2. Background on Algebraic Theories

Classical universal algebra starts with a signature $\Sigma$ giving a set of operations equipped with arities which are finite cardinals. A $\Sigma$ -algebra A assigns to every n-ary operation f a mapping $f_A:A^n\to A$ . Then one defines terms and equations, interprets terms on an algebra and says when an equation is satisfied by an algebra. An equational theory E is a set of equations and an algebra satisfies E if it satisfies all equations from E. The forgetful functor $U:\operatorname{\bf Alg}\!(E)\to\operatorname{\bf Set}$ from the category $\operatorname{\bf Alg}\!(E)$ of algebras satisfying E has a left adjoint F and $\operatorname{\bf Alg}\!(E)$ is equivalent to the category $\operatorname{\bf Alg}\!(T)$ of algebras for the monad $T=UF$ . The algebra Fn consists of equivalence classes of n-ary terms and morphisms $Fm\to Fn$ give m-tuples of n-ary terms. We can consider them as (n,m)-ary operations where the input arity n is the usual arity and the output arity m is the multiplicity. Then the superposition of terms is replaced by the composition of these operations. If $\mathcal {N}$ is the full subcategory of $\operatorname{\bf Set}$ consisting of finite cardinals and $\mathcal {T}$ the full subcategory of $\operatorname{\bf Alg}\!(E)$ consisting of free algebras on finite cardinals, then the domain-codomain restriction of F gives an identity-on-objects functor $J:\mathcal {N}\to\mathcal {T}$ . The dual $\mathcal {T}^{\operatorname{op}}$ of $\mathcal {T}$ is a Lawvere theory whose algebras A are functors $\hat{A}:\mathcal {T}^{\operatorname{op}}\to\operatorname{\bf Set}$ preserving finite products. This is the same as $\hat{A}J^{\operatorname{op}}=\operatorname{\bf Set}(K-, A)$ where $K:\mathcal {N}\to\operatorname{\bf Set}$ is the inclusion and $A=\hat{A}(1)$ . Then $\operatorname{\bf Alg}\!(E)$ is equivalent to the category of algebras of $\mathcal {T}^{\operatorname{op}}$ . Hence, classical universal algebra can be equivalently captured by equational theories, finitary monads (= monads preserving filtered colimits), or Lawvere theories. All this is well known due to Birkhoff, Lawvere and Linton (see Adámek et al. (Reference Adámek, Rosický and Vitale2011)).

In ordered universal algebra, given a signature $\Sigma$ , a $\Sigma$ -algebra is a poset A equipped with monotone mappings $f_A:A^n\to A$ for every n-ary operation f from $\Sigma$ . But, instead of equations, one takes inequations $p\leq q$ of terms (see Bloom (Reference Bloom1976)). An algebra satisfies this inequation if $p_A\leq q_A$ in the poset of monotone mappings $A^n\to A$ . It is natural to take enriched signatures where arities are finite posets. Then the resulting enriched inequational theories correspond to finitary enriched monads on $\operatorname{\bf Pos}$ (Adámek et al., Reference Adámek, Ford, Milius and Schröder2021). Recall that the category $\operatorname{\bf Pos}$ of posets and monotone mapping is locally finitely presentable as a cartesian closed category. The free algebra FX on a finite poset X consists of equivalence classes of X-ary terms and morphisms $FY\to FX$ can be taken as (X,Y)-ary operations. The inequation $p\leq q$ of X-ary terms, which is called an inequation in the context X, then means that the pair p,q of terms is an (X,2)-ary operation where 2 is a two-element chain. In this way, inequational theories can be replaced by equational theories in (X,Y)-ary operations (see Rosický (Reference Rosický2021, Example 4.11(1))). If $\mathcal {F}$ is the full subcategory of $\operatorname{\bf Pos}$ consisting of finite poset and $\mathcal {T}$ the full subcategory of $\operatorname{\bf Alg}\!(E)$ consisting of free algebras on finite posets then the domain-codomain restriction of F gives an identity-on-objects enriched functor $J:\mathcal {F}\to\mathcal {T}$ . The dual $\mathcal {T}^{\operatorname{op}}$ of $\mathcal {T}$ is an enriched Lawvere theory (Power, Reference Power2005). Hence, ordered universal algebra can be equivalently captured by inequational theories, enriched equational theories, finitary monads, or enriched Lawvere theories.

