2. Finitary and strongly finitary functors

In the present section, we recall finitary and strongly finitary endofunctors of $\mathsf{Pos}$ . We observe that the latter means that the functor is enriched, finitary and preserves reflexive coinserters.

Notation 2. The full subcategory of $\mathsf{Pos}$ on natural numbers is denoted by $\mathscr{N}$ , and the full embedding by:

\begin{equation*} I: \mathscr{N} \to \mathsf{Pos}. \end{equation*}

Example 3 Coinserters are colimits of the scheme $\mathscr{D}$ given by a parallel pair:

weighted by $\varphi: \mathscr{D}^{\mathit{op}} \to \mathsf{Pos}$ defined as follows:

Thus, a diagram in $\mathsf{Pos}$ is a parallel pair:

\begin{equation*} f_0,f_1: A \to B \end{equation*}

of monotone maps (considered as an ordered pair $(f_0,f_1)$ , of course). And the coinserter is a morphism $c: B \to C$ universal w.r.t. the inequality $c \cdot f_0 \leq c \cdot f_1$ :

That is,

1. (1) for every morphism $u: B \to D$ with $u \cdot f_0 \leq u \cdot f_1$ , there exists a unique morphism $v: C \to D$ with $u = v \cdot c$ , and

2. (2) the map $u \mapsto v$ is monotone: given $u' = v' \cdot c$ , then $u \leq u'$ implies $v \leq v'$ .

Remark 4. Every finite poset P is a canonical coinserter of a parallel pair:

of morphisms in $\mathscr{N}$ . Let n be the number of elements of P and k, the number of comparable pairs in P. Thus, we can assume that P has elements $0, \dots, n-1$ , and we can index all comparable pairs as follows:

\begin{equation*} {p_0(t)} \leq {p_1(t)} \text{ for } t = 0, \dots, k-1. \end{equation*}

This defines functions $p_0, p_1: k \to n$ . The coinserter of this pair is carried by the identity map:

Notation 5. Denote by

\begin{equation*} J: \mathsf{Pos}_\mathsf{f} \hookrightarrow \mathsf{Pos} \end{equation*}

a full embedding of a subcategory $\mathsf{Pos}_\mathsf{f}$ representing all finite posets up to isomorphism. We choose $\mathsf{Pos}_\mathsf{f}$ so that it contains $\mathscr{N}$ as a full subcategory.

We now turn to strongly finitary functors.

Definition 7. (Kelly and Lack 1993) An endofunctor H of $\mathsf{Pos}$ is called strongly finitary if it is the left Kan extension of its restriction to $\mathscr{N}$ . More precisely:

\begin{equation*} H = {\mathrm{Lan}}_{{I}} {H \cdot I}. \end{equation*}

In our enriched setting, sifted colimits are introduced analogously. A weight $\varphi: \mathscr{D}^{\mathit{op}} \to \mathsf{Pos}$ is called sifted if the colimit functor ${{{\varphi}}\mathop{*}{{{-}}}}: [\mathscr{D},\mathsf{Pos}] \to \mathsf{Pos}$ preserves finite (conical) products. Sifted colimits then are colimits weighted by sifted weights.

We now verify that the weight $\overline{\varphi}$ is sifted. In other words, reflexive coinserters commute with (conical) finite products in $\mathsf{Pos}$ . The proof is analogous to the proof that reflexive coequalizers commute with finite products in $\mathsf{Set}$ ((Adámek et al. Reference Adámek, Rosicky and Vitale2010, Example 1.2)).

Proof.

1. (1) We first describe the coinserter of a reflexive pair $u_0, u_1: X \to Y$ in $\mathsf{Pos}$ . We define a preorder $\sqsubseteq$ on Y for which the posetal reflection $e: (Y,\sqsubseteq) \to (Y,\sqsubseteq)/{\sim}$ , where $y \sim \overline{y}$ iff $y \sqsubseteq \overline{y} \sqsubseteq y$ , is the desired coinserter. Put $y \sqsubseteq \overline{y}$ iff there exist $x_0,\dots,x_n \in X$ such that $y = u_0(x_0)$ , $\overline{y} = u_1(x_n)$ and

   \begin{equation*} u_1(x_i) \leq u_0(x_{i+1}) \qquad \text{ for $i=0,\dots,n-1$.} \end{equation*}
   Having a monotone map $d: Y \to X$ with $u_0 \cdot d = {\mathit{id}}_Y = u_1 \cdot d$ , we see that $\sqsubseteq$ is a preorder whose posetal reflection is the coinserter of $u_0$ and $u_1$ . Moreover, given a sequence $x_0,\dots,x_n$ for y and $\overline{y}$ as above, we can extend it for every $m > n$ to a corresponding sequence $x_0,\dots,x_m$ : put $x_i = d(\overline{y})$ for $i = n+1,\dots,m$ .

2. (2) For every other reflexive pair $u_0', u_1': X' \to Y'$ with the corresponding coinserter $e': (Y',~\sqsubseteq'~) \to (Y', \sqsubseteq')/{\approx}$ , it is our task to prove that we have the following coinserter:

   
   Indeed, the pair $u_0 \times u_0'$ and $u_1 \times u_1'$ is clearly reflexive, thus its coinserter is described as in (1) above. Let $\preceq$ be the resulting preorder on $Y \times Y'$ and let elements (y,y’) and $(\overline{y},\overline{y}')$ be given. Whenever a sequence $(x_0,x_0'),\dots,(x_n,x_n')$ in $X \times X'$ demonstrates that $(y,y') \preceq (\overline{y},\overline{y}')$ , then we have $y \sqsubseteq \overline{y}$ and $y' \sqsubseteq \overline{y}'$ : use the sequences in X and X’, respectively, obtained by the projection of $(x_0,x_0'),\dots, (x_n,x_n')$ . It remains to prove the converse: from $y \sqsubseteq \overline{y}$ and $y' \sqsubseteq \overline{y}'$ , we derive $(y,y') \preceq (\overline{y},\overline{y}')$ . Here, we use that the sequence in (1) can be prolonged to any larger length. Thus, for $y \sqsubseteq \overline{y}$ , we have a sequence $x_0,\dots,x_n$ in X as in (1), and for $y' \sqsubseteq \overline{y}'$ , we have a sequence of the same length $x_0',\dots,x_n'$ in X’. But then, $(x_0,x_0'),\dots,(x_n,x_n')$ is the desired sequence in $X \times X'$ demonstrating that $(y,y') \preceq (\overline{y},\overline{y}')$ .

1. (1) H is strongly finitary,

2. (2) H preserves sifted colimits,

3. (3) H is finitary and preserves reflexive coinserters.

Proof. Every poset is a filtered colimit of its finite subposets, each of which is a coinserter as in Remark 4.

Consequently, starting with the subcategory $\mathscr{N}$ , we obtain all of $\mathsf{Pos}$ by reflexive coinserters and filtered colimits. Moreover, every functor $\mathsf{Pos}(In,{-})$ preserves these colimits (see Remark 6 and Proposition 10). In the terminology of (Kelly 2005, Theorem 5.29), this states that the embedding $I: \mathscr{N}\to \mathsf{Pos}$ has a codensity presentation formed by filtered colimits and reflexive coinserters. By that theorem, properties (1)–(3) are equivalent.

