Notation 3.13. Let $\mathbf {K}$ be a class of expanded $\mathcal {L}$ -algebras, then we write $\mathbf {K}\restriction \mathcal {L}$ for the class of its $\mathcal {L}$ -reducts. If it is clear from the context, we also write $\mathbf {K}$ for $\mathbf {K}\restriction \mathcal {L}$ .

Proof. Let $(x_i)_{i<\omega }$ be an enumeration of $\mathsf {Var}$ . Since A is core-generated, we have that for all $x_i\in \mathsf {Var}$ there is a polynomial $t_i$ such that $h(x_i) = t_i(a^i_0,\dots ,a^i_{n_i})$ with $a^i_j\in \mathsf {core}(A)$ for all $j{\leqslant } n_i$ . Let now $\{ y^i_j : i<\omega , \; j{\leqslant } n_i \}$ be another enumeration of $\mathsf {Var}$ and define $g \in \mathsf {Hom}^c(\mathsf {Fm},A)$ such that $g(y^i_j) = a^i_j$ for all $ i<\omega $ , $j{\leqslant } n_i$ . In particular, this means that for all formulas $\psi (x_0, \dots , x_m)\in \mathsf {Fm}$ we have

$$\begin{align*}h(\psi(x_0, \dots x_m)) = g(\psi( t_0(y^0_0,\dots,y^0_{n_0}), \dots, t_m(y^m_0,\dots,y^m_{n_m}) ) ).\end{align*}$$

By construction we have that g is a core assignment. Let $\sigma $ be the substitution defined by letting $\sigma (x_i) = t_i(y^i_0,\dots ,y^i_{n_i})$ for all $x_i\in \mathsf {Var}$ , then from the display above we derive that $h(\phi )=g(\sigma (\phi ))$ for all $\phi \in \mathsf {Fm}$ .

Proof. We consider the left-to-right direction. Suppose that $\Theta \not \models _{\mathbf {Q}} \varepsilon \approx \delta $ . Since $\mathbf {Q}$ is core-generated, there is a core-generated expanded algebra $A \in \mathbf {Q}$ and some $h\in \mathsf {Hom}(\mathsf {Fm},A)$ such that $A\models h(\Theta )$ and $A\not \models h(\varepsilon ) \approx h(\delta )$ (recall Definition 2.17). From Lemma 3.20 it follows that there is a core assignment $g\in \mathsf {Hom}^c(\mathsf {Fm},A)$ and a substitution $\sigma $ such that $h(\phi )=g(\sigma (\phi ))$ for all $\phi \in \mathsf {Fm}$ . Then it follows that $A \models g(\sigma [\Theta ])$ but $A \not \models g(\sigma (\varepsilon \approx \delta ))$ , whence $\sigma (\Theta ) \not \models ^c_{\mathbf {Q}} \sigma (\varepsilon \approx \delta )$ .

We consider the right-to-left direction. Suppose that $\sigma (\Theta ) \not \models ^c_{\mathbf {Q}} \sigma (\varepsilon \approx \delta )$ , thus we can find an algebra $A\in \mathbf {Q}$ and an assignment $h \in \mathsf {Hom}^c(\mathsf {Fm}, A)$ such that $A \models h(\sigma [\Theta ])$ and $A \not \models h(\sigma (\varepsilon \approx \delta ))$ . Then we have that $h \circ \sigma \in \mathsf {Hom}(\mathsf {Fm},A)$ , whence $ \Theta \not \models _{\mathbf {Q}} \varepsilon \approx \delta $ .

Proof. This follows immediately from the fact that an assignment $h\in \mathsf {Hom}(\mathsf {Fm}, A)$ is a core assignment if and only if $h(x)\in \mathsf {core}(A)$ for all $x\in \mathsf {Var}$ .

Proof. For Claim (1) recall that the validity of universal Horn formulas is preserved by the operators $\mathbb {O}\in \{ \mathbb {I}, \mathbb {S},\mathbb {P},\mathbb {P}_{\mathrm {U}} \}$ . Then Lemma 3.23 implies that the validity of formulas in core semantics is preserved by $\mathbb {O}\in \{ \mathbb {I}, \mathbb {S},\mathbb {P},\mathbb {P}_{\mathrm {U}} \}$ . Preservation of validity by $\mathbb {C}$ follows immediately by the definitions of core semantics and core superstructure. Claim (2) is immediate by the definition of core semantics and the fact that, if $B\in \mathbb {H}(\mathbf {K})$ , then there is $A\in \mathbf {K}$ and a surjective homomorphism $h:A\to B$ such that $\mathsf {core}(B)=h[\mathsf {core}(A)]$ .

Proof. The direction from left-to-right follows immediately by the definition of $\mathrm {Th}_{\mathrm {qe}}^c(\mathbf {Q})$ . We consider the direction from right-to-left. Let A be core-generated and suppose $A\notin \mathbf {Q}_{\mathrm {CG}}$ . It follows that $A\notin \mathbf {Q}$ and so by Maltsev’s Theorem $A \not \models \mathrm {Th}_{\mathrm {qe}}(\mathbf {Q})$ . Let $\bigwedge _{i{\leqslant } n} \varepsilon _i \approx \delta _i \to \alpha \approx \beta \in \mathrm {Th}_{\mathrm {qe}}(\mathbf {Q})$ be such that $A\not \models \bigwedge _{i{\leqslant } n} \varepsilon _i \approx \delta _i \to \alpha \approx \beta $ . By Proposition 3.21 it follows that there is a substitution $\sigma $ such that $A\not \models ^c \sigma (\bigwedge _{i{\leqslant } n} \varepsilon _i \approx \delta _i \to \alpha \approx \beta )$ . Now, since the relation $\models _{\mathbf {Q}}$ is closed under uniform substitution, it follows in particular that $\sigma (\bigwedge _{i{\leqslant } n} \varepsilon _i \approx \delta _i \to \alpha \approx \beta ) \in \mathrm {Th}_{\mathrm {qe}}(\mathbf {Q})$ . Moreover, since obviously $\mathrm {Th}_{\mathrm {qe}}(\mathbf {Q})\subseteq \mathrm {Th}_{\mathrm {qe}}^c(\mathbf {Q}) $ , we obtain that $\sigma (\bigwedge _{i{\leqslant } n} \varepsilon _i \approx \delta _i \to \alpha \approx \beta ) \in \mathrm {Th}_{\mathrm {qe}}^c(\mathbf {Q})$ . We conclude that $A \not \models ^c \mathrm {Th}_{\mathrm {qe}}^c(\mathbf {Q})$ .