In metric universal algebra, one has to take into account that the category $\operatorname{\bf Met}$ of metric spaces (distances $\infty$ are allowed) and nonexpanding maps is only locally $\aleph_1$ -presentable as a symmetric monoidal closed category. Thus, one has to take countable metric spaces as arities. Given such an enriched signature $\Sigma$ , a $\Sigma$ -algebra is a metric space A equipped with nonexpanding mappings $f_A:A^X\to A$ for every X-ary operation f from $\Sigma$ . Now, instead of equations, one takes quantitative equations $p=_\varepsilon q$ of terms where $\varepsilon\geq 0$ is a real number (see Mardare et al. (Reference Mardare, Panangaden and Plotkin2016, Reference Mardare, Panangaden and Plotkin2017)). An algebra satisfies this quantitative equation if $d(p_A,q_A)\leq\varepsilon$ in the metric space of nonexpanding mappings $A^X\to A$ . These equations are called basic quantitative equations (in context X). This means that the pair p,q of terms is an $(X,2_\varepsilon)$ -ary operation where $2_\varepsilon$ is a two-element metric space with the distance $\varepsilon$ between the two points. In this way, basic quantitative theories can be replaced by equational theories in (X,Y)-ary operations (see Rosický (Reference Rosický2021, Example 4.11(2))). If $\mathcal {C}$ is the full subcategory of $\operatorname{\bf Met}$ consisting of countable metric spaces and $\mathcal {T}$ the full subcategory of $\operatorname{\bf Alg}\!(E)$ consisting of free algebras on countable metric spaces then the domain-codomain restriction of F gives an identity-on-objects enriched functor $J:\mathcal {F}\to\mathcal {T}$ . The dual $\mathcal {T}^{\operatorname{op}}$ of $\mathcal {T}$ is an enriched $\aleph_1$ -ary enriched Lawvere theory, and these theories correspond to enriched monads preserving $\aleph_1$ -filtered colimits (see Power (Reference Power2005)). Hence, metric universal algebra can be equivalently captured by basic quantitative theories, enriched equational theories, $\aleph_1$ -ary monads, or enriched $\aleph_1$ -ary Lawvere theories.

In general, let $\mathcal {V}$ be a symmetric monoidal closed category with the unit object I and the underlying category $\mathcal {V}_0$ . We assume that $\mathcal {V}$ is locally $\lambda$ -presentable as as a symmetric monoidal closed category which means that the underlying category $\mathcal {V}_0$ is locally $\lambda$ -presentable, the tensor unit I is $\lambda$ -presentable and $X\otimes Y$ is $\lambda$ -presentable whenever X and Y are $\lambda$ -presentable. We will denote by $\mathcal {V}_\lambda$ the (representative) small, full subcategory consisting of $\lambda$ -presentable objects.

Following Reference Bourke and GarnerBourke and Garner (Reference Bourke and Garner2019), let $\mathcal {A}$ be a small, full, dense sub- $\mathcal {V}$ -category of $\mathcal {V}$ with the inclusion $K:\mathcal {A}\to\mathcal {V}$ . Objects of $\mathcal {A}$ are called arities. Then an $\mathcal {A}$ -pretheory is an identity-on-objects $\mathcal {V}$ -functor $J:\mathcal {A}\to\mathcal {T}$ . A $\mathcal {T}$ -algebra is an object A of $\mathcal {V}$ together with a $\mathcal {V}$ -functor $\hat{A}:\mathcal {T}^{\operatorname{op}}\to\mathcal {V}$ whose composition with $J^{\operatorname{op}}$ is $\mathcal {V}(K-,A)$ (in Bourke and Garner (Reference Bourke and Garner2019), $\mathcal {T}$ -algebras are called concrete $\mathcal {T}$ -models). Every $\mathcal {A}$ -pretheory induces a $\mathcal {V}$ -monad $T:\mathcal {V}\to\mathcal {V}$ given by its $\mathcal {V}$ -category $\operatorname{\bf Alg}\!(\mathcal {T})$ of algebras. Conversely, a $\mathcal {V}$ -monad T induces an $\mathcal {A}$ -pretheory $J:\mathcal {A}\to\mathcal {T}$ where $\mathcal {T}$ is the full subcategory of $\operatorname{\bf Alg}\!(T)$ consisting of free algebras on objects from $\mathcal {A}$ and J is the domain-codomain restriction of the free algebra functor. An $\mathcal {A}$ -pretheory is an $\mathcal {A}$ -theory if it is given by its monad. Then $\hat{A}$ is the hom-functor $\operatorname{\bf Alg}\!(\mathcal {T})(-,A)$ restricted on free algebras over $\mathcal {A}$ . Conversely, a monad T is $\mathcal {A}$ -nervous if it is given by its $\mathcal {A}$ -theory. In this way, we get a one-to-one correspondence between $\mathcal {A}$ -theories and $\mathcal {A}$ -nervous monads (see Bourke and Garner (Reference Bourke and Garner2019, Corollary 21)).