Remark 15. Every strongly finitary endofunctor H of $\mathsf{Pos}$ generates a free monad whose underlying functor $\widehat{H}$ is also strongly finitary. Indeed, following Trnkova et al. (1975), $\widehat{H}$ is a conical colimit in $[\mathsf{Pos},\mathsf{Pos}]$ of the following $\omega$ -chain:

That is, the chain $W: \omega \to [\mathsf{Pos},\mathsf{Pos}]$ has objects:

\begin{equation*} W_0 = {\mathit{Id}} \text{ and } W_{n+1} = HW_n + {\mathit{Id}} \end{equation*}

and morphisms:

\begin{equation*} w_0: {\mathit{Id}} \to H + {\mathit{Id}} \text{ the coproduct injection} \end{equation*}

and

\begin{equation*} w_{n+1} = Hw_n + {\mathit{id}}. \end{equation*}

Thus, if H is strongly finitary, so is each $W_n$ (by the preceding example). Consequently, $\widehat{H} = \mathop{\mathrm{colim}}\limits W_n$ is strongly finitary.

Notation 16. A monad whose endofunctor is strongly finitary is called a strongly finitary monad. We denote by:

\begin{equation*} \mathsf{Mnd_{sf}}(\mathsf{Pos}) \end{equation*}

the ordinary category of strongly finitary monads and monad morphisms.

3. From Varieties of Ordered Algebras to Strongly Finitary Monads

In the current section, we prove that the free-algebra monad of every variety is strongly finitary, and that it has a coinserter presentation via free monads on signatures.

Notation 19. Let $\Sigma$ be a signature, that is, a collection of sets $\Sigma_n$ (of n-ary operation symbols) indexed by $n \in \mathbb{N}$ . An ordered $\Sigma$ -algebra is a poset A together with a monotone map $\sigma_A: A^n \to A$ for every $n \in \mathbb{N}$ and $\sigma \in \Sigma_n$ . The category of ordered $\Sigma$ -algebras and homomorphisms (i.e., monotone functions preserving the given operations) is denoted by $\mathsf{Alg}(\Sigma)$ .

By a variety of ordered $\Sigma$ -algebras, we understand a full subcategory of $\mathsf{Alg}(\Sigma)$ specified by a set $\mathscr{E}$ of inequations. That is, a $\Sigma$ -algebra belongs to the variety iff it satisfies every inequation from $\mathscr{E}$ .

1. (1) Every variety is ordinarily reflective in $\mathsf{Alg}(\Sigma)$ . Indeed, since $H_\Sigma$ is a finitary endofunctor on a locally finitely presentable category, $\mathsf{Alg}(\Sigma) \cong H_\Sigma\text{-}\mathsf{Alg}$ is also locally finitely presentable (Adámek and Rosický Reference Adámek and Rosický1994, Remark 2.78). In particular, it is complete and cowellpowered. The factorization system (epi, embedding) on $\mathsf{Pos}$ lifts, since $H_\Sigma$ preserves epimorphisms, to $\mathsf{Alg}(\Sigma)$ . Since a variety $\mathcal{V}$ is easily seen to be closed under products and subalgebras carried by embeddings, the surjective reflections follow (Adámek et al. Reference Adámek, Herrlich and Strecker1990, Theorem 16.8).

2. (2) Since $\mathcal{V}$ is closed in $\mathsf{Alg}(\Sigma)$ under powers of the poset $\mathbf{2} = \{ 0 \leq 1 \}$ , it follows from Kelly (2005) (Section 1.11) that the above ordinary reflection is in fact enriched.

Construction 25. (See Reference Bloom Bloom 1976 ) For every variety $\mathcal{V}$ of ordered algebras, the free algebra $T_\mathcal{V} X$ of $\mathcal{V}$ on a poset X can be constructed as follows.

For every poset X, let $\mathcal{E}_X$ be the collection of all inequations $s \leq t$ satisfied by all algebras of $\mathcal{V}$ , where $s,t \in T_\Sigma X$ are terms in variables from $|X|$ . Then $\mathcal{E}_X$ is a preorder, that is, a reflexive and transitive relation on the underlying set $|T_\Sigma X|$ of $T_\Sigma X$ . Moreover, it is admissible: given an n-ary symbol $\sigma \in \Sigma$ and n pairs $s_i \leq t_i$ ( $i<n$ ) in $\mathcal{E}_X$ , it follows that the pair $\sigma(s_i) \leq \sigma(t_i)$ also lies in $\mathcal{E}_X$ . Indeed, given an algebra $(A,\alpha) \in \mathcal{V}$ and an interpretation $f: X \to A$ , we know that the homomorphism $f^\sharp: T_\Sigma X \to A$ fulfills $f^\sharp (s_i) \leq f^\sharp (t_i)$ for all i, thus

\begin{equation*} f^\sharp (\sigma(s_i)) = \sigma_{A}(f^\sharp(s_i)) \leq \sigma_{A}(f^\sharp(t_i)) = f^\sharp(\sigma(t_i)). \end{equation*}

Consequently, for the induced equivalence relation:

\begin{equation*} \mathcal{E}_X^o = \mathcal{E}_X \cap \mathcal{E}_X^{-1} \end{equation*}

we obtain a $\Sigma$ -algebra $T_\mathcal{V} X$ on the quotient set:

\begin{equation*} |T_\mathcal{V} X| = |T_\Sigma X|/\mathcal{E}_X^o \end{equation*}

(of all equivalence classes [t] of terms $t \in T_\Sigma X$ ). The operations are as expected:

\begin{equation*} \sigma_{T_\mathcal{V} X} ([t_0], \dots, [t_{n-1}]) = [\sigma(t_0,\dots,t_{n-1})] \end{equation*}

for every n-ary $\sigma$ and all n-tuples $t_0,\dots,t_{n-1} \in T_\Sigma X$ . Finally, we consider $T_\mathcal{V} X$ as a poset via

\begin{equation*} [s] \leq [t] \text{ iff } (s,t) \in \mathcal{E}_X. \end{equation*}

The following theorem was stated by (Bloom Reference Bloom1976, Theorem 2.2). We present a full proof since we need it later, and the original proof was only a sketch.

Proof.

1. (1) $T_\mathcal{V} X$ is a well-defined ordered $\Sigma$ -algebra. This follows easily from the fact that $\mathcal{E}_X$ is an admissible preorder.