Fact 4.4. Let $\vdash $ be a standard logic, then:

1. (1) if Conditions 4.2(A1) and 4.2(A4), or 4.2(A2) and 4.2(A3) hold, then $\vdash $ is algebraized by $(\mathbf {Q}, \tau , \Delta )$ ;

2. (2) if $\vdash $ is algebraizable, then there are two finitary transformers $\tau , \Delta $ witnessing this fact.

Fact 4.5. If the tuples $(\mathbf {Q}_0, \tau _0, \Delta _0)$ and $(\mathbf {Q}_1, \tau _1, \Delta _1)$ both witness the algebraizability of a standard logic $\vdash $ , then:

1. (1) $\mathbf {Q}_0= \mathbf {Q}_1$ ;

2. (2) $\Delta _0(\varepsilon ,\delta ) \dashv \vdash \Delta _1(\varepsilon ,\delta )$ , for all $\varepsilon \approx \delta \in \mathsf {Eq}$ ;

3. (3) $\tau _0(\phi ) \equiv _{\mathbf {Q}_i} \tau _1(\phi )$ with $i\in \{0,1\}$ , for all $\phi \in \mathsf {Fm}$ .

Proof. Notice that the two witnesses of algebraizability give rise to two different consequence relations, which we shall denote by $\models ^0_{\mathbf {Q}_0}$ and $\models ^1_{\mathbf {Q}_1}$ .

Clause (2): , for all $\varepsilon \approx \delta \in \mathsf {Eq}$ . We prove $\Delta _0(\varepsilon ,\delta ) \Vdash \Delta _1(\varepsilon ,\delta ) $ . Let $\phi \in \Delta _1(\varepsilon ,\delta ) $ , then we clearly have that

$$\begin{align*}\tau_0(\phi(\varepsilon,\varepsilon)), \phi(\varepsilon,\varepsilon)\approx \phi(\varepsilon,\delta) \models^0_{\mathbf{Q}_0} \tau_0(\phi(\varepsilon,\delta)).\end{align*}$$

By 4.6(W2) it follows:

$$\begin{align*}\Delta_0(\tau_0(\phi(\varepsilon,\varepsilon))), \Delta_0(\phi(\varepsilon,\varepsilon), \phi(\varepsilon,\delta)) \Vdash \Delta_0(\tau_0(\phi(\varepsilon,\delta))),\end{align*}$$

hence, by 4.6(W3),

(a) $$ \begin{align} \phi(\varepsilon,\varepsilon), \Delta_0(\phi(\varepsilon,\varepsilon), \phi(\varepsilon,\delta)) \Vdash \phi(\varepsilon,\delta). \end{align} $$

Now, we have that $\varnothing \models ^1_{\mathbf {Q}_1} \varepsilon \approx \varepsilon $ , hence by $\phi \in \Delta _1(\varepsilon ,\varepsilon )$ and 4.6(W2), we obtain

(b) $$ \begin{align} \varnothing \Vdash \phi(\varepsilon,\varepsilon). \end{align} $$

Moreover, it also follows that $\varepsilon \approx \delta \models ^0_{\mathbf {Q}_0} \phi (\varepsilon ,\varepsilon )\approx \phi (\varepsilon ,\delta )$ , hence by 4.6(W2),

(c) $$ \begin{align} \Delta_0(\varepsilon,\delta)\Vdash \Delta_0(\phi(\varepsilon,\varepsilon),\phi(\varepsilon,\delta)). \end{align} $$

Finally, by (a)–(c), it follows that $\Delta _0(\varepsilon ,\delta ) \Vdash \phi (\varepsilon ,\delta ) $ , hence $\Delta _0(\varepsilon ,\delta ) \Vdash \Delta _1(\varepsilon ,\delta ) $ . The converse direction is proven analogously.

