colimit injections c n : W n X → HX in Pos have the property that if a parallel pair u, v : HX → A satisfies c n · u ≤ c n · v for all n < ω, then we have u ≤ v. It follows thatĤ is enriched if H is.
(2) The category of H-algebras is isomorphic to the Eilenberg-Moore category Pos H [4].
(3) Lack [13] shows that the forgetful functor is monadic. The corresponding monad M on Sig assigns to every signature Σ the signature P Σ · J : |Pos f | → Pos.
(4) It follows that every enriched finitary monad T on Pos can be regarded as an algebra for the monad M. Therefore, T is a coequalizer in FinMnd(Pos) of a parallel pair of monad morphisms between free M-algebras on signatures ∆, Σ: This is the equational presentation of T considered by Kelly and Lack [10].
Example 2.6. (1) In the case where V = Set and C = Pos, FinMnd(Pos) is the category of (non-enriched) finitary monads on Pos. Consider the above coequalizer in the special case that ∆ consists of a single operation δ of context Γ. That is, ∆ Γ = {δ} and all ∆Γ forΓ = Γ are empty. By the Yoneda lemma, l and r simply choose two elements of H Σ Γ, say t l and t r . The above coequalizer means that T is presented by the signature Σ and the equation t l = t r . For ∆ arbitrary, we do not get one equation, but a set of equations (one for every operation symbol in ∆) and T is presented by Σ and the corresponding set of equations, grouped by their respective contexts.
(2) The case V = C = Pos yields as FinMnd(Pos) the category of enriched finitary monads on Pos. That is, the underlying endofunctor T is locally monotone.
Remark 2.7. The fact that every finitary (possibly enriched) monad on Pos has an equational presentation depends heavily on the fact that signatures are not reduced to the discrete ones. In contrast, we make do with discrete signatures in the rest of the paper, and then obtain a characterization of finitary (possibly enriched) monads using inequational presentations. While it is clear that the two specification formats are mutually convertible, inequational presentations seem natural for varieties of algebras on Pos.
Of course, it is possible to translate Σ-algebras for non-discrete signatures Σ as varieties of algebras for discrete ones (see Example 3.17 (7)). Using the result of Kelly and Power, such a translation would lead to a correspondence between finitary monads and varieties. This paper can be viewed as a detailed realization of this.

Varieties of Ordered Algebras
Recall that Pos f is a fixed set of finite posets that represent all finite posets up to isomorphism. If Γ ∈ Pos f has the underlying set {x 0 , . . . , x n−1 }, then we call the x i the variables of Γ. Recall that all monotone functions from A to B form a set Pos 0 (A, B) and a poset Pos(A, B) with the pointwise order.
Notation 3.1. The category Pos is cartesian closed, with hom-objects Pos(X, Y ) given by all monotone functions X → Y , ordered pointwise. That is, given monotone functions We denote by |X| the underlying set of a poset X. We also often consider |X| to be the discrete poset on that set.
Notation 3.3. We denote by Alg Σ the category of Σ-algebras. Its morphisms A → B are the homomorphisms in the expected sense; i.e. they are monotone functions h : A → B such that for every context Γ and every operation symbol σ ∈ Σ Γ , the square commutes. Similarly, we have the category Alg c Σ of all coherent Σ-algebras. For their homomorphisms we have the commutative squares Example 3.4. Let Σ be the signature given by where ¾ is a 2-chain and ½ is a singleton. A Σ-algebra consists of a poset A with a (not necessarily monotone) unary operation @ A and a partial binary operation + A whose definition domain is formed by all comparable pairs. Moreover, A is coherent iff both @ A and + A are monotone, the latter in the sense that a Similarly to the more general signatures discussed in Section 2, signatures Σ in our present sense can be represented as polynomial functors H Σ (for Σ-algebras) and K Σ (for coherent Σ-algebras), respectively, introduced next. These functors arise by specializing the corresponding instances of the polynomial functor P Σ according to Observation 2.4 to discrete signatures.