2. (2) $\mathcal{V}$ has a free algebra on X which is given by an admissible preorder $\sqsubseteq$ on $T_\Sigma X$ (i.e., for the induced equivalence relation $\sim$ , the underlying poset is $|T_\mathcal{V} X| = |T_\Sigma X|/\sim$ and the operations are induced by those of $T_\Sigma X$ ). This statement follows from Remark 24, which implies that a free algebra $T_\mathcal{V} X$ exists, and the unique homomorphism:

   \begin{equation*} e_X: T_\Sigma X \to T_\mathcal{V} X \end{equation*}
   extending the universal arrow is epic. Indeed, the desired preorder is simply
   \begin{equation*} s \sqsubseteq t \text{ iff } e_X(s) \leq e_X(t). \end{equation*}

3. (3) The preorder $\mathcal{E}_X$ of the above construction coincides with $\sqsubseteq$ of (2). Indeed, if $(s,t) \in \mathcal{E}_X$ , then the algebra $T_\mathcal{V} X$ satisfies $s \leq t$ (since it lies in $\mathcal{V}$ ) and taking the universal map $(\eta_\mathcal{V})_X: X \to T_\mathcal{V} X$ as the interpretation, we have

   \begin{equation*} e_X = (\eta_\mathcal{V})_X^\sharp \end{equation*}
   (because $e_X$ is a $\Sigma$ -homomorphism). Since $e_X(s) \leq e_X(t)$ , we conclude that $s \sqsubseteq t$ . Conversely, if $s \sqsubseteq t$ , which means $e_X(s) \leq e_X(t)$ , we verify that every algebra $A \in \mathcal{V}$ satisfies $s \leq t$ . Let $f: X \to A$ be an interpretation, then the corresponding homomorphism $f^\sharp: T_\Sigma X \to A$ factorizes through the reflection of $T_\Sigma X$ in $\mathcal{V}$ in $\mathsf{Alg}(\Sigma)$ :
   
   Since h is monotone, the inequality $e_X(s) \leq e_X(t)$ implies $f^\sharp(s) \leq f^\sharp(t)$ , as required.

Notation 27. For every variety $\mathcal{V}$ of $\Sigma$ -algebras, we denote by

\begin{equation*} c_\mathcal{V}: \mathbf{T}_\Sigma \to \mathbf{T}_\mathcal{V} \end{equation*}

the monad morphism whose components are the canonical quotient maps:

\begin{equation*} |T_\Sigma X| \to |T_\Sigma X|/\mathcal{E}_X^o. \end{equation*}

Proof. For classical varieties, see Mac Lane (1998), Theorem VI.8.1. The proof for varieties of ordered algebras can be used verbatim (with the equation $\lambda_B = \mu_B$ being replaced by $\lambda_B \leq \mu_B$ ).

1. (1) The continuation monad $\langle {A,A} \rangle$ on $\mathsf{Pos}$ associated with a poset A assigns to a poset X the power of A to the set $\mathsf{Pos}_o (X,A)$ of all monotone maps $f: X \to A$ :

   \begin{equation*} \langle {A,A} \rangle X = \prod_{\mathsf{Pos}_o (X,A)} A. \end{equation*}
   This is the monad resulting from the adjunction given by the right adjoint $\mathsf{Pos}({-},A): \mathsf{Pos} \to \mathsf{Pos}^{\mathit{op}}$ . Denote by $\pi_f: \langle {A,A} \rangle X \to A$ the projection corresponding to $f: X \to A$ . To every morphism $h: X \to Y$ , the monad assigns the morphism $\langle {A,A} \rangle h$ determined by the following commutative triangles:
   
   The unit is $\langle f \rangle_{f \in \mathsf{Pos}_o (X,A)}: X \to \langle {A,A} \rangle X$ , and the multiplication $\mu_X$ is determined by the following commutative triangles:
   

2. (2) It follows from Dubuc (Reference Dubuc1970) that for every monad $\mathbf{T}$ and every poset A, there is a bijection between monad morphisms $\mathbf{T} \to \langle {A,A} \rangle$ and algebras of $\mathsf{Pos}^\mathbf{T}$ on A. This bijection assigns to an algebra $\alpha: TA \to A$ the monad morphism:

   \begin{equation*} \widehat{\alpha}: \mathbf{T} \to \langle {A,A} \rangle \end{equation*}
   with components $\widehat{\alpha}_X$ determined by the following commutative squares:
   
   Thus, if $\mathbf{T} = \mathbf{T}_\Sigma$ , then $\widehat{\alpha}_X$ assigns to a term $t \in T_\Sigma X$ the tuple $(f^\sharp(t))_{f:X \to A}$ .

3. (3) Let $b: \mathbf{S} \to \mathbf{T}$ be a monad morphism. Every algebra $(A,\alpha)$ in $\mathsf{Pos}^\mathbf{T}$ then yields an algebra $(A, \alpha \cdot b_A)$ in $\mathsf{Pos}^\mathbf{S}$ . The following triangle:

   
   commutes. Indeed, for every poset X and every $f \in \mathsf{Pos}_o (X,A)$ , we have
   \begin{equation*} \pi_f(\widehat{\alpha}_X \cdot b_X) = \alpha \cdot Tf \cdot b_X = \alpha \cdot b_A \cdot Sf. \end{equation*}
   The same result is obtained by:
   \begin{equation*} \pi_f ((\widehat{\alpha \cdot b_A})_X) = \alpha \cdot b_A \cdot Sf. \end{equation*}

4. (4) In particular, let $\mathbf{T} = \mathbf{T}_\Sigma$ for a signature $\Sigma$ . Given a term u in $T_\Sigma n$ , it corresponds to a monad morphism:

   \begin{equation*} \widetilde{u}: \mathbf{T}_{\Omega_n} \to \mathbf{T}_\Sigma \end{equation*}
   where $\Omega_n$ is a signature of a single operation $\omega$ of arity n. Its component $\widetilde{u}_X: T_{\Omega_n} X \to T_\Sigma X$ assigns to a term t over X (containing the unique operation symbol $\omega$ ) the $\Sigma$ -term obtained by replacing each $\omega$ by the term u. Thus, if a $\Sigma$ -algebra $(A,\alpha)$ satisfies an inequation $u_0 \leq u_1$ , the inequation $(\widehat{\alpha} \cdot \widetilde{u}_0)_X \leq (\widehat{\alpha} \cdot \widetilde{u}_1)_X$ holds for all posets X. Shortly: $\widehat{\alpha} \cdot \widetilde{u}_0 \leq \widehat{\alpha} \cdot \widetilde{u}_1$ .