Under a $\lambda$ -ary $\mathcal {V}$ -theory we will mean a $\mathcal {V}_\lambda$ -theory. Following Reference Bourke and GarnerBourke and Garner (Reference Bourke and Garner2019), $\lambda$ -ary $\mathcal {V}$ -theories correspond to $\mathcal {V}$ -monads on $\mathcal {V}$ preserving $\lambda$ -filtered colimits. They are called $\lambda$ -ary $\mathcal {V}$ -monads. Hence, $\lambda$ -ary monads are precisely $\mathcal {V}_\lambda$ -nervous monads.

3. Surjections

The underlying functor $\mathcal {V}_0(I,-):\mathcal {V}_0\to\operatorname{\bf Set}$ has a left adjoint $-\cdot I$ sending a set X to the coproduct $X\cdot I$ of X copies of I in $\mathcal {V}_0$ . Objects $X\cdot I$ will be called discrete. Every object V of $\mathcal {V}$ determines a discrete object $V_0=\mathcal {V}_0(I,V)\cdot I$ and morphism $\delta_V:V_0\to V$ given by the counit of the adjunction. Every morphism $f:V\to W$ determines the morphism $f_0=\mathcal {V}_0(I,f)\cdot I$ between the underlying discrete objects.

A morphism $f:A\to B$ will be called a surjection if $\mathcal {V}_0(I,f)$ is surjective. Let $\operatorname{\it Surj}$ denote the class of all surjections in $\mathcal {V}_0$ and let $\operatorname{\it Inj}$ be the class of morphisms of $\mathcal {V}_0$ having the unique right lifting property with respect to every surjection. Morphisms from $\operatorname{\it Inj}$ will be called injections. Recall that g is an injection iff for every surjection f and every commutative square

there is a unique $t:B\to C$ such that $tf=u$ and $gt=v$ .

Proof. Take an injection $g:C\to D$ and $u,v:B\to C$ such that $gu=gv$ . Consider a commutative square

where $\nabla$ is the codiagonal. Since $\nabla$ is a split epimorphism, it is a surjection and thus there is a unique diagonal $B\to C$ . Hence, $u=v$ . We have proved that g is a monomorphism.

Proof. If I is connected, then $\operatorname{\it Surj}$ is closed under coproducts in $\mathcal {V}_0^\to$ . If regular epimorphisms are surjections, then surjections are closed under coequalizers in $\mathcal {V}_0^\to$ . Indeed, let $f_0$ and $f_1$ be surjections and

be a coequalizer in $\mathcal {V}_0^\to$ . Then

is a coequalizer in $\mathcal {V}_0$ , hence v is a surjection. Following 3.2(1), f is a surjection.

The factorization system $(\operatorname{\it Surj},\operatorname{\it Inj})$ is $\lambda$ -convenient in the sense of Reference Di Liberti and RosickýDi Liberti and Rosický (Reference Di Liberti and Rosický2022) if

1. (1) $\mathcal {V}$ is $\operatorname{\it Surj}$ -cowellpowered, i.e., every object of $\mathcal {V}$ has only a set of surjective quotients, and

2. (2) $\operatorname{\it Inj}$ is closed under $\lambda$ -directed colimits, i.e., every $\lambda$ -directed colimit of injections has the property that a colimit cocone

   1. (a) consists of injections, and

   2. (b) for every cocone of injections, the factorizing morphism is an injection.