Clause (1): $\mathbf {Q}_0= \mathbf {Q}_1$ . We first prove that $\mathbf {Q}_0$ and $\mathbf {Q}_1$ satisfy the same quasiequations under core semantics. We show only that $\mathrm {Th}_{\mathrm {qe}}^c(\mathbf {Q}_0) \subseteq \mathrm {Th}_{\mathrm {qe}}^c(\mathbf {Q}_1)$ , as the other direction follows analogously. Let $\bigwedge _{i{\leqslant } n} \varepsilon _i \approx \delta _i \to \varepsilon \approx \delta \in \mathrm {Th}_{\mathrm {qe}}^c(\mathbf {Q}_0) $ , then it follows that $\bigwedge _{i{\leqslant } n} \varepsilon _i \approx \delta _i \models ^0_{\mathbf {Q}_0} \varepsilon \approx \delta $ and this yields that $ \bigcup _{i{\leqslant } n} \Delta _0(\varepsilon _i,\delta _i) \Vdash \Delta _0(\varepsilon , \delta )$ by 4.6(W2). By point (3) above, it follows that $\bigcup _{i{\leqslant } n} \Delta _1(\varepsilon _i,\delta _i) \Vdash \Delta _1 (\varepsilon , \delta ) $ , hence by 4.6(W2) we get $\bigwedge _{i{\leqslant } n} \varepsilon _i \approx \delta _i \models ^1_{\mathbf {Q}_1} \varepsilon \approx \delta $ . The latter entails $\bigwedge _{i{\leqslant } n} \varepsilon _i \approx \delta _i \to \varepsilon \approx \delta \in \mathrm {Th}_{\mathrm {qe}}^c(\mathbf {Q}_1) $ and thus $\mathrm {Th}_{\mathrm {qe}}^c(\mathbf {Q}_0)\subseteq \mathrm {Th}_{\mathrm {qe}}^c(\mathbf {Q}_1)$ . By reasoning analogously we obtain that $\mathrm {Th}_{\mathrm {qe}}^c(\mathbf {Q}_1)\subseteq \mathrm {Th}_{\mathrm {qe}}^c(\mathbf {Q}_0) $ , hence $\mathrm {Th}_{\mathrm {qe}}^c(\mathbf {Q}_0)= \mathrm {Th}_{\mathrm {qe}}^c(\mathbf {Q}_1) $ . It then follows by Proposition 3.26 above that $(\mathbf {Q}_0)_{\mathrm {CG}}=(\mathbf {Q}_1)_{\mathrm {CG}}$ and since both $\mathbf {Q}_0 $ and $\mathbf {Q}_1 $ are core-generated we conclude that $\mathbf {Q}_0=\mathbf {Q}_1$ .

Clause (3): $\tau _0(\phi )\equiv ^c_{\mathbf {Q}_i}\tau _1(\phi ) $ with $i\in \{0,1\}$ , for all $\phi \in \mathsf {Fm}$ . By Clause (1), it suffices to prove that $\tau _0(\phi ) \equiv _{\mathbf {Q}_0}^0 \tau _1(\phi )$ . By 4.6(W3), we have and by Clause (2) this is equivalent to . It then follows by 4.6(W2) that $\tau _0(\phi ) \equiv _{\mathbf {Q}_0}^0 \tau _1(\phi )$ .

Proof. We prove (1). Suppose $\Vdash $ is a weak logic and let $(\mathbf {Q}, \tau , \Delta )$ satisfy 4.6(W1) and 4.6(W4). We verify that 4.6(W2) and 4.6(W3) hold as well. By 4.6(W4) $\Theta \models ^c_{\mathbf {Q}} \varepsilon \approx \delta $ is equivalent to $\tau [\Delta (\Theta )]\models ^c_{\mathbf {Q}}\tau (\Delta (\varepsilon ,\delta )) $ , which by 4.6(W1) is equivalent to $\Delta (\Theta )\Vdash \Delta (\varepsilon ,\delta ) $ , proving 4.6(W2). Also, for any formula $\phi $ , we have that $ \tau (\phi )\equiv ^c_{\mathbf {Q}} \tau (\phi )$ , hence by 4.6(W4) $\tau [\Delta ( \tau (\phi ))]\equiv ^c_{\mathbf {Q}} \tau (\phi )$ and by 4.6(W1) , proving 4.6(W3). If $(\mathbf {Q}, \tau , \Delta )$ satisfies 4.6(W2) and 4.6(W3), then we proceed analogously.

We prove (2). For any $\phi \in \mathsf {Fm}$ , we have by 4.6(W3) that , thus by $\Vdash $ being finitary there is some $\tau _0(\phi )\subseteq \tau (\phi ) $ such that $|\tau _0(\phi )|<\aleph _0$ and . Moreover, for any equation $\varepsilon \approx \delta \in \mathsf {Eq}$ , we have by 4.6(W4) that $\varepsilon \approx \delta \equiv ^c_{\mathbf {Q}} \tau [ \Delta (\varepsilon ,\delta )]$ , so we obtain by finitarity a finite subset $\Delta _0(\varepsilon ,\delta ) \subseteq \Delta (\varepsilon ,\delta )$ such that $\varepsilon \approx \delta \equiv ^c_{\mathbf {Q}} \tau [ \Delta _0(\varepsilon ,\delta )]$ . Finally, it follows by the choice of $\tau _0,\Delta _0$ that , hence $\tau (\phi )\equiv ^c_{\mathbf {Q}} \tau _0(\phi )$ . Similarly, from $\tau [ \Delta (\varepsilon ,\delta )] \equiv ^c_{\mathbf {Q}} \tau [ \Delta _0(\varepsilon ,\delta )]$ we obtain . Thus $\tau _0,\Delta _0,$ and $\mathbf {Q}$ witness the algebraizability of $\Vdash $ .

Notation 4.10. If A is an $\mathcal {L}$ -algebra and $\Sigma (x)$ a set of equations in the variable x, we let $\Sigma (A)=\{a\in A : A \models \varepsilon (a) \approx \delta (a) \text { for all } \varepsilon \approx \delta \in \Sigma \}$ .

Proof. Let $\mathbf {Q}$ be a quasivariety of $\mathcal {L}$ -algebras and let $\mathbf {K}$ be the class of structures $(A,\mathsf {core}(A))$ with $A\in \mathbf {Q}$ and $ \mathsf {core}(A)=\Sigma (A) $ . Let $\{\alpha _i\approx \beta _i : i{\leqslant } n\}$ enumerate all equations in $\Sigma $ . Then consider the formulas

Clearly, both $\Psi $ and each $\Phi _i$ are universal Horn sentences, thus by Fact 2.14 they are preserved by the quasivariety operators $\mathbb {I}$ , $\mathbb {S}$ , $\mathbb {P,}$ and $\mathbb {P}_{\mathrm {U}}$ . It follows that $\mathsf {core}(A)=\Sigma (A)$ for all $A\in \mathbb {Q}(\mathbf {K})$ , and thus that $\mathbf {K}$ is already a quasivariety.