Construction 30 We describe the free-algebra monad of the variety given by a single inequation $u_0 \leq u_1$ in signature $\Sigma$ . Let $u_0,u_1$ be terms with variables $x_0,\dots, x_{n-1}$ . For the signature $\Omega_n$ of a single operation of arity n, they can be viewed (via Yoneda lemma) as natural transformations:

\begin{equation*} u_0,u_1: H_{\Omega_n} \to T_\Sigma. \end{equation*}

The corresponding monad morphisms:

\begin{equation*} \widetilde{u}_0,\widetilde{u}_1: \mathbf{T}_{\Omega_n} \to \mathbf{T}_\Sigma. \end{equation*}

have a coinserter in $\mathsf{Mnd_{sf}}(\mathsf{Pos})$ which we denote as follows:

We verify that this is precisely $c_\mathcal{V}$ of Notation 27 for the variety presented by $u_0 \leq u_1$ :

Proof. The variety $\mathcal{V}$ presented by $u_0 \leq u_1$ yields a free-algebra monad $\mathbf{T}_\mathcal{V}$ . The proposition will be proved by verifying that $c_\mathcal{V}$ is a coinserter of $\widetilde{u}_0$ , $\widetilde{u}_1$ in $\mathsf{Mnd_{sf}}(\mathsf{Pos})$ . From the definition of $c_\mathcal{V}$ , we conclude

\begin{equation*} c_\mathcal{V} \cdot \widetilde{u}_0 \leq c_\mathcal{V} \cdot \widetilde{u}_1. \end{equation*}

1. (a) Given a strongly finitary monad $\mathbf{S} = (S,\mu^S,\eta^S)$ and a monad morphism $b: \mathbf{T}_\Sigma \to \mathbf{S}$ with

   \begin{equation*} b \cdot \widetilde{u}_0 \leq b \cdot \widetilde{u}_1, \end{equation*}
   we prove that b factorizes through $c_\mathcal{V}$ via a monad morphism. For every poset X, the free algebra $(SX, \mu^S_X)$ for $\mathbf{S}$ yields, since b is a monad morphism, the following algebra for $\mathbf{T}_\Sigma$ on SX:
   \begin{equation*} \beta_X := T_\Sigma SX \xrightarrow{b_{SX}} SSX \xrightarrow{\mu^S_X} SX \end{equation*}
   From $\alpha_X \cdot (\widetilde{u}_0)_X \leq \alpha_X \cdot (\widetilde{u}_1)_X$ we deduce, using Remark 29 (4), that the $\Sigma$ -algebra $(SX, \beta_X)$ satisfies the inequation $u_0 \leq u_1$ . Since the free algebra $(TX, \mu^T_X)$ of $\mathcal{V}$ on X corresponds to the $\Sigma$ -algebra:
   \begin{equation*} T_\Sigma TX \xrightarrow{(c_\mathcal{V})_{TX}} TTX \xrightarrow{\mu^T_X} TX, \end{equation*}
   we obtain a unique $\Sigma$ -homomorphism $\overline{b}_X$ with $\overline{b}_X \cdot \eta^T_X = \eta^S_X$ :
   
   We verify that these morphisms $\overline{b}_X$ form a monad morphism:
   \begin{equation*} \overline{b}: \mathbf{T} \to \mathbf{S} \text{ with } b = \overline{b} \cdot c_\mathcal{V}. \end{equation*}

2. (1) The equality $b_X = \overline{b}_X \cdot (c_\mathcal{V})_X: T_\Sigma X \to SX$ holds because both sides are homomorphisms of $\Sigma$ -algebras and we have

   \begin{equation*} b_X \cdot \eta^\Sigma_X = \eta^S_X = \overline{b}_X \cdot \eta^T_X = \overline{b}_X \cdot (c_\mathcal{V})_X \cdot \eta^\Sigma_X. \end{equation*}

3. (2) $\overline{b}_X$ is natural in X. In fact, every morphism $f: X \to Y$ yields a $\Sigma$ -homomorphism:

   \begin{equation*} Tf: (TX, \mu^T_X \cdot (c_\mathcal{V})_{TX}) \to (TY, \mu^T_Y \cdot (c_\mathcal{V})_{TY}) \end{equation*}
   Thus, $\overline{b}_Y \cdot Tf$ is also a $\Sigma$ -homomorphism, and so is $Sf \cdot \overline{b}_X: (TX, \mu^T_X \cdot (c_\mathcal{V})_{TX}) \to (SY, \alpha_Y)$ . Since the domain of both composites is a free algebra of $\mathcal{V}$ on X, for proving that they are equal we just need to verify
   \begin{equation*} \overline{b}_Y \cdot Tf \cdot \eta^T_X = Sf \cdot \overline{b}_X \cdot \eta^T_X. \end{equation*}
   See the following diagram:
   

4. (3) The equality $ \overline{b} \cdot \eta^T = \eta^S $ follows from the right-hand triangle in the diagram defining $\overline{b}_X$ above.

5. (4) We finally prove $ \overline{b} \cdot \mu^T = \mu^S \cdot S\overline{b} \cdot \overline{b}T. $ Consider the following diagram:

   
   The outward rectangle is the definition of $\overline{b}_X$ . The left-hand parts commute by (1) and (2). Consequently, the desired right-hand square commutes since it does when precomposed by the epimorphism $(c_\mathcal{V})_{TX}$ .

6. (b) Finally for every monad morphism $b': \mathbf{T}_\Sigma \to \mathbf{S}$ factorized as $b' = \overline{b'} \cdot c_\mathcal{V}$ we are to verify that

   \begin{equation*} b \leq b' \text{ implies } \overline{b} \leq \overline{b'}. \end{equation*}
   This is trivial since the components of $c_\mathcal{V}$ are surjective.

Construction 32. The above proposition immediately generalizes to sets of inequations. For every variety $\mathcal{V}$ of $\Sigma$ -algebras, the free-algebra monad $\mathbf{T}_\mathcal{V}$ is a canonical quotient $c_\mathcal{V}: \mathbf{T}_\Sigma \to \mathbf{T}_\mathcal{V}$ of the free- $\Sigma$ -algebra monad, see Notation 27. We construct monad morphisms $\widetilde{u}_0, \widetilde{u}_1: \mathbf{T}_\Omega \to \mathbf{T}_\Sigma$ for some signature $\Omega$ forming a coinserter in $\mathsf{Mnd_{sf}}(\mathsf{Pos})$ as follows:

Given a set

\begin{equation*} u_0^i \leq u_1^i, \qquad i \in I \end{equation*}

of inequations specifying the variety $\mathcal{V}$ , let $n_i$ be the number of variables on both sides. We define a signature $\Omega = \{ \gamma_i \}_{i \in I}$ , where $\gamma_i$ has arity $n_i$ . By Yoneda lemma, we obtain natural transformations $u_0,u_1: H_{\Omega} \to T_\Sigma$ , since we have $H_\Omega \cong \coprod_{i \in I} \mathsf{Pos}(n_i,{-})$ . Let $\widetilde{u}_0,\widetilde{u}_1: \mathbf{T}_{\Omega} \to \mathbf{T}_\Sigma$ be the corresponding monad morphisms. In the category $\mathsf{Mnd_{sf}}(\mathsf{Pos})$ , we form a coinserter:

The proof is completely analogous to the proof of Proposition 31.

Indeed, this follows from the above coinserter in $\mathsf{Mnd_{sf}}(\mathsf{Pos})$ via $\mathbf{T}_\mathcal{V} = \mathbf{T}$ .