respectively, where we regard the sets Σ Γ and Pos 0 (Γ, X) as discrete posets. Thus, the elements of both H Σ X and K Σ X are pairs (σ, f ) where σ is an operation symbol of arity Γ and f : Γ → X is monotone. The action on monotone maps h : X → Y is then the same for both functors: Conversely, every H Σ -algebra α : H Σ A → A can be viewed as a Σ-algebra, putting σ A (f ) = α(σ, f ). More conceptually, we have bijective correspondences between the following (families of) maps: Thus, Alg Σ is isomorphic to the category Alg H Σ of algebras for H Σ whose morphisms from (A, α) to (B, β) are those monotone maps h : A → B for which the square below commutes: Indeed, this is equivalent to h being a homomorphism of Σ-algebras. Shortly, Moreover, this isomorphism is concrete, i.e. it preserves the underlying posets (and monotone maps). That is, if U : Alg Σ → Pos andŪ : Alg H Σ → Pos denote the forgetful functors, the above isomorphism I : Alg Σ → Alg H Σ makes the following triangle commutative: (2) Similarly, every coherent Σ-algebra defines an algebra for K Σ , and conversely. Indeed, giving an algebra structure α : Equivalently, we have for every σ of arity Γ a monotone map σ A : This leads to an isomorphism Alg c Σ ∼ = Alg K Σ , which is concrete: where I c , U c andŪ c denote the isomorphism and the forgetful functors, respectively. where embeddings are maps m : A → B such that for all a, a ′ ∈ A we have a ≤ a ′ iff m(a) ≤ m(a ′ ). That is, embeddings are order-reflecting monotone functions. Given an ω-chain of embeddings in Pos, its colimit is simply their union (with inclusion maps as the colimit cocone).
Proposition 3.8. Every poset X generates a free Σ-algebra T Σ X. Its underlying poset is the union of the following ω-chain of embeddings in Pos: where w 0 is the right-hand coproduct injection X → H Σ X + X and w n+1 = Hw n + id X : Proof. Observe first that the polynomial functor H Σ can be rewritten, up to natural isomorphism, as because every Σ Γ is discrete. It follows that H Σ is finitary, being a coproduct of functors Pos 0 (Γ, −) (each Pos 0 (Γ, −) is finitary because Γ is finite). It follows that the free H Σalgebra over X is the colimit of the ω-chain (W n ) from (3.1) in Pos, where W 0 = X and W n+1 = H Σ W n + X with connecting maps w n as described [1]. The desired result thus follows from the concrete isomorphism Alg Σ ∼ = Alg H Σ .
A similar result can be proved for coherent Σ-algebras and the associated functor K Σ , using the fact that like Pos 0 (Γ, −), also the internal hom-functor Pos(Γ, −) is finitary: Every poset X generates a free coherent Σ-algebra T c Σ X. Its underlying poset is the union of the following ω-chain of embeddings in Pos: The universal morphism η X : X → T c Σ X is the inclusion of W 0 into the union. Definition 3.10. We define terms as usual in universal algebra, ignoring the order structure of arities; we write T (Γ) for the set of Σ-terms in variables from Γ. Explicitly, the set T (Γ) of terms is the least set containing |Γ| such that given an operation σ with arity ∆ and a function f : |∆| → T (Γ), we obtain a term σ(f ) ∈ T (Γ).
We denote by u Γ : Γ → T (Γ) the inclusion map. We will often silently assume that the elements of |∆| are listed in some fixed sequence x 1 , . . . , x n , and then write σ(t 1 , . . . , t n ) in lieu of σ(f ) where f (x i ) = t i for i = 1, . . . , n. In particular, in examples we will normally use arities ∆ with |∆| = {1, . . . , k} for some k, and then assume the elements of ∆ to be listed in the sequence 1, . . . , k. We will often abbreviate (t 1 , . . . , t n ) as (t i ), in particular writing σ(t i ) in lieu of σ(t 1 , . . . , t n ). Every σ ∈ Σ Γ yields the term σ(u Γ ) ∈ T (Γ), which by abuse of notation we will occasionally write as just σ.