Proof. Clauses (1), (3), and (4) follow immediately from Theorem 4.7, so we prove (2). We write $\mathsf {core}_0(A)$ and $\mathsf {core}_1(A)$ for the core predicates induced by $\Sigma _0$ and $\Sigma _1$ on A, respectively. First, notice that by (1) we have $\mathbf {Q}_0 = \mathbf {Q}_1$ , so let $\mathbf {Q} = \mathbf {Q}_i$ , $i \in \{0,1\}$ . Let $\alpha _0\approx \beta _0\in \Sigma _0$ , then $\models ^{0}_{\mathbf {Q}} \alpha _0\approx \beta _0$ hence by 4.6(W2) we obtain $\varnothing \Vdash \Delta _0(\alpha _0,\beta _0)$ and thus $\varnothing \Vdash \Delta _1(\alpha _0,\beta _0)$ . It follows that $\models _{\mathbf {Q}}^{1}\alpha _0\approx \beta _0$ , meaning that $(A,\mathsf {core}_1(A)) \models \alpha _0\approx \beta _0$ . Since this holds for all equations $\alpha _0\approx \beta _0\in \Sigma _0$ , it follows that $\mathsf {core}_1(A) \subseteq \mathsf {core}_0(A)$ . The other direction is proven analogously.

Notation 4.19. Let $t(x)$ be unary $\mathcal {L}$ -term, then we define recursively $t^1(x)=t(x)$ and $t^{n+1}(x)=t(t^n(x))$ for all $1\leq n < \omega $ .

Proof. Let $\mathbf {V}$ be a variety of $\mathcal {L}$ -algebras and let $\mathbf {K}$ be the class of structures $(A,\mathsf {core}(A))$ with $A\in \mathbf {V}$ and $ \mathsf {core}(A)=\Sigma (A) $ . By Proposition 4.12 it follows that $\mathbf {K}$ is a quasivariety. We show that $\mathbf {K}$ is also closed under homomorphic images. Let $A\in \mathbf {V}$ and consider a surjective homomorphism $h:A\to B$ , it suffices to show that $\mathsf {core}(B)=h[\mathsf {core}(A)]$ . Since $\mathsf {core}(A)=\Sigma (A)$ and $\mathsf {core}(B)=\Sigma (B)$ , it follows from the fact that homomorphic images preserve the validity of equations that $h[\mathsf {core}(A)]\subseteq \mathsf {core}(B)$ . Consider now some $b\in \mathsf {core}(B)$ and notice that since h is surjective there is some $a\in A$ with $h(a)=b$ . Then we have that $h(\delta (a))=\delta (h(a))=h(a)$ , given that $h(a)=b\in \mathsf {core}(B)=\Sigma (B)$ . Since $\delta $ is a selector term, we have $A\models \delta (a)\approx \delta ^2(a)$ and so $\delta (a) \in \mathsf {core}(A)$ and $b \in h[\mathsf {core}(A)]$ .

Notation 5.2. Let $\Gamma \subseteq \mathsf {Fm}$ be a set of formulas, we let $\mathsf {At}[\Gamma ]$ be the closure of $\Gamma $ under all atomic substitutions $\sigma \in \mathsf {At}(\mathcal {L})$ . If $\Theta $ is a set of equations we similarly denote by $\mathsf {At}[\Theta ]$ the closure of $\Theta $ under atomic substitutions.

Proof. Let $(\mathbf {Q},\tau ,\Delta )$ witness the loose algebraizability of $\Vdash $ . In particular $\mathbf {Q}$ is a core-generated quasivariety of expanded algebra, and thus the collection of its algebraic reducts $\mathbf {Q}\restriction \mathcal {L}$ is a quasivariety of algebras. We claim that $(\mathbf {Q}\restriction \mathcal {L},\tau ,\Delta )$ witnesses the algebraizability of $\mathrm {Schm}(\Vdash )$ . Notice that by Fact 4.4 it suffices to verify that $(\mathbf {Q}\restriction \mathcal {L}, \tau , \Delta )$ satisfies 4.2(A1) and 4.2(A4). Firstly, we consider 4.2(A1):

$$ \begin{align*} \Gamma \Vdash_{\mathrm{s}} \phi &\Longleftrightarrow \forall \sigma \in \mathsf{Subst}(\mathcal{L}),\; \sigma[\Gamma] \Vdash \sigma(\phi) & \text{(by definition)} \\ &\Longleftrightarrow \forall \sigma \in \mathsf{Subst}(\mathcal{L}),\;\tau(\sigma[\Gamma]) \models^c_{\mathbf{Q}\restriction\mathcal{L}} \tau(\sigma(\phi)) & \text{(by 4.6(W2))} \\ &\Longleftrightarrow \forall \sigma \in \mathsf{Subst}(\mathcal{L}),\;\sigma(\tau[\Gamma]) \models^c_{\mathbf{Q}\restriction\mathcal{L}} \sigma(\tau(\phi)) & (\text{by structurality of } \tau) \\ &\Longleftrightarrow \tau[\Gamma] \models_{\mathbf{Q}\restriction\mathcal{L}} \tau(\phi) & \text{(by Proposition~3.21).} \end{align*} $$

Next, consider 4.2(A4). To this end, suppose that $\tau (\Delta (x,y)) \not \equiv _{\mathbf {Q}\restriction \mathcal {L}} x \approx y$ , then by Proposition 3.21 and the fact that $\mathbf {Q}$ is core-generated, we get that there exists a substitution $\sigma $ such that $\sigma (\tau (\Delta (x,y))) \not \equiv ^c_{\mathbf {Q}} \sigma (x \approx y)$ . Then, we obtain by the structurality of $\tau $ and $\Delta $ that $\tau (\Delta (\sigma (x), \sigma (y))) \not \equiv ^c_{\mathbf {Q}} \sigma (x) \approx \sigma (y)$ , contradicting the algebraizability of $\Vdash $ . This completes the proof.