Denote by $\mathcal{V}$ the category of partial algebras $(A,\alpha)$ where A is a poset and $\alpha$ a monotone function assigning to every pair $a_0 \leq a_1$ in A an element of A. Morphisms to $(B,\beta)$ are monotone functions $h: A \to B$ such that

\begin{equation*} h \alpha (a_0,a_1) = \beta(h(a_0),h(a_1)) \end{equation*}

holds for all $a_0 \leq a_1$ . This is a “variety in context” as introduced in Adámek et al. (Reference Adámek, Ford, Milius and Schröder2021), from which it follows that the forgetful functor $U: \mathcal{V} \to \mathsf{Pos}$ is finitary monadic, see Theorem 3.24 in op. cit. The corresponding monad $\mathbf{T}$ assigns to a poset X the poset TX defined by induction as follows:

1. (1) elements of X are terms; they are ordered as in X, and

2. (2) given terms $u_0 \leq u_1$ , then $\alpha(u_0,u_1)$ is a term and the ordering is pointwise: for terms $v_0 \leq v_1$ we have $\alpha(u_0,u_1) \leq \alpha(v_0,v_1)$ iff $u_i \leq v_i$ for $i=0,1$ .

This monad is not strongly finitary because for the 2-chain P given by $x_0 \leq x_1$ it does not preserve its canonical reflexive coinserter (recall Remark 4):

Indeed, every coinserter is surjective, whereas Tc is not: the element $\alpha(x_0,x_1)$ of TP does not lie in the image of Tc.

4. From Strongly Finitary Monads to Varieties

We now prove that the results of Section 3 can be reversed: for every strongly finitary monad $\mathbf{T}$ , a variety is presented whose free-algebra monad is $\mathbf{T}$ .

Recall that, given a monad $\mathbf{T}$ , every morphism $f: X \to TY$ yields a homomorphism $f^*: (TX,\mu_X) \to (TY,\mu_Y)$ by $f^* = \mu_Y \cdot Tf$ . Recall from Remark 22 that we associate with every n-ary operation symbol $\sigma$ the term $\sigma(x_i)_{i<n}$ over V.

1. (1) $\sigma(x_i) \leq \tau(x_i)$ for all $\sigma \leq \tau$ in Tn;

2. (2) $k^*(\sigma)(x_i) = \sigma(k_0(x_i),\dots,k_{m-1}(x_i))$ for all m-tuples $k: m \to Tn$ , $k = (k_0,\dots,k_{m-1})$ and all $\sigma \in Tm$ ;

3. (3) $x_i = \eta_n(x_i)$ for all $i = 0,\dots,n-1$ .

Construction 37 Every algebra $\alpha: TA \to A$ in $\mathsf{Pos}^\mathbf{T}$ yields a $\Sigma$ -algebra in $\mathcal{V}_\mathbf{T}$ : given an n-ary symbol $\sigma \in Tn$ and an n-tuple $f: n \to A$ , let $f^+ = \alpha \cdot Tf: (Tn, \mu_n) \to (A, \alpha)$ be the corresponding homomorphism for $\mathbf{T}$ . We define $\sigma_A: A^n \to A$ by:

\begin{equation*} \sigma_A(f) = f^+(\sigma). \end{equation*}

To verify that this $\Sigma$ -algebra satisfies (1) in Definition 36, observe that for every n-tuple $f: n \to A$ the corresponding $\Sigma$ -homomorphism $f^\sharp: T_\Sigma V \to A$ fulfills

\begin{equation*} f^\sharp(\sigma(x_i)) = f^+(\sigma) \text{ for all } \sigma \in Tn. \tag{5} \end{equation*}

This equality holds since $\sigma(x_i)$ is the result of the operation $\sigma$ in the algebra $T_\Sigma n$ (Example 17) applied to $(x_i)$ , thus, $f^\sharp(\sigma(x_i)) = \sigma_A(f(x_i))$ . Given $\sigma \leq \tau$ in Tn, then $f^+(\sigma) \leq f^+(\tau)$ since $f^+ = \alpha \cdot Tf$ is monotone, thus $f^\sharp(\sigma(x_i)) \leq f^\sharp(\tau(x_i))$ holds.

To verify (2), we need to prove

\begin{equation*} f^\sharp(k^*(\sigma)(x_i)) = f^\sharp(\sigma( k_0(x_i), \dots, k_{m-1}(x_i) )) \end{equation*}

for every n-tuple $f: n \to A$ . Due to (5) above, the left-hand side is

\begin{equation*} f^+(k^*(\sigma)). \end{equation*}

Since $f^\sharp$ is a homomorphism, the right-hand side is

\begin{equation*} \sigma_A(f^\sharp(k_0(x_i)), \dots, f^\sharp(k_{m-1} (x_i) ) ) \end{equation*}

which due to (5) is equal to

\begin{equation*} \sigma_A(f^+ \cdot k) = (f^+ \cdot k)^+(\sigma) \end{equation*}

Thus, we only need to observe that:

\begin{equation*} f^+ \cdot k^* = (f^+ \cdot k)^+: (Tn, \mu_n) \to (A, \alpha). \tag{6} \end{equation*}

Indeed, both sides are homomorphisms in $\mathsf{Pos}^\mathbf{T}$ , and they are equal when precomposed with the universal map:

\begin{equation*} f^+ \cdot k^* \cdot \eta_n = f^+ \cdot k = (f^+ \cdot k)^+ \cdot \eta_n. \end{equation*}

The verification of (3) is trivial: for $\sigma = \eta_n(x_i)$ we have (due to $\alpha \cdot \eta_A = {\mathit{id}}$ ) that $\sigma_A$ is the i-th projection: given an n-tuple $f: n \to A$ , we get

\begin{equation*} \sigma_A(f) = \alpha \cdot T\,f \cdot \eta_n (x_i) = \alpha \cdot \eta_A \cdot f (x_i) = f (x_i). \end{equation*}

1. (1) Let $h \cdot \alpha = \beta \cdot Th$ . Then

   \begin{equation*} h \cdot f^+ = (h \cdot f)^+: (Tn, \mu_n) \to (A,\alpha) \end{equation*}
   because both sides are homomorphisms of $\mathsf{Pos}^\mathbf{T}$ extending $h \cdot f$ . For every $\sigma \in \Sigma_n$ and every n-tuple $f: n \to A$ we have
   \begin{align*} h(\sigma_A(f)) & = h \cdot f^+(\sigma), \text{ by definition of $\sigma_A$} \\ & = (h \cdot f)^+, \text{ as $h \cdot f^+ = (h \cdot f)^+$} \\ & = \sigma_B(h \cdot f), \text{ by definition of $\sigma_B$.} \end{align*}
   Thus, h is a $\Sigma$ -homomorphism.

2. (2) Let h be a $\Sigma$ -homomorphism. To prove that h is a homomorphism of T-algebras, consider the diagram below for an arbitrary $n \in \mathbb{N}$ and $f: n \to A$ . (Recall that n is the discrete poset on $\{0,\dots,n-1\}$ .) Since $\mathbf{T}$ is finitary, it is sufficient to show that the desired square commutes when precomposed by Tf:

   
   Indeed, given $\sigma \in Tn$ we have
   \begin{align*} \beta \cdot Th \cdot Tf (\sigma) & = (h \cdot f)^+(\sigma), \text{ by definition of $(h \cdot f)^+$} \\ & = \sigma_B(h \cdot f), \text{ by definition of $\sigma_B$} \\ & = h(\sigma_A(f)), \text{ since} h \text{is a $\Sigma$-homomorphism} \\ & = h(f^+(\sigma)), \text{ by definition of $\sigma_A$} \\ & = h(\alpha \cdot Tf(\sigma)), \text{ by definition of $f^+$.} \end{align*}

Proof.