Example 3.11. Let Σ be a signature with a single operations symbol σ whose arity is a 2-chain. Then T (Γ) is the set of usual terms for a binary operation on the variables from Γ. Whereas T Σ Γ contains only those terms which are variables or have the form σ(t, t) for terms t or σ(x, y) for x ≤ y in Γ. The order of T Σ Γ is such that the only comparable distinct terms are the variables. (2) f # (σ(g)) is defined for σ ∈ Σ ∆ and g : Example 3.13. (1) For the signature in Example 3.4, we have terms in T {x, y} such as @x, y + @y, etc. Given a Σ-algebra A and an interpretation f : {x, y} → A (say, with {x, y} ordered discretely), we see that @x is always interpreted as f # (@x) = @ A (f (x)), whereas f # (y + @x) is defined if and only if f (y) ≤ @ A (f (x)), and then f # (y + @x) = f (y) + A @ A (f (x)).
(2) Every operation symbol σ ∈ Σ Γ considered as a term (see Definition 3.10) satisfies Definition 3.14. An inequation in context Γ is a pair (s, t) of terms in T (Γ), written in the form Γ ⊢ s ≤ t.  An algebra A satisfies this inequation iff a ≤ @ A (a) holds for every a ∈ A. In such algebras, the interpretation of the term x + @x is defined everywhere. As a slightly more advanced example, consider the inequality (in the same signature) According reading of inequalities as per Definition 3.14, this inequality implies that x + @x is always defined, which amounts precisely to (3.2).
Definition 3.16. A variety of Σ-algebras is a full subcategory of Alg Σ specified by a set E of inequations in context. We denote it by Alg(Σ, E). Analogously, a variety of coherent Σ-algebras is a full subcategory of Alg c Σ specified by a set of inequations in context.
Example 3.17. We present some varieties of algebras.
(1) We have seen a variety V specified by (3.2) in Example 3.15.
(2) The subvariety of all coherent algebras in V can be specified as follows. Consider the contexts Γ 1 and Γ 2 given by x and the inequations Γ 1 ⊢ @x ≤ @y and It is clear that Σ-algebras satisfying (3.2) and (3.3) form precisely the full subcategory of V consisting of coherent algebras.
(3) In general, all coherent Σ-algebras form a variety of Σ-algebras. For every context Γ, form the contextΓ with variables x and x ′ for every variable x of Γ, where the order is the least one such that the functions e, e ′ : Γ →Γ given by e(x) = x and e ′ (x) = x ′ are embeddings such that e ≤ e ′ . For every Γ and every σ ∈ Σ Γ consider the following inequation in contextΓ:Γ ⊢ σ(e) ≤ σ(e ′ ).
It is satisfied by precisely those Σ-algebras A for which σ A is monotone.
(4) Recall that an internal semilattice in a category with finite products is an object A together with morphisms + : A × A → A and 0 : 1 → A such that (a) 0 is a unit for +, i.e. the following triangles commute (b) + is associative, commutative, and idempotent: Here swap = π r , π ℓ : A × A → A × A is the canonical isomorphism commuting product components, and ∆ = id, id : A → A × A is the diagonal.
Internal semilattices in Pos form a variety of coherent Σ-algebras. To see this, consider the signature Σ with Σ 2 = {+} and Σ ∅ = {0}, where 2 denotes the two-element discrete poset. The set E is formed by (in)equations specifying that + is monotone, associative, commutative, and idempotent with unit 0. Note that this does not imply that x + y is the join of x, y in X w.r.t. its given order (cf. Example 3.27).
(5) A related variety is that of classical join-semilattices (with 0). To specify those, we take the signature Σ from the previous item; but now we need just two inequations in context specifying that 0 and + are the least element and the join operation, respectively: It then follows that + is monotone, associative, commutative and idempotent, whence these equations need not be contained in E.