Proof. We first show that (1) entails (2). By definition, if $\Vdash $ is strictly algebraizable, then it is also loosely algebraizable. We show that it is also finitely representable. Let $(\mathbf {Q},\Sigma ,\tau ,\Delta )$ be the strict algebraic semantics of $\Vdash $ . In particular we have that $\mathsf {core}(A)=\Sigma (A)$ for all $A\in \mathbf {Q}$ . We then have the following equivalences:

$$ \begin{align*} &\Gamma \Vdash\phi \Longleftrightarrow \tau[\Gamma] \models^c_{\mathbf{Q}} \tau(\phi) & \text{(by 4.6(W1))} \\ & \Longleftrightarrow \tau[\Gamma] \cup \mathsf{At}[\Sigma] \models_{\mathbf{Q}} \tau(\phi) & \text{(by Lemma~3.23)} \\ & \Longleftrightarrow \forall \sigma \in \mathsf{Subst}(\mathcal{L}),\; \sigma(\tau(\Gamma)) \cup \sigma(\mathsf{At}[\Sigma]) \models^c_{\mathbf{Q}} \sigma(\tau(\phi))& \text{(by Proposition~3.21)} \\ & \Longleftrightarrow \forall \sigma \in \mathsf{Subst}(\mathcal{L}),\; \tau(\sigma(\Gamma)) \cup \sigma(\mathsf{At}(\Sigma)) \models^c_{\mathbf{Q}} \tau(\sigma(\phi))& (\text{by structurality of} \tau) \\ & \Longleftrightarrow \forall \sigma \in \mathsf{Subst}(\mathcal{L}),\; \sigma[\Gamma] \cup \Delta(\sigma(\mathsf{At}(\Sigma))) \Vdash \sigma(\phi) & \text{(by 4.6(W2) and 4.6(W3))} \\ & \Longleftrightarrow \forall \sigma \in \mathsf{Subst}(\mathcal{L}),\; \sigma[\Gamma] \cup \sigma(\mathsf{At}(\Delta(\Sigma))) \Vdash \sigma(\phi) & (\text{by structurality of } \Delta) \\ & \Longleftrightarrow \Gamma \cup \mathsf{At}(\Delta(\Sigma)) \Vdash_{\mathrm{s}} \phi, & \end{align*} $$

and thus $\Lambda =\Delta (\Sigma )$ witnesses the fact that $\Vdash $ is finitely representable.

The direction from (2) to (3) follows immediately by Proposition 5.4.

Finally, we show that (3) entails (1). Suppose that $\mathrm {Schm}(\Vdash )$ is algebraized by $(\mathbf {Q}, \tau , \Delta )$ and that $\Vdash $ is finitely represented via a set of formulas $\Lambda $ . Let $\mathsf {core}(A)=\tau [\Lambda ](A)$ for all $A\in \mathbf {Q}$ and consider $\mathbf {Q'} := \mathbb {Q}(\{\langle \mathsf {core}(A) \rangle : A \in \mathbf {Q}\})$ , i.e., $\mathbf {Q'}$ is the quasivariety of expanded algebras generated by the algebras of the form $\langle \mathsf {core}(A) \rangle $ , for $A \in \mathbf {Q}$ . We then derive the following equivalences:

$$ \begin{align*} \Gamma \Vdash \phi &\iff \Gamma \cup \mathsf{At}[\Lambda] \Vdash_{\mathrm{s}} \phi & \text{(by assumption)} \\ &\iff \tau[\Gamma] \cup \tau[\mathsf{At}[\Lambda]] \models_{\mathbf{Q}} \tau(\phi)& \text{(by 4.2(A1))} \\ &\iff \tau[\Gamma] \models_{\mathbf{Q}}^c \tau(\phi). & \text{(by Lemma~3.23)} \\ &\iff \tau[\Gamma] \models_{\mathbf{Q}'}^c \tau(\phi), & \text{(by Proposition~3.24)} \end{align*} $$

which establish 4.6(W1). We next verify that 4.6(W4) holds as well. Suppose that $\tau (\Delta (x,y)) \not \equiv ^c_{\mathbf {Q'}} x \approx y$ , then since $\mathsf {core}(A)=\tau [\Lambda ](A)$ we obtain that

$$\begin{align*}\tau(\Delta(x,y) ) \cup \tau[\Lambda] \not \equiv_{\mathbf{Q'}} \{ x \approx y\} \cup \tau[\Lambda], \end{align*}$$

which contradicts the fact that $\mathrm {Schm}(\Vdash )$ is algebraized by $(\mathbf {Q}, \tau , \Delta )$ .

Proof. We first prove the left-to-right direction. Since $\vdash \in \mathrm {St}(\Vdash )$ we have by definition that $\vdash $ is algebraized by some $(\mathbf {K},\tau ,\Delta )$ , where $\tau $ and $\Delta $ also witness the strict algebraizability of $\Vdash $ . In particular, notice that by the strict algebraizability of $\Vdash $ and Corollary 5.6 we have that $\Vdash $ is represented by $\Delta [\Sigma ]$ over its schematic fragment, i.e., that $\Lambda =\Delta [\Sigma ]$ . Now, we consider $\mathbf {K}$ as a class of expanded algebras with core $\Sigma (A)$ for all $A\in \mathbf {K}$ and we let $\mathbf {C}=\mathbb {Q}(\{ \langle \Sigma (A)\rangle : A\in \mathbf {K} \})$ . Clearly, we have by Proposition 3.26 that $\Theta \models ^c_{\mathbf {C}}\varepsilon \approx \delta $ holds if and only if $\Theta \models ^c_{\mathbf {K}}\varepsilon \approx \delta $ does. Also, we have by strict algebraizability of $\Vdash $ and $\vdash $ that $\Theta \models ^c_{\mathbf {K}}\varepsilon \approx \delta $ is equivalent to $\Theta \models ^c_{\mathbf {Q}}\varepsilon \approx \delta $ , since $\vdash \in \mathrm {St}(\Vdash )$ and $\Lambda =\Delta [\Sigma ]$ . Thus we obtain that $\mathrm {Th}_{\mathrm {qe}}^c(\mathbf {C})=\mathrm {Th}_{\mathrm {qe}}^c(\mathbf {Q})$ and so by Proposition 3.26 $\mathbf {C}=\mathbf {Q}$ , proving that $\mathbf {K}$ is a standard companion of $\mathbf {Q}$ .