1. (1) For every poset X, we prove that the free algebra $(TX,\mu_X)$ on X in $\mathsf{Pos}^\mathbf{T}$ , considered as a $\Sigma$ -algebra, is free on X in $\mathcal{V}_\mathbf{T}$ w.r.t. $\eta_X$ as the universal map. To verify this, we can restrict ourselves to finite posets X. Then, this holds for all posets since T preserves filtered colimits: express $X = \mathop{\mathrm{colim}}\limits_{i \in I} X_i$ as a filtered colimit of finite posets, then $TX = \mathop{\mathrm{colim}}\limits_{i \in I} TX_i$ , and from Remark 38 we conclude that the $\Sigma$ -algebra TX is a filtered colimit of $TX_i$ ( $i \in I$ ) in $\mathsf{Alg}(\Sigma)$ . Thus, from $TX_i$ being a free $\Sigma$ -algebra on $X_i$ in $\mathcal{V}_\mathbf{T}$ , we conclude that TX is a free $\Sigma$ -algebra on X.

2. (2) We first consider the discrete poset n. Given an algebra A and an n-tuple $f: n \to A$ , we prove that for $(Tn, \mu_n)$ considered as a $\Sigma$ -algebra there exists a unique homomorphism $\overline{f}: Tn \to A$ with $f = \overline{f} \cdot \eta_n$ . Existence: define $\overline{f}(\sigma) = \sigma_A(f(i))_{i < n}$ for every $\sigma \in Tn$ . Then, $f = \overline{f} \cdot \eta_n$ follows from A fulfilling the equations (3): the operation $(\eta_n(x_i))_A$ is the i-th projection, thus $\overline{f}(\eta_n(x_i)) = f(x_i)$ . Equations (1) guarantee that $\overline{f}$ is monotone. Finally, $\overline{f}$ is a $\Sigma$ -homomorphism: $\overline{f} \cdot \tau_{Tm} = \tau_A \cdot \overline{f}^m$ holds for all m and all $\tau \in Tm$ . That is, given $k: m \to Tn$ we verify

   \begin{equation*} \overline{f} \cdot \tau_{Tn} (k) = \tau_A \cdot \overline{f} (k). \end{equation*}
   The right-hand side is $\tau_A ( (k(x_j))_A (f(x_i)))$ due to the definition of $\overline{f}$ . Since A satisfies equations (2) when applied to k and $\tau$ , we see that the right-hand side is $k^* (\tau)_A (f(x_i))$ . The left-hand side is, since $\tau_{Tn}(k) = k^*(\tau)$ , the same due to the definition of $\overline{f}$ . Uniqueness: let $\overline{f}$ be a $\Sigma$ -homomorphism with $f = \overline{f} \cdot \eta_n$ . Recalling that $\eta_n^* = {\mathit{id}}_{Tn}$ , for every $\sigma \in Tn$ we get $\sigma_{Tn} (\eta_n(x_i)) = \sigma$ . Since $\overline{f}$ is a homomorphism, this implies that $\overline{f}(\sigma) = \sigma_A(\overline{f} \cdot \eta_n(x_i)) = \sigma(f(x_i))$ , as above.

3. (3) We now consider an arbitrary finite poset P on the set $\{x_0, \dots, x_{n-1}\}$ . Then its canonical coinserter (Remark 4) yields, since T is strongly finitary, the following coinserter:

   
   The free algebras Tk and Tn of $\mathsf{Pos}^\mathbf{T}$ are by (2) also free $\Sigma$ -algebras in $\mathcal{V}_\mathbf{T}$ . Given an algebra A of $\mathcal{V}_\mathbf{T}$ and a monotone function $f: P \to A$ , we thus have a unique $\Sigma$ -homomorphism $f': Tn \to A$ with $f = f' \cdot \eta_n$ . To prove that f’ is also a $\Sigma$ -homomorphism $f': TP \to A$ , it is sufficient to verify
   \begin{equation*} f' \cdot Tp_0 \leq f' \cdot Tp_1: Tk \to A. \end{equation*}
   
   Thus, we need to prove that for each $x \in Tk$ we have $f'(Tp_0(x)) \leq f'(Tp_1(x))$ . Indeed, this holds for all $x = \eta_k(y_i)$ with variables $y_i \in k$ :
   \begin{align*} f' \cdot Tp_0(\eta_k(y_i)) & = f(p_0(y_i)), \text{ by the diagram above} \\ & \leq f(p_1(y_i)), \text{ by} f \text{being monotone} \\ & = f' \cdot Tp_1(\eta_k(y_i)), \text{ by the diagram above.} \end{align*}
   And thus, we only need to observe that the set of all $x \in Tk$ with the desired property is closed under the $\Sigma$ -operations. For every $\sigma \in \Sigma_n$ and every n-tuple $(x_i)_{i<n}$ with $f' \cdot Tp_0(\sigma_{Tk}(x_i)) \leq f' \cdot Tp_1(\sigma_{Tk}(x_i))$ , we have (since $Tp_i$ are homomorphisms of $\mathsf{Pos}^\mathbf{T}$ ):
   \begin{align*} f' \cdot Tp_0(\sigma_{Tk}(x_i)) & = f'(\sigma_{Tn}(Tp_0(x_i))), \text{ by Remark ??} \\ & = \sigma_A(f'(Tp_0(x_i))), \text{ since} f' \text{is a $\Sigma$-homomorphism} \\ & \leq \sigma_A(f'(Tp_1(x_i))), \text{ since $\sigma_A$ is monotone} \\ & = f' \cdot Tp_1(\sigma_{Tk}(x_i)) \text{ as above.} \end{align*}

4. (4) The full embedding $E: \mathsf{Pos}^\mathbf{T} \to \mathcal{V}_\mathbf{T}$ of Remark 38 is concrete. That is, if $U: \mathsf{Pos}^\mathbf{T} \to \mathsf{Pos}$ and $V: \mathcal{V}_\mathbf{T} \to \mathsf{Pos}$ denote the forgetful functors, the triangle:

   
   commutes. Both U and V are monadic functors by Lemma 28. It follows from (1) that the corresponding monads are isomorphic.

Notation 40. Let $\mathsf{Var}(\mathsf{Pos})$ denote the ordinary category of varieties of ordered algebras and concrete functors. These are functors $F: \mathcal{V}_1 \to \mathcal{V}_2$ which commute (strictly) with the forgetful functors $U_i: \mathcal{V}_i \to \mathsf{Pos}$ :

Theorem 41. The ordinary category of varieties is dually equivalent to the ordinary category of strongly finitary monads:

\begin{equation*} \mathsf{Var}(\mathsf{Pos}) \simeq \mathsf{Mnd_{sf}}(\mathsf{Pos})^{\mathit{op}}. \end{equation*}

Proof.