If I is a generator, then surjections are epimorphisms and (1) follows from the fact that every locally presentable category is co-wellpowered (see Adámek and Rosický (Reference Adámek and Rosický1994, Theorem 1.58)).

4 Discrete Theories

Then the object $X_0$ of $\mathcal {V}$ is $\lambda$ -presentable whenever X is $\lambda$ -presentable.

This means that operations of arity X are induced by those of arity $X_0$ .

Consider an arbitrary X in $\mathcal {V}$ and express it as a $\lambda$ -directed colimit of $\lambda$ -presentable objects $X_m$ ( $m\in M$ ). Since I is $\lambda$ -presentable, $X_0=\operatorname{colim} X_{m0}$ and $\delta_X=\operatorname{colim}\delta_{X_m}$ . Hence, $UF(\delta_X)=\operatorname{colim} UF(\delta_{X_m})$ and, since surjections are closed under $\lambda$ -directed colimits (see 3.1), $UF(\delta_X)$ is a surjection.

Consider a surjective morphism $f:Y\to X$ in $\mathcal {V}$ and $f_0:Y_0\to X_0$ be its underlying morphism. Then $f\delta_Y=\delta_Xf_0$ and thus $UF(f)UF(\delta_Y)=UF(\delta_X)UF(f_0)$ . Since epimorphisms in $\operatorname{\bf Set}$ split, $f_0$ is a split epimorphism and $UF(f_0)$ is surjective. Hence, UF(f) is surjective.

II. Let T be a $\lambda$ -ary $\mathcal {V}$ -monad on $\mathcal {V}$ preserving surjections. Let $\mathcal {T}$ be the corresponding $\lambda$ -ary $\mathcal {V}$ -theory. We will show that $\mathcal {T}$ is discrete. Consider $f:FI\to FX$ and $\tilde{f}:I\to UFX$ its adjoint transpose. Since $\delta_X$ is surjective and T preserves surjections, $UF(\delta_X)$ is surjective. Thus, there is $g:I\to UFX_0$ such that $UF(\delta_X)g=\tilde{f}$ . Let $\tilde{g}:FI\to FX_0$ be the adjoint transpose of g. Then $f=F(\delta_X)\tilde{g}$ and thus $\mathcal {T}$ is discrete.

5. Birkhoff Subcategories

Observation 5.2. Recall that $(\operatorname{\it Surj},\operatorname{\it Inj})$ is proper if surjections are epimorphisms and injections are monomorphisms. Let T be the $\mathcal {V}$ -monad induced by $\mathcal {T}$ . The underlying category $\operatorname{\bf Alg}\!(\mathcal {T})_0$ is the category of algebras for the underlying monad $T_0$ . Let $U_0:\operatorname{\bf Alg}\!(\mathcal {T})_0\to\mathcal {V}_0$ be the forgetful functor. We will say that a morphism $f:A\to B$ in $\operatorname{\bf Alg}\!(\mathcal {T})_0$ is a surjection (injection) if U(f) is a surjection (injection). Following Manes ( Reference Manes1976 , Chapter 3, Proposition 4.17), (surjections, injections) form a proper factorization system on $\operatorname{\bf Alg}\!(\mathcal {T})_0$ . Here we need that $\mathcal {T}$ is discrete because then, following 4.5, $T_0$ preserves surjection. Given a morphism $f:A\to B$ of $\mathcal {T}$ -algebras and

$$U_0A \xrightarrow{\ e } C \xrightarrow{\ m } U_0B$$

$(\operatorname{\it Surj},\operatorname{\it Inj})$ -factorization of $U_0(f)$ then f factorizes as

$$A \xrightarrow{\ \overline{e} } \overline{C} \xrightarrow{\ \overline{m} } B$$

where $U_0\overline{C}=C$ , $U_0(\overline{e})=e$ and $U_0(\overline{m})=m$ .