(6) Bounded joins: Take the signature Σ consisting of a unary operation ⊥ and an operation j (bounded join) of arity {0, 1, 2} where 0 ≤ 2 and 1 ≤ 2 (but 0 ≤ 1). We then define a variety V by inequations in context That is, j(x, y, z) is the join of elements x, y having a joint upper bound z. It follows that the value of j(x, y, z), when it is defined, does not actually depend on z, which instead just serves as a witness for boundedness of {x, y}. The operation ⊥ and its inequality specify that algebras are either empty or have a least element, i.e. the empty set has a join provided that it is bounded. Thus, V consists of the partial orders having all bounded finite joins, which we will refer to as bounded-join semilattices, and morphisms in V are monotone maps that preserve all existing finite joins.
Remark 3.18. We will now discuss limits and directed colimits in Alg Σ.
(1) It is easy to see that for every endofunctor H on Pos the category Alg H of algebras for H is complete. Indeed, the forgetful functor V : Alg H → Pos creates limits. This means that for every diagram D : D → Alg H with V D having a limit cone (ℓ d : L → V Dd) d∈obj(D) , there exists a unique algebra structure α : HL → L making each ℓ d a homomorphism in Alg H. Moreover, the cone (ℓ d ) is a limit of D.
(2) Analogously, it is easy to see that for every finitary endofunctor H of Pos the category Alg H has filtered colimits created by V .
(3) We conclude from Alg Σ ∼ = Alg H Σ that limits and filtered colimits of Σ-algebras exist and are created by the forgetful functor into Pos, and similarly for Alg c Σ.
(2) if h(f # (t)) is defined and h is an embedding, then f # (t) is defined, too. Proof.
(1) We proceed by induction on the structure of t. If t is a variable, then the claim is immediate from the definition of (−) # . For the inductive step, let t ∈ T (Γ) be a term of the form t = σ(t 1 , . . . , t n ) such that f # (t) defined, where σ ∈ Σ ∆ and |∆| = n. Then, by definition of (−) # , it follows that f # (t i ) is defined for all i ≤ n and Combining this with our assumption that h : A → B is a homomorphism, we obtain that Moreover, since f # (t i ) is defined for all i ≤ n, the inductive hypothesis implies that ) is defined and equal to h · f # (σ(t 1 , . . . , t n )), as desired.
(2) Suppose now that h is an embedding. We use a similar inductive proof. In the inductive step suppose that (h · f ) # (t) is defined. Then by the definition of (−) # , it holds for all i ≤ j in ∆. By induction we know that all f # (t i ) are defined and by item (1) that Proof. Let V be a variety of Σ-algebras. Let D : D → Alg Σ be a filtered diagram having colimit c d : Dd → A (d ∈ obj D). It suffices to show that every inequation in context Γ ⊢ s ≤ t satisfied by every algebra Dd is also satisfied by A. Let f : Γ → A be a monotone interpretation. Since Γ is finite, f factorizes, for some d ∈ obj D, through c d via a monotone mapf : Γ → Dd: in symbols, c d ·f = f . Since Dd satisfies the given inequation in context, we know thatf # (s) andf # (t) are defined and thatf # (s) ≤f # (t) in Dd. By Lemma 3. 19 we conclude that are defined. Using the monotonicity of c d we obtain as desired. Indeed, the forgetful functor of a variety V is a composite of the inclusion V ֒→ Alg Σ and the forgetful functor of Alg Σ, which both create filtered colimits. (1) Alg(Σ, E) is closed under products in Alg Σ. Indeed, given A = i∈I A i with projections π i : A → A i and a monotone interpretation f : Γ → A, we prove for every term s ∈ T (Γ) that f # (s) is defined if and only if so is (π i · f ) # (s) for all i ∈ I. This is done by structural induction: for s ∈ |Γ| there is nothing to prove. Suppose that Equivalently (since the π i are monotone and jointly order-reflecting, i.e. for every We now prove that A satisfies every inequation Γ ⊢ s ≤ t in E, as claimed. Let f : Γ → A be a monotone interpretation. We have that (π i · f # )(s) and (π i · f # )(t) are defined and π i · f # (s) ≤ π i · f # (t) for all i ∈ I, using Lemma 3.19 and since all A i satisfy the given inequation in context. Using again that the π i are jointly order-reflecting, we obtain f # (s) ≤ f # (t), as required.