We consider the right-to-left direction. Suppose $\vdash $ is algebraized by $(\mathbf {K},\tau ,\Delta )$ and let $\mathbf {K}$ be a standard companion of $\mathbf {Q}$ , we show that $\vdash $ is a standard companion of $\Vdash $ . We view $\mathbf {K}$ also as a quasivariety of expanded algebras by letting, $\mathsf {core}(A)=\Sigma (A)$ for each $A\in \mathbf {K}$ . We have the following equivalences:

$$ \begin{align*} \Gamma \Vdash \phi & \Longleftrightarrow \tau(\Gamma) \models^c_{\mathbf{Q}} \tau(\phi) & \text{(by 4.6(W1))} \\ & \Longleftrightarrow \tau(\Gamma) \models^c_{\mathbf{K}} \tau(\phi) & \text{(by Definition~5.9 and Proposition~3.26)} \\ & \Longleftrightarrow \tau(\Gamma) \cup \mathsf{At}[\Sigma] \models_{\mathbf{K}} \tau(\phi) & \text{(by Lemma~3.23)} \\ & \Longleftrightarrow \Gamma \cup \Delta(\mathsf{At}[\Sigma]) \vdash \phi & \text{(by 4.2(A2) and 4.2(A3))} \\ & \Longleftrightarrow \Gamma \cup \mathsf{At}(\Delta[\Sigma]) \vdash \phi & (\text{by structurality of } \Delta) \end{align*} $$

and thus, since $\Lambda =\Delta [\Sigma ]$ (as we argued in the previous direction), it follows that $\vdash $ is a standard companion of $\Vdash $ . This completes our proof.

Proof. We notice that $A\in \mathbb {C}_\Sigma (\mathbf {K})$ if and only if $A\models \forall x(\mathsf {core}(x)\leftrightarrow \bigwedge \Sigma (x))$ , which can be rephrased by means of Horn formulas as in Proposition 4.12. Also, for all quasiequations $\bigwedge _{i{\leqslant } n} \varepsilon _i \approx \delta _i \to \varepsilon \approx \delta \in \mathrm {Th}_{\mathrm {qe}}^c(\mathbf {K})$ we have that this is equivalent to saying

$$\begin{align*}A\models \forall x_1,\dots \forall x_n ( \bigwedge_{1{\leqslant} i{\leqslant} n}\mathsf{core}(x_i) \land \bigwedge_{i{\leqslant} n} \varepsilon_i \approx \delta_i \to \varepsilon\approx \delta ), \end{align*}$$

where these are also Horn formulas. It follows immediately by Fact 2.14 that $\mathbb {C}_\Sigma (\mathbf {K})$ is closed under $ \mathbb {I} $ , $ \mathbb {S} $ , $ \mathbb {P} $ , and $ \mathbb {P}_{\mathrm {U}} $ .

Notation 5.13. Let $\tau :\mathsf {Fm}\to \mathsf {Eq}$ and let $\mathbf {Q}$ be a quasivariety, then we write $\mathsf {Log}_\tau (\mathbf {Q})$ for the set of all pairs $(\Gamma ,\phi )$ such that $\tau (\Gamma )\models _{\mathbf {Q}} \tau (\phi )$ .

Proof. Let $\vdash _0,\vdash _1 \in \mathrm {St}(\Vdash )$ , then $\vdash _0\cap \vdash _1$ is a standard logic and it is straightforward to verify that $\vdash _0\cap \vdash _1\in \mathrm {St}(\Vdash )$ . Similarly, if $\vdash _0,\vdash _1 \in \mathrm {St}(\Vdash )$ and they are algebraized, respectively, by $(\mathbf {K}_0,\tau ,\Delta )$ and $(\mathbf {K}_1,\tau ,\Delta )$ , then $\mathbf {K}_0\cap \mathbf {K}_1$ is a quasivariety, and the logic $\vdash _2$ defined by letting

$$\begin{align*}\Gamma\vdash_2 \phi \;\Longleftrightarrow\; \tau(\Gamma)\models^c_{\mathbf{K}_0\cap \mathbf{K}_1} \tau(\phi) \end{align*}$$

is the supremum of $\vdash _0$ and $\vdash _1$ in $\mathrm {St}(\Vdash )$ . This shows that $(\mathrm {St}(\Vdash ), \subseteq )$ is a lattice.

We show that $\mathrm {Schm}(\Vdash )$ is maximal in $\mathrm {St}(\Vdash )$ . Recall that, by Corollary 5.6, we have that $\mathrm {Schm}(\Vdash )=\mathsf {Log}_\tau (\mathbf {Q})$ . Suppose now that $\vdash \;\in \mathrm {St}(\Vdash )$ then by monotonicity $\Gamma \vdash \phi $ entails $\Gamma \cup \Lambda \vdash \phi $ , and thus we have $\Gamma \Vdash \phi $ . This shows that $\vdash \;\subseteq \; \Vdash $ . Since $\vdash $ and $\mathrm {Schm}(\Vdash )$ are both closed under uniform substitution, it follows that $\vdash \subseteq \mathrm {Schm}(\Vdash )$ .