Since $\operatorname{\bf Alg}\!(\mathcal {T})_0$ is locally presentable (strong epimorphisms, monomorphisms) is a factorization system on $\operatorname{\bf Alg}\!(\mathcal {T})_0$ (see Reference Adámek and Rosický Adámek and Rosický (1994 , 1.61)). Since injections are monomorphisms, strong epimorphisms in $\operatorname{\bf Alg}\!(\mathcal {T})_0$ are surjections.

Following Manes (Reference Manes1976, Chapter 3, 3.1), a Birkhoff subcategory of $\operatorname{\bf Alg}\!(\mathcal {T})_0$ is a full replete subcategory of $\operatorname{\bf Alg}\!(\mathcal {T})_0$ closed under products, subalgebras and $U_0$ -split quotients. Here, a morphism $g:K\to L$ in $\operatorname{\bf Alg}\!(\mathcal {T})_0$ is $U_0$ -split if $U_0(f)$ is a split epimorphism. Hence, for a Birkhoff subcategory $\mathcal {L}$ of $\operatorname{\bf Alg}\!(\mathcal {T})_0$ , the reflections $\rho_K:K\to K^\ast$ are strong epimorphisms.

Hence, Birkhoff subcategories of $\operatorname{\bf Alg}\!(\mathcal {T})_0$ are $\mathcal {V}$ -subcategories and, in what follows, we will take them as subcategories of $\operatorname{\bf Alg}\!(\mathcal {T})$ . The following definition goes back to Hatcher (Reference Hatcher1970), Herrlich and Ringel (Reference Herrlich and Ringel1972).

Proof. $\operatorname{\bf Alg}\!(E)$ is clearly closed under products and subalgebras. Let $f:A\to B$ be a $U_0$ -split quotient of an algebra A satisfying E, i.e., there is $s:UB\to UA$ such that $U(f)s=\operatorname{id}_{UB}$ . Let $p,q:FY\to FX$ give an equation $p=q$ from E. Consider $h:FX\to B$ and $\tilde{h}:X\to UB$ be the adjoint transpose of h. Let $g=s\tilde{h}$ and $\tilde{g}:FX\to A$ be the adjoint transpose of g. Since $U(f)g=U(f)s\tilde{h}=\tilde{h}$ , we have $f\tilde{g}=h$ . Since $\tilde{g}p=\tilde{g}q$ , we get

$$hp=f\tilde{g}p=f\tilde{g}q=hq.$$

Thus B satisfies E.

Recall that an object V in $\mathcal {V}$ is $\mu$ -generated with respect to $\operatorname{\it Inj}$ if $\mathcal {V}_0(V,-):\mathcal {V}_0\to\operatorname{\bf Set}$ preserves $\mu$ -directed colimits of injections.

Thus $\mathcal {L}$ is a $\mu$ -Birkhoff subcategory of $\operatorname{\bf Alg}\!(\mathcal {T})$ if and only if it is a full subcategory closed under products, subalgebras and $\mu$ -pure quotients.

Recall that $\mathcal {V}_0$ is $\operatorname{\it Inj}$ -locally $\mu$ -generated if it has a set $\mathcal {X}$ of $\mu$ -generated objects with respect to $\operatorname{\it Inj}$ such that every object of $\mathcal {V}_0$ is a $\mu$ -directed colimit of objects from $\mathcal {X}$ and injections (see Di Liberti and Rosický (Reference Di Liberti and Rosický2022)).

The following definition and theorem were motivated by Milius and Urbat (Reference Milius, Urbat, Boja ńczyk and Simpson2019).

Proof. I. Necessity: Let E consist of $\mu$ -clustered equations. We have to show that $\operatorname{\bf Alg}\!(E)$ is closed in $\operatorname{\bf Alg}\!(\mathcal {T})$ under quotients $f:A\to B$ such that Uf is $\mu$ -pure. So, let $f:A\to B$ be such a homomorphism. Consider an equation $p=q$ from E where $p,q:FY\to FX$ and $X=\coprod_i X_i$ with $X_i$ $\mu$ -generated. Let $h:FX\to B$ and $\tilde{h}:X\to UB$ be its adjoint transpose. If $u_i:X_i\to X$ are coproduct injections, then there are $v_i:X_i\to UA$ such that $U(f)v_i=\tilde{h}u_i$ for every i. We get $v:X\to UA$ such that $U(f)v=\tilde{h}$ . Therefore,

$$hp=f\tilde{v}p=f\tilde{v}q=hq.$$

Hence, B satisfies E.