(2) Alg(Σ, E) is closed under subalgebras in Alg Σ. Indeed, let m : B ֒→ A be a Σhomomorphism carried by an embedding. For every inequation Γ ⊢ s ≤ t in E we prove that B satisfies it. For a monotone interpretation f : Γ → B, we see that (m · f ) # (s) and (m · f ) # (t) are defined and (m · f ) # (s) ≤ (m · f ) # (t) since A satisfies the given inequation in context. By Lemma 3. 19 we obtain that f # (s) and f # (t) are defined and Since m is an embedding, it follows that f # (s) ≤ f # (t).  (1) The functor U has a left adjoint because it is the composite of the embedding E : V → Alg Σ and the forgetful functor V : Alg Σ → Pos: the functor E has a left adjoint by Proposition 3.22 and V has one by Proposition 3.8.
(2) Let f, g : A → B be a U -split pair of homomorphisms in V. That is, there are monotone maps c, i, j as in the following diagram For every σ ∈ Σ Γ , there exists a unique operation σ C : Pos 0 (Γ, C) → C making c a homomorphism: Then c is a homomorphism since σ C (c · k) = c · σ B (k) for every k : Γ → B: Conversely, if C has an algebra structure making c a homomorphism, then the above formula holds since c · i = id: Furthermore, C lies in V. To verify this, we just prove that whenever an inequation Γ ⊢ s ≤ t is satisfied by B, then the same holds for the algebra C. Given a monotone interpretation h : Γ → C such that h # (s) and h # (t) are defined, we prove h # (s) ≤ h # (t).
For the monotone interpretation i · h : Γ → B we have that (i · h) # (s) and (i · h) # (t) are defined and that (i·h) # (s) ≤ (i·h) # (t) since B lies in V. Since c is a homomorphism, we conclude using Lemma 3.19 and that c · i = id C that is defined and similarly for h # (t). Then we have as desired using the monotonicty of c.
Finally, we prove that c is a coequalizer of f and g in V. Let d : Moreover, d ′ : C → D is a homomorphism since c is a surjective homomorphism such that d ′ · c = d is also a homomorphism. This clearly is the unique homomorphic factorization of d through c.  (4). It is well known (and easy to show) that the free internal semilattice on a poset X is formed by the poset C ω X of its finitely generated convex subsets.
Here, a subset S ⊆ X is convex if x, y ∈ S implies that every z such that x ≤ z ≤ y lies in S, too, and finitely generated means that S is the convex hull of a finite subset of X.
The order on C ω X is the Egli-Milner order, which means that for S, T ∈ C ω X we have S ≤ T iff ∀s ∈ S. ∃t ∈ B. s ≤ t ∧ ∀t ∈ T. ∃s ∈ S. s ≤ t.
The constant 0 is the empty set, and the operation + is the join w.r.t. inclusion, explicity, S+T is the convex hull of S∪T for all S, T ∈ C ω X. One readily shows that + is monotone w.r.t. the Egli-Milner order and that C ω X with the universal monotone map x → {x} is a free internal semilattice on X. Thus we see that C ω is a monad on Pos and Pos Cω is (isomorphic to) the category of internal semilattices in Pos.
(2) Denote by D ω the monad of free join semilattices. It assigns to every poset X the set of finitely generated, downwards closed subsets of X ordered by inclusion. Here a downwards closed subset S ⊆ X is finitely generated if there are x 1 , . . . , x n ∈ S, n ∈ N, such that S = n i=1 x i ↓. The category Pos Dω is equivalent to that of join-semilattices, see Example 3.17 (5).