1. (1) Let $F: \mathcal{V}_1 \to \mathcal{V}_2$ be a concrete functor. The comparison functors $K_i: \mathcal{V}_i \to \mathsf{Pos}^{\mathbf{T}_{\mathcal{V}_i}}$ are isomorphisms of categories by Lemma 28. These isomorphisms are concrete: if $U_i': \mathsf{Pos}^{\mathbf{T}_{\mathcal{V}_i}} \to \mathsf{Pos}$ denotes the underlying functor, then $U_i = U_i' \cdot K_i$ . From F, we thus obtain a concrete functor:

   \begin{equation*} \overline{F} = K_2 \cdot F \cdot K_1^{-1}: \mathsf{Pos}^{\mathbf{T}_{\mathcal{V}_1}} \to \mathsf{Pos}^{\mathbf{T}_{\mathcal{V}_2}}. \end{equation*}
   
   The passage $F \mapsto \overline{F}$ is bijective (with the inverse passage $K_2^{-1} \cdot ({-}) \cdot K_1$ ) and preserves composition and identity morphisms.

2. (2) Given monads $\mathbf{T}_1$ , $\mathbf{T}_2$ , monad morphisms $\rho: \mathbf{T}_2 \to \mathbf{T}_1$ bijectively correspond to concrete functors from $\mathsf{Pos}^{\mathbf{T}_1}$ to $\mathsf{Pos}^{\mathbf{T}_2}$ : the bijection takes $\rho$ to $H_\rho: \mathsf{Pos}^{\mathbf{T}_1} \to \mathsf{Pos}^{\mathbf{T}_2}$ assigning to an algebra $\alpha: T_1 A \to A$ in $\mathsf{Pos}^{\mathbf{T}_1}$ the algebra:

   \begin{equation*} T_2 A \xrightarrow{\rho_A} T_1 A \xrightarrow{\alpha} A \end{equation*}
   in $\mathsf{Pos}^{\mathbf{T}_2}$ . This passage $\rho \mapsto H_\rho$ moreover preserves composition and identity morphisms. See Barr and Wells (Reference Barr and Wells1985), Theorem 3.6.3.

3. (3) Define a functor:

   \begin{equation*} R: \mathsf{Var}(\mathsf{Pos}) \to \mathsf{Mnd_{sf}}(\mathsf{Pos})^{\mathit{op}} \end{equation*}
   on objects by
   \begin{equation*} R(\mathcal{V}) = \mathbf{T}_\mathcal{V} \end{equation*}
   and on morphisms $F: \mathcal{V}_1 \to \mathcal{V}_2$ by the following rule:
   \begin{equation*} R(F) = \rho \text{ iff } H_\rho = \overline{F}. \end{equation*}
   It follows from (1) and (2) that R is a well-defined full and faithful functor. Theorem 39 tells us that every strongly finitary monad is isomorphic to $R(\mathcal{V})$ for some variety $\mathcal{V}$ . Therefore, R is an equivalence of categories.

5. Lifting and Extending Finitary Monads from $\mathsf{Set}$ to $\mathsf{Pos}$

The examples of varieties of ordered algebras presented so far are all liftings of varieties of classical algebras (over $\mathsf{Set}$ ). In the present section, we prove that this is no coincidence: there are no other examples. Since varieties of ordered algebras are in a bijective correspondence with strongly finitary monads on $\mathsf{Pos}$ (and varieties of classical algebras are in a bijective correspondence with finitary monads on $\mathsf{Set}$ ), an equivalent statement is the theorem below. (To lighten up the notation, we write $\mathsf{Set}$ instead of $\mathsf{Set}_o$ when talking about ordinary categories below.)

1. (1) A lifting of an ordinary monad $\widetilde{\mathbf{T}} = (\widetilde{T}, \widetilde{\mu}, \widetilde{\eta})$ on $\mathsf{Set}$ to $\mathsf{Pos}$ is a monad $\mathbf{T} = (T, \mu, \eta)$ on $\mathsf{Pos}$ such that the square:

   
   commutes, and the equalities $U \eta = \widetilde{\eta} U$ and $U \mu = \widetilde{\mu} U$ hold.

2. (2) An extension of $\widetilde{\mathbf{T}}$ is a monad $\mathbf{T}$ on $\mathsf{Pos}$ such that the square:

   
   commutes, and the equalities $\eta D = D \widetilde{\eta}$ and $\mu D = D \widetilde{\mu}$ hold.

Proof. In view of Theorem 41 it is sufficient to present, for every variety $\mathcal{V}$ of ordered $\Sigma$ -algebras, a variety $\widetilde{\mathcal{V}}$ of non-ordered algebras such that $\mathbf{T}_\mathcal{V}$ is a lifting of $\mathbf{T}_{\widetilde{\mathcal{V}}}$ . Here, $\mathbf{T}_\mathcal{V}$ is the ordinary $\mathcal{V}$ -free-algebra monad on $\mathsf{Pos}$ , and $\mathbf{T}_{\widetilde{\mathcal{V}}}$ the $\widetilde{\mathcal{V}}$ -free-algebra (ordinary) monad on $\mathsf{Set}$ . Recall that we consider an arbitrary set as the poset with the trivial order.

1. (1) For our standard set $V = \{ x_0, x_1, x_2,\dots \}$ of variables in Definition 21, we have defined a set $\mathcal{E}_V^o = \mathcal{E}_V \cap \mathcal{E}_V^{-1}$ of equations in Construction 25: they are those equations $s = t$ which every ordered algebra in $\mathcal{V}$ satisfies. (Since this is equivalent to satisfying both $s \leq t$ and $t \leq s$ .) We denote by $\widetilde{\mathcal{V}}$ the variety of non-ordered algebras presented by $\mathcal{E}_V^o$ . This clearly implies that every algebra in $\widetilde{\mathcal{V}}$ satisfies, for every set X, all equations $s = t$ for pairs in $\mathcal{E}_X^o$ . Moreover, $\mathcal{E}_X^o$ is clearly a congruence on the non-ordered $\Sigma$ -algebra $\widetilde{T}_\Sigma X$ of all $\Sigma$ -terms on X.