II. Sufficiency: Conversely, let $\mathcal {L}$ be a $\mu$ -Birkhoff subcategory of $\operatorname{\bf Alg}\!(\mathcal {T})$ . Following 5.3, $\mathcal {L}$ is $\mathcal {V}$ -reflective in $\operatorname{\bf Alg}\!(\mathcal {T})$ . Moreover, following 5.2, the reflections $\rho_K:K\to K'$ are strong epimorphisms.

Let $U':\mathcal {L}\to\mathcal {V}_0$ be the restriction of $U_0$ on $\mathcal {L}$ and F’ its left adjoint. Let E be given by pairs (p,q) where $p,q:FY\to FX$ with $X=\coprod X_i$ , where $X_i$ are $\mu$ -generated with respect to $\operatorname{\it Inj}$ $X_i$ , such that every $A\in\mathcal {L}$ satisfies the equation $p=q$ . This is equivalent to $\rho_{FX}p=\rho_{FX}q$ . Clearly, $\mathcal {L}\subseteq\operatorname{\bf Alg}\!(E)$ . Following I., $\operatorname{\bf Alg}\!(E)$ is a $\mu$ -Birkhoff subcategory of $\operatorname{\bf Alg}\!(\mathcal {T})$ . Hence, $\operatorname{\bf Alg}\!(E)$ is $\mathcal {V}$ -reflective in $\operatorname{\bf Alg}\!(\mathcal {T})$ and the reflections $\rho'_K:K\to K''$ are strong epimorphisms.

Let $U'':\operatorname{\bf Alg}\!(E)\to\mathcal {V}_0$ be the restriction of U on $\operatorname{\bf Alg}\!(E)$ and F” its left adjoint. We get $\rho':F\to F''$ and $\tau:F''\to F'$ such that $\tau\rho'=\rho_F$ . Since both $\mathcal {L}$ and $\operatorname{\bf Alg}\!(E)$ are monadic, it suffices to prove that $\tau$ is a natural isomorphism.

Consider an arbitrary X in $\mathcal {V}$ and express UFX as a $\mu$ -directed colimit

$$(z_m:Z_m\to UFX)_{m\in M}$$

of injections between objects $Z_m$ which are $\mu$ -generated with respect to $\operatorname{\it Inj}$ . Let $\eta_Z:Z\to UFZ$ , Z in $\mathcal {V}$ , be the adjunction units and $\tilde{z}_m: FZ_m\to FX$ be given as $U(\tilde{z}_m)\eta_{Z_m}=z_m$ . Let

$$t_X:F(\coprod Z_m)\to FX$$

be the induced morphism, i.e., $t_XF(u_m)=\tilde{z}_m$ where $(u_m:Z_m\to \coprod Z_m)_{m\in M}$ is the coproduct. Then, $U(t_X)$ is a $\mu$ -pure epimorphism. Indeed, every morphism $f:Z\to UFX$ with Z $\mu$ -generated with respect to $\operatorname{\it Inj}$ factors through $z_m$ for some $m\in M$ as $f=z_mg$ where $g:Z\to Z_m$ . Hence,

$$\tilde{f}=\tilde{z}_mF(g)=t_XF(u_m)F(g)$$

and thus

$$f=U(\tilde{f})\eta_Z=U(t_X)UF(u_mg)\eta_Z.$$

We have to prove that $\tau_X$ is an isomorphism for every X. Consider the commutative diagram

where $\varepsilon:FU\to\operatorname{Id}$ is the counit of the adjunction $F\dashv U$ . The arrow s is given by $\rho'_{\coprod Z_m}$ being the reflection of $F\coprod Z_m$ to $\operatorname{\bf Alg}\!(E)$ . Since $U(\varepsilon_{F''X})$ is a split epimorphism and $U(t_{UF''X})$ is a $\mu$ -pure epimorphism, the composition $U(\varepsilon_{F''X}t_{UF''X})$ is a $\mu$ -pure epimorphism. Thus,U(s) is a $\mu$ -pure epimorphism. Thus, it suffices to show that $F''\coprod Z_m$ belongs to $\mathcal {L}$ , i.e., that $\tau_Z$ is an isomorphism where $Z=\coprod Z_m$ .