(3) Similarly, the monad D b ω generated by the variety of bounded-join semilattices (Example 3.17(6)) assigns to a poset X the set of finitely generated downwards closed bounded subsets of X, ordered by inclusion.
It then follows that the evaluation map f # : T (Γ) → |T X| coincides with f * on operation symbols (converted to terms as per Definition 3.10): for all σ ∈ T Γ. Indeed, for |Γ| = {x 1 , . . . , x n } we have It now follows that the Σ-algebra T X lies in V T . It satisfies the inequations of type (1) because f * is monotone: Moreover, it satisfies the inequations of type (2) since for every monotone map k : ∆ → T Γ we know that f # (k * (σ)) is defined by Example 3.13 (2), and we have Proof. (1) We first prove that the algebra T X of Example 4.3 is a free algebra of V T w.r.t. the monad unit η X : X → T X.
Since σ = k(x i ) this gives the desiredf · k when we let x range over ∆. As for uniqueness, suppose thatf : T Γ → A is a homomorphism such that f =f · η Γ . The above square commutes for ∆ = Γ which applied to η Γ ∈ Pos(Γ, T Γ) yields for every σ ∈ |T Γ|:f as required.
(1b) Now, let X be an arbitrary poset. Express it as a filtered colimit X = colim i∈I Γ i of contexts. The free algebra on X is then a filtered colimit of the corresponding diagram of the Σ-algebras T Γ i (i ∈ I). Indeed, that T X = colim T Γ i in Pos follows from T preserving filtered colimits. That this colimit lifts to V follows from the forgetful functor of V creating filtered colimits, see Proposition 3.20.
(2) To conclude the proof, we apply Remark 4.1. Our given monad and the monad T V of the associated variety share the same object assignment X → T X = T V X for an arbitrary poset X, and the same universal map η X , as shown in part (1). It remains to prove that for every morphism f : X → T Y in Pos the homomorphism h * = µ Y · T h extending h in Pos T is a Σ-homomorphism h * : T X → T Y of the corresponding Σalgebras of Example 4.3. Then T and T V also share the operator h → h * . Thus given σ ∈ Σ Γ we are to prove that the following square commutes: Proof. For the first claim, let T be enriched. Then the Σ-algebra T X of Example 4.3 is coherent: Given an operation symbol σ ∈ Σ Γ and monotone interpretations f ≤ g in Pos(Γ, T X), we have T f ≤ T g, and hence f * = µ T X · T f ≤ µ T X · T g = g * because T is enriched. Therefore, f * (σ) ≤ g * (σ). That is, For every algebra A of the variety V T we have the unique Σ-homomorphism k : T A → A such that k · η A = id A (since T A is a free Σ-algebra in V T ; see Theorem 4.4 (1)). The coherence of T A implies the coherence of A: given f 1 ≤ f 2 in Pos(Γ, A), we verify σ A (f 1 ) ≤ σ A (f 2 ) by applying the commutative square ; by monotonicity of composition in Pos and of σ T A as established above, this implies σ A (f 1 ) ≤ σ A (f 2 ) as desired.
Conversely, let V be a variety of coherent Σ-algebras. Given f 1 ≤ f 2 in Pos(X, Y ), we prove that the free-algebra monad Since for x ∈ X we know that f 1 (x) ≤ f 2 (x), the poset E contains all elements η X (x). Moreover, E is closed under the operations of T V X: Suppose that σ ∈ Σ Γ and that h : Γ → T V X is a monotone map such that h[Γ] ⊆ E; we have to show that σ T V X (h) ∈ E. Applying the commutative square to h, we obtain using in the inequality that σ T V Y is monotone and, by assumption, We thus see that E is a Σ-subalgebra of T V X. Since T V X is the free algebra of V w.r.t. η X and the subalgebra E contains η X [X], it follows that E = T V X. This proves that T f 1 ≤ T f 2 , as desired.

Enriched Lawvere Theories
Power [23] proves that enriched finitary monads on Pos bijectively correspond to Lawvere Pos-theories. This is another way of proving Corollary 4.7. However, we believe that a precise verification of all details would not be simpler than our proof. Here we indicate this alternative proof.