2. (2) Denote by $T_{\widetilde{\mathcal{V}}} X$ the free algebra of $\widetilde{\mathcal{V}}$ on the set X. It can be constructed as the quotient of the non-ordered algebra $\widetilde{T}_\Sigma X$ modulo the congruence $\mathcal{E}_X^o$ :

   \begin{equation*} T_{\widetilde{\mathcal{V}}} X = \widetilde{T}_\Sigma X / \mathcal{E}_X^o. \end{equation*}
   The proof is completely analogous to that of Theorem 26. We thus conclude that for an arbitrary poset X our choice of $T_\mathcal{V} X$ and $T_{\widetilde{\mathcal{V}}} |X|$ can be such that the underlying set of $T_\mathcal{V} X$ is $T_{\widetilde{\mathcal{V}}} |X|$ and all operations are equal. The universal arrows $(\eta_\mathcal{V})_X: X \to |T_\mathcal{V} X|$ and $(\eta_{\widetilde{\mathcal{V}}})_{|X|}: |X |\to T_{\widetilde{\mathcal{V}}} |X|$ are both given by forming the equivalence classes of $x \in X$ modulo $\mathcal{E}_X^o$ , thus $\eta_{\widetilde{\mathcal{V}}}$ is the underlying map of $\eta_\mathcal{V}$ . The multiplication $(\mu_\mathcal{V})_X: T_\mathcal{V} T_\mathcal{V} X \to T_\mathcal{V} X$ is an interpretation of every term $t \in T_\mathcal{V} X$ over the poset $T_\mathcal{V} X$ of $\Sigma$ -terms as a term $\mu_\mathcal{V}(t)$ over X modulo $\mathcal{E}_X^o$ . This interpretation is independent of the ordering of X, shortly, the underlying function of $(\mu_\mathcal{V})_X$ is the corresponding interpretation $(\widetilde{\mu}_\mathcal{V})_{|X|}$ of terms modulo $\mathcal{E}_X^o$ w.r.t. $\widetilde{\mathbf{T}}$ .

Indeed, given a variety $\mathcal{V}$ , let $\widetilde{\mathbf{T}}$ be an ordinary monad of $\mathsf{Set}$ such that $\mathbf{T}_\mathcal{V}$ is a lifting of it. The comparison functor is an isomorphism $K: \mathcal{V} \to\mathsf{Pos}^{\mathbf{T}_\mathcal{V}}$ concrete over $\mathsf{Pos}$ (Lemma 28). And we have a classical variety $\widetilde{\mathcal{V}}$ with an analogous concrete isomorphism $\widetilde{K}: \widetilde{\mathcal{V}} \to \mathsf{Set}^{\widetilde{\mathbf{T}}}$ over $\mathsf{Set}$ . Define a concrete functor $H: \mathsf{Pos}^\mathbf{T} \to \mathsf{Set}^{\widetilde{\mathbf{T}}}$ over $\mathsf{Set}$ by the obvious rule: it sends an algebra $\alpha: TA \to A$ to $\alpha: \widetilde{T}|A| \to |A|$ . The desired functor is $\widetilde{K}^{-1} \cdot H \cdot K: \mathcal{V} \to \widetilde{\mathcal{V}}$ .

In contrast, ordered monoids with e the least element (Example 23 (4)) yield a monad $\mathbf{T}$ which is not an extension of $\widetilde{\mathbf{T}}$ : for every set $X \neq \emptyset$ the monoid T(DX) is not discrete. We now demonstrate that this is not a coincidence.

1. (1) equational varieties (and concrete functors) and

2. (2) strongly finitary monads on $\mathsf{Pos}$ which are extensions of monads on $\mathsf{Set}$ (and monad morphisms).

Proof. Using the equivalence of Theorem 41, it is sufficient to prove that a variety $\mathcal{V}$ of ordered algebras is equational iff the monad $\mathbf{T}_\mathcal{V}$ is an extension of a finitary monad on $\mathsf{Set}$ .

1. (1) Let $\mathbf{T}_\mathcal{V}$ be a strongly finitary extension of a finitary monad $\widetilde{\mathbf{T}}$ . If an inequation $u \leq v$ holds for all algebras of $\mathcal{V}$ , we prove that $v \leq u$ also holds. Then $\mathcal{V}$ is presented by the corresponding equations $u = v$ . Let X be the set of variables that occur in u or v. Then $T_\mathcal{V}(DX)$ is discretely ordered, since $T_\mathcal{V} D = D \widetilde{T}$ , and it satisfies $u \leq v$ . The trivial interpretation $\eta_{DX}: DX \to T_\mathcal{V}(DX)$ , with $\eta_{DX}^\sharp = {\mathit{id}}$ , thus implies $u \leq v$ in $T_\mathcal{V}(DX)$ , hence $u = v$ . Since $T_\mathcal{V}(DX)$ is the free algebra on DX, this implies that every algebra of $\mathcal{V}$ satisfies $u = v$ .

2. (2) Let $\mathcal{V}$ be a variety of ordered algebras specified by equations. For every set X, the free algebra $T_\mathcal{V}(DX)$ is discretely ordered. Indeed, this is an ordered algebra in $\mathcal{V}$ and we have the discretely ordered $\Sigma$ -algebra A underlying $T_\mathcal{V}(DX)$ . Now A satisfies every equation $u = v$ satisfied by $T_\mathcal{V}(DX)$ : given an interpretation $f: V \to |T_\mathcal{V}(DX)|$ of variables, the corresponding homomorphism $f^\sharp: T_\Sigma V \to T_\mathcal{V}(DX)$ is the same for A. Thus A lies in $\mathcal{V}$ . This implies $A = T_\mathcal{V}(DX)$ .

In other words, free algebras of $\mathcal{V}$ on discrete posets are discretely ordered, and $\eta_{DX} = D \widetilde{\eta}_X$ . From this, $\mu_{DX} = D\widetilde{\mu}_X$ easily follows.

Example 50. A monad on $\mathsf{Pos}$ which is both an extension and a lifting of a finitary monad on $\mathsf{Set}$ need not be strongly finitary. Let $\Sigma = \{+,*\}$ be a signature of two binary operations. The full subcategory on all ordered algebras for which the implication:

\begin{equation*} x \leq y \implies x+y \leq x*y \end{equation*}

holds yields the following monad $\mathbf{T}$ on $\mathsf{Pos}$ . Given a poset X, the poset TX consists of all terms with variables in X using $+$ and $*$ , where the order on TX is the smallest one such that

1. (1) $x \leq y$ in X implies $x \leq y$ in TX,

2. (2) $+$ and $*$ are monotone, and

3. (3) $t + s \leq t*s$ for all terms $t \leq s$ in TX.

Thus, $\mathbf{T}$ is a lifting of the monad $\widetilde{\mathbf{T}}$ on $\mathsf{Set}$ of free $\Sigma$ -algebras. If X is discretely ordered, then so is TX, and thus we see that $\mathbf{T}$ is also an extension of $\widetilde{\mathbf{T}}$ .

The monad $\mathbf{T}$ is not strongly finitary. For example, it does not preserve the canonical coinserter (recall Remark 4) of the chain $\mathbf{2}$ given by $0 < 1$ :

Indeed, in $T\mathbf{2}$ we have $0 + 1 < 0 * 1$ . In contrast, this does not hold in the coinserter of $Tp_0$ and $Tp_1$ . We can describe the order of that coinserter as the smallest one that, besides conditions (1)-(3) above, also fulfills $t\leq s$ for terms such that s is obtained by changing some 0 in t to 1. The down-set of the term $0*1$ in that coinserter consists of the following terms:

\begin{equation*} 0+0 < 0*0 < 0*1. \end{equation*}

Thus, $\mathbf{T}$ does not preserve the coinserter of $p_0$ and $p_1$ .