Since $\rho_Z$ is a strong epimorphism, $\tau_Z$ is a strong epimorphism. Thus, it suffices to show that $\tau_Z$ is a monomorphism. Consider $p,q:FY\to F''Z$ such that $\tau_Zp=\tau_Zq$ . We have to show that $p=q$ . Consider the pullback

and take the compositions $p',q':P\to FZ$ of the product projections with r. Let $p'',q'':FUP\to FZ$ be the compositions of p’ and q’ with $\varepsilon_P:FUP\to P$ . Since

$$\rho_{FZ}p''=\tau_Z\rho'_Zp''=\tau_Zpe\varepsilon_P=\tau_Zqe\varepsilon_P=\rho_{FZ}q'',$$

$p''=q''$ belongs to E. Thus, we have $\rho'_Zp''=\rho'_Zq''$ , i.e., $pe=qe$ . Since $\rho'_Z$ is a strong epimorphism, it is a surjection (see 5.2). Following 3.2, e is a surjection, hence an epimorphism. Thus, $p=q$ and we have proved that $\tau_Z$ is a monomorphism.

Express $X=\coprod_{m\in M}X_m$ as a $\mu$ -directed colimit of subcoproducts $X_N=\coprod_{m\in N} X_m$ where $|N|<\mu$ and $X_m$ are non-initial. Following 5.14, X is a $\mu$ -directed colimit $x_N:X_N\to X$ of split monomorphisms $x_{NN'}:X_N\to X_{N'}$ where $N\subseteq N'$ . Hence, UFX is a $\mu$ -directed colimit $UF(x_N):UFX_N\to UFX$ of split monomorphisms $UF(x_{NN'}):UFX_N\to UFX_{N'}$ . Following 3.5, $UF(x_{NN'})$ are injections. Let $\tilde{p},\tilde{q}:Y\to UFX$ be the adjoint transposes of $p,q:FY\to FX$ . Since Y is $\mu$ -generated with respect to $\operatorname{\it Inj}$ , there is $N\subseteq M$ , $|N|<\mu$ and $p',q':Y\to UFX_N$ such that $\tilde{p}=x_Np'$ and $\tilde{q}=x_Nq'$ .

Now, $p=q$ is an equation from E, iff $U(\rho_{FX})\tilde{p}=U(\rho_{FX})\tilde{q}$ . Since $x_N$ is a split monomorphism, this is equivalent to $U(\rho_{FX_N})p'=U(\rho_{X_N})q'$ , hence to $\rho_{X_N}p^\ast=\rho_{X_N}q^\ast$ where $p^\ast,q^\ast:FY\to FX_N$ are adjoint transposes of p’,q’. Since $X_N$ is $\mu$ -generated with respect to $\operatorname{\it Inj}$ (see Di Liberti and Rosický (Reference Di Liberti and Rosický2022, Lemma 2.13)), the proof is finished.

As a consequence of 5.15, we get the Birkhoff theorems over $\operatorname{\bf Met}$ from Mardare et al. (Reference Mardare, Panangaden and Plotkin2017), see Adámek (Reference Adámek2022, Theorem 2.14). Here, under an $\omega$ -ary theory over $\operatorname{\bf Met}$ we mean an $\mathcal {F}$ -theory where $\mathcal {F}$ consists of finite metric spaces, i.e., of finitely generated metric spaces with respect to $\operatorname{\it Inj}$ .