Dual to Remark 2.2, cotensors P ⋔ X in an enriched category T (over Pos) are characterized by an enriched natural isomorphism T (−, P ⋔ X) ∼ = Pos(P, T (−, X)). If we restrict ourselves to finite posets P we speak about finite cotensors.
Definition 5.1 [23]. A Lawvere Pos-theory is a small enriched category T with finite cotensors together with an enriched identity-on-objects functor ι : Pos op f → T which preserves finite cotensors.
Example 5.2. Let V be a variety, and denote by T V its free-algebra monad on Pos. The following theory T V is the restriction of the Kleisli category of T V to Pos f : objects are all contexts, and morphisms from Γ to Γ ′ form the poset Pos(Γ ′ , T V Γ). A composite of f : Γ ′ → T V Γ and g : Γ ′′ → T V Γ ′ is f * · g : Γ ′′ → T V Γ where (−) * is the Kleisli extension (see Remark 4.1(3)). The proof of Theorem 5.3 implies that these are, up to isomorphism, all models of T V and this yields an equivalence between V and Mod T V .
Thus, Corollary 4.7 can be proved by verifying that every Lawvere Pos-theory T is naturally isomorphic to T V for a variety of algebras, and the passage from T to V is inverse to the passage V → T V of Example 5.4.
In addition, Nishizawa and Power [20] generalize the concept of Lawvere theory to a setting in which one may obtain an alternative proof of the non-coherent case (Corollary 4.5); we briefly indicate how. Again we believe that that proof would not be simpler than ours. The setting of op. cit. includes a symmetric monoidal closed category V that is locally finitely presentable in the enriched sense and a locally finitely presentable V-category A . For our purposes, V = Set and A = Pos. Example 5.7. Every variety of (not necessarily coherent) algebras yields a theory T analogous to Example 5.2: the hom-set T (Γ, Γ ′ ) is Pos 0 (Γ ′ , T V Γ).
Remark 5.8. Here, a model of a theory T is an ordinary functor A : T → Set such that A · ι : Pos op f → Set is naturally isomorphic to Pos(−, X)/Pos op f for some poset X. The category Mod T of models has ordinary natural transformations as morphisms.
Theorem 5.9 [20,Cor. 5.2]. There is a bijective correspondence between ordinary finitary monads on Pos an Lawvere Pos-theories in the sense of Definition 5.6.

Conclusion and Future Work
Classical varieties of algebras are well known to correspond to finitary monads on Set. We have investigated the analogous situation for the category of posets. It turns out that there are two reasonable variants: one considers either all (ordinary) finitary monads, or just the enriched ones, whose underlying endofunctor is locally monotone. (An orthogonal restriction, not considered here, is to require the monad to be strongly finitary, which corresponds to requiring the arities of operations to be discrete [2].) We have defined the concept of a variety of ordered algebras using signatures where arities of operation symbols are finite posets. We have proved that these varieties bijectively correspond to (1) all finitary monads on Pos, provided that algebras are not required to have monotone operations, and (2) all enriched finitary monads on Pos for varieties of coherent algbras, i.e. those with monotone operations.
In both cases, 'term' has the usual meaning in universal algebra, and varieties are classes presented by inequations in context.
Although we have concentrated entirely on posets, many features of our paper can clearly be generalized to enriched locally λ-presentable categories and the question of a semantic presentation of (ordinary or enriched) λ-accessible monads. For example, what type of varieties corresponds to countably accessible monads on the category of metric spaces with distances at most one (and nonexpanding maps)? Such varieties will be related to Mardare et al.'s quantitative varieties [17] (aka. c-varieties [18,19]), probably extended by allowing non-discrete arities of operation symbols.
Jiří Rosický (private communication) has suggested another possibility of presenting finitary monads on Pos: by applying the functorial semantics of Linton [14] to functors into Pos and taking the appropriate finitary variation in the case where those functors are finitary. We intend to pursue this idea in future work.