We show that $\mathsf {Log}_\tau (\mathbb {C}_\Sigma (\mathbf {Q}))$ is minimal in $\mathrm {St}(\Vdash )$ . If $\vdash \; \in \mathrm {St}(\Vdash )$ , then by Lemma 5.10 $\vdash $ is algebraized by $(\mathbf {K},\tau ,\Delta )$ , for some standard companion $\mathbf {K}$ of $\mathbf {Q}$ . By Definition 5.11, it follows that $\mathbf {K}\subseteq \mathbb {C}_{\Sigma }(\mathbf {Q})$ and so $\mathsf {Log}_\tau (\mathbb {C}_\Sigma (\mathbf {Q}))\subseteq \mathsf {Log}_\tau (\mathbf {K})=\; \vdash $ .

Fact 5.18 (Blok–Pigozzi).

Let $\vdash $ be a standard algebraizable logic with equivalent algebraic semantics $\mathbf {Q}$ , then $\vdash $ has a deduction–detachment theorem if and only if $\mathbf {Q}$ has EDPRC.

Proof. We first prove the right-to-left direction. Suppose $\Vdash _{\mathrm {s}}$ has DDT, and notice that since $\Vdash $ is strictly algebraizable then it is finitely represented by a set of formulas $\Lambda $ (by Theorem 5.5). We have the following equivalences:

$$ \begin{align*} \Gamma\cup \{\phi\}\Vdash \psi &\Longleftrightarrow \Gamma\cup \{\phi\}\cup \mathsf{At}[\Lambda]\Vdash_{\mathrm{s}} \psi & \text{(by representability)} \\ &\Longleftrightarrow \Gamma\cup \mathsf{At}[\Lambda]\Vdash_{\mathrm{s}} I(\phi,\psi) & \text{(by assumption)} \\ &\Longleftrightarrow \Gamma \Vdash I(\phi,\psi), & \text{(by representability)} \end{align*} $$

which establish that $\Vdash $ has DDT.

Consider now the left-to-right direction. We assume that $\Vdash $ has DDT and that this is witnessed by the finite set of formulas $I(x,y)$ , then:

$$ \begin{align*} \Gamma\cup \{\phi\}\Vdash_{\mathrm{s}} \psi &\Longleftrightarrow \forall\sigma\in\mathsf{Subst}(\mathcal{L}), \; \sigma(\Gamma\cup \{\phi\})\Vdash \sigma(\psi) & \text{(by Definition~5.1)} \\ &\Longleftrightarrow \forall\sigma\in\mathsf{Subst}(\mathcal{L}), \; \sigma(\Gamma)\Vdash I(\sigma(\phi),\sigma(\psi)) & \text{(by assumption)} \\ &\Longleftrightarrow \Gamma \Vdash_{\mathrm{s}} I(\phi,\psi), & \text{(by Definition~5.1)} \end{align*} $$

which proves our claim.

Proof. Immediate from Fact 5.18 and Proposition 5.19.

Notation 6.3. Given any algebra A we let $\mathsf {Con}(A)$ be the set of all congruences $\theta $ over A, and $\mathsf {Con}_{\mathbf {Q}}(A)$ be the set of all $\mathbf {Q}$ -congruences of A, i.e., those congruences $\theta $ over A such that $A/\theta \in \mathbf {Q}$ .

Fact 6.7 (Isomorphism theorem).

Let $\vdash $ be a standard logic and $\mathbf {Q}$ a quasivariety, then the following are equivalent:

1. (1) $\vdash $ is algebraizable with equivalent algebraic semantics $\mathbf {Q}$ ;

2. (2) $\mathsf {Fi}^+_{\vdash }(A) \cong \mathsf {Con}^+_{\mathbf {Q}}(A)$ , for any algebra A;

3. (3) $\mathsf {Th}^+(\vdash ) \cong \mathsf {Th}^+(\models _{\mathbf {Q}})$ .

Proof. Clause (2) follows immediately from the definition of core $\mathbf {Q}$ -congruence. We thus consider clause (1). Let $G\subseteq A$ be a core filter of $\Vdash $ , and let $\Gamma \Vdash _{\mathrm {s}} \phi $ , $h[\Gamma ]\subseteq G$ for some $h\in \mathsf {Hom}(\mathsf {Fm},A)$ . Now, since A is core-generated, by Lemma 3.20 we can find a substitution $\sigma $ and a core assignment g, such that $g(\sigma (x))=h(x)$ for all $x\in \mathsf {Var}$ . In particular, it follows that $g[\sigma [\Gamma ]]\subseteq G$ . Now, since $\Gamma \Vdash _{\mathrm {s}} \phi $ , it follows by uniform substitution that $\sigma [\Gamma ]\Vdash _{\mathrm {s}} \sigma (\phi )$ and therefore $\sigma [\Gamma ]\Vdash \sigma (\phi )$ . Now, since g is a core assignment and G a core filter with $g[\sigma [\Gamma ]]\subseteq G$ , it follows that $g(\sigma (\phi ))\in G$ , whence $h(\phi )\in G$ . This shows that G is a deductive filter with respect to $\Vdash _{\mathrm {s}}$ .

Proof. We already pointed out in Remark 6.12 that $\mathsf {Fi}_{\Vdash }(A)$ and $\mathsf {Con}^c_{\mathbf {Q}}(A)$ are (complete) lattices under the subset ordering, whence we simply prove (1) and (2).