Manes (Reference Manes1976) defined a $\operatorname{\it Surj}$ -Birkhoff subcategory $\mathcal {L}$ of $\operatorname{\bf Alg}\!(\mathcal {T})_0$ as a full replete subcategory closed under products, $\operatorname{\it Inj}$ -subalgebras and $U_0$ -split quotients. Here, a morphism $g:K\to L$ in $\operatorname{\bf Alg}\!(\mathcal {T})_0$ is an $\operatorname{\it Inj}$ -subalgebra of L if g is an injection. Since the monad T preserves surjections, Manes (Reference Manes1976, Chapter 3, Theorem 4.23) shows that a full subcategory $\mathcal {L}$ of $\operatorname{\bf Alg}\!(\mathcal {T})_0$ is a $\operatorname{\it Surj}$ -Birkhoff subcategory if and only if it is a reflective subcategory such that $U_0(\rho_K)$ is a surjection for every reflection $\rho_K:K\to K^\ast$ . FollowingManes (Reference Manes1976, Chapter 3, Theorem 3.3) $\mathcal {L}$ is monadic over $\mathcal {V}_0$ . Hence, every Birkhoff subcategory of $\operatorname{\bf Alg}\!(\mathcal {T})_0$ is a $\operatorname{\it Surj}$ -Birkhoff subcategory.

Following 4.3(2), $\operatorname{\bf Alg}\!(\Sigma)\cong\operatorname{\bf Alg}\!(\mathcal {T})$ for a discrete $\mu$ -ary $\operatorname{\bf Met}$ -theory $\mathcal {T}$ . Given a $\mu$ -basic quantitative equational theory E then, following 4.3(2) and I. of the proof of 5.11, $\operatorname{\bf Alg}\!(E)$ is a $\mu$ - $\operatorname{\it Surj}$ -Birkhoff subcategory of $\operatorname{\bf Alg}\!(\Sigma)$ .

Conversely, let $\mathcal {L}$ be a $\mu$ - $\operatorname{\it Surj}$ -Birkhoff subcategory of $\operatorname{\bf Alg}\!(\Sigma)$ . We extend $\Sigma$ to a new signature $\Sigma^\ast$ by adding, for every $\varepsilon>0$ , binary operations $f_\varepsilon$ and $g_\varepsilon$ satisfying the quantitative equations

1. (1) $f_\varepsilon(x,y)=_\varepsilon g_\varepsilon(x,y)$ and

2. (2) $x=_\varepsilon y\vdash f_\varepsilon(x,y)=x,\quad x=_\varepsilon y\vdash g_\varepsilon(x,y)=y.$

Then subalgebras of $\Sigma^\ast$ -algebras satisfying (1) and (2) are $\operatorname{\it Inj}$ -subalgebras. Indeed, let B be a subalgebra of a $\Sigma^\ast$ -algebra A satisfying (1) and (2) and consider $a,b\in B$ such that $d_B(a,b)>d_A(a,b)$ . Following (2) for $\varepsilon=d_A(a,b)$ , we have

$$(f_\varepsilon)_B(a,b)=(f_\varepsilon)_A(a,b)=a,\quad (g_\varepsilon)_B(a,b)=(g_\varepsilon)_A(a,b)=b,$$

which contradicts (1).

Let $\mathcal {L}^\ast$ consist of all $\Sigma^\ast$ -algebras satisfying (1) and (2) whose $\Sigma$ -reduct is in $\mathcal {L}$ . Then $\mathcal {L}^\ast$ is a $\mu$ -Birkhoff subcategory of $\operatorname{\bf Alg}\!(\Sigma^\ast)$ . Let $\mathcal {T}^\ast$ be the discrete $\mu$ -ary $\operatorname{\bf Met}$ -theory given by $\Sigma^\ast$ , (1) and (2). Following 5.15, $\mathcal {L}^\ast=\operatorname{\bf Alg}\!(E^\ast)$ where equations $p=q$ from $E^\ast$ have $p,q:F^\ast Y \to F^\ast X$ with X and Y being $\mu$ -generated with respect to $\operatorname{\it Inj}$ . Let E’ consist of those equations from $E^\ast$ which do not contain the added operations $f_\varepsilon$ and $g_\varepsilon$ . Clearly, $\operatorname{\bf Alg}\!(E')\subseteq\mathcal {L}$ . Consider $A\in\mathcal {L}$ and interpret $f_\varepsilon$ and $g_\varepsilon$ on A as follows: $(f_\varepsilon)_A(x,y)=x$ and $(g_\varepsilon)_A(x,y)=y$ if $d(x,y)=\varepsilon$ and $(f_\varepsilon)_A=(g_\varepsilon)_A$ otherwise. Since we get an $E^\ast$ -algebra, $\mathcal {L}\subseteq\operatorname{\bf Alg}\!(E')$ .