We prove (1). Suppose $\Gamma \Vdash \phi $ and let $ g\in \mathsf {Hom}^c(\mathsf {Fm}, A) $ be such that $g[\Gamma ]\subseteq h^{-1}(F)$ . Then $h(g[\Gamma ])\subseteq F$ and, since $h\circ g\in \mathsf {Hom}^c(\mathsf {Fm}, A)$ , it follows that $h(g[\phi ])\subseteq F$ and thus $g(\phi )\in h^{-1}(F)$ .

We prove (2). The fact that $h^{-1}(\theta )\in \mathsf {Con}(A)$ follows from Remark 6.4. We next show that $A/h^{-1}(\theta )\in \mathbf {Q}$ . Consider a quasiequation $\bigwedge _{i{\leqslant } n} \varepsilon _i \approx \delta _i \to \alpha \approx \beta \in \mathrm {Th}_{\mathrm {qe}}^c(\mathbf {Q})$ . Since $A/\theta \in \mathbf {Q}$ we have that $A/\theta \models ^c \bigwedge _{i{\leqslant } n} \varepsilon _i \approx \delta _i \to \alpha \approx \beta $ . Now, let $g\in \mathsf {Hom}^c(\mathsf {Fm}, A) $ be such that $g(\varepsilon _i)/h^{-1}(\theta ) = g(\delta _i)/h^{-1}(\theta )$ for all $i{\leqslant } n$ , then it follows that $h\circ g(\varepsilon _i)/\theta = h\circ g(\delta _i)/\theta $ for all $i{\leqslant } n$ . Since h is a strong endomorphism of A we obtain that $h\circ g(\alpha )/\theta = h\circ g(\beta )/\theta $ . It then follows that $ g(\alpha )/h^{-1}(\theta ) = g(\beta )/h^{-1}(\theta )$ . This shows that $A/h^{-1}(\theta )\models ^c \bigwedge _{i{\leqslant } n} \varepsilon _i \approx \delta _i \to \alpha \approx \beta $ and so $A/h^{-1}(\theta )\models ^c \mathrm {Th}_{\mathrm {qe}}^c(\mathbf {Q})$ . Finally, notice that since A is core generated, also $A/h^{-1}(\theta )$ is core generated, and thus we have from Proposition 3.26 that $A/h^{-1}(\theta ) \in \mathbf {Q}$ . We conclude that $h^{-1}(\theta )\in \mathsf {Con}^c_{\mathbf {Q}}(A)$ .

Proof. Firstly, we define the following map from filters to congruences:

and the following map from congruences to filters:

Now, notice that since $\Vdash $ is loosely algebraized by $(\mathbf {Q},\tau ,\Delta )$ , then it follows from Proposition 5.4 that also $\Vdash _{\mathrm {s}}$ is algebraized by $(\mathbf {Q},\tau ,\Delta )$ . Since the two maps above are exactly the same that occur in the proof of the isomorphism theorem for standard logics (cf. Remark 6.8), it follows immediately that they describe an isomorphism $\mathsf {Fi}^+_{\Vdash _{\mathrm {s}}}(A) \cong \mathsf {Con}^+_{\mathbf {Q}}(A)$ . Thus, it suffices to show that $\theta _{(-)}$ sends core filters to core congruences, and that $F_{(-)}$ sends core congruences to core filters.

First, let G be a core filter over A, we show that $\theta _G$ is a core- $\mathbf {Q}$ -congruence of A. Consider $A/\theta _G$ and notice that this is well-defined because $\theta _G$ is a congruence. Now, consider the expansion of $A/\theta _G$ defined by letting $\mathsf {core}(A/\theta _G)=\mathsf {core}(A)/\theta _G$ , we claim that $(A/\theta _G, \mathsf {core}(A)/\theta _G)\in \mathbf {Q}$ . Let $\bigwedge _{i{\leqslant } n} \varepsilon _i \approx \delta _i \models ^c_{\mathbf {Q}} \alpha \approx \beta $ and suppose $h\in \mathsf {Hom}^c(\mathsf {Fm},A/\theta _G)$ is such that $h(\epsilon _i)=h(\delta _i)$ for all $i{\leqslant } n$ . By algebraizability we have that $\bigcup _{i{\leqslant } n}\Delta (\varepsilon _i \approx \delta _i) \Vdash \Delta (\alpha ,\beta )$ and so $h(\Delta (\alpha ,\beta ))\subseteq G$ . It follows that $(h(\alpha ),h(\beta ))\in \theta _G$ and so $A/\theta _G\models ^c \alpha \approx \beta $ . This shows that $(A/\theta _G, \mathsf {core}(A)/\theta _G)\models ^c \bigwedge _{i{\leqslant } n} \varepsilon _i \approx \delta _i \to \alpha \approx \beta $ . Since $\mathbf {Q}$ is a core-generated quasivariety of expanded algebras, it follows from Proposition 3.26 that $(A/\theta _G, \mathsf {core}(A)/\theta _G)\in \mathbf {Q}$ . We conclude that the map $\theta _{(-)}$ sends core filters to core- $\mathbf {Q}$ -congruences.

Next, we show that for any core- $\mathbf {Q}$ -congruence $\theta $ over A the filter $F_\theta $ is a core filter. Suppose $\Gamma \Vdash \phi $ and $ h\in \mathsf {Hom}^c(\mathsf {Fm}, A) $ is such that $h[\Gamma ]\subseteq F_\theta $ . Then, we have by definition that $\tau ^A(c)\subseteq \theta $ for every $c\in h[\Gamma ]$ . By algebraizability, notice that we have $\tau (\Gamma )\models ^c_{\mathbf {Q}}\tau (\phi )$ , thus we obtain $h(\tau (\phi ))=\tau ^A(h(\phi ))\subseteq \theta $ and so $h(\phi )\in F_\theta $ . This shows that $F_\theta $ is a core deductive filter and completes the proof.