Universal Algebraic Logic
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H. Andréka, Z. Gyenis, I. Németi, and I. Sain,
Universal Algebraic Logic,
Birkhauser, 2022
ERRATA:
 p. 334, l.10: in the second line of the proof of 2.2.8, Sg should be fraktur.
 p. 153, line 3 of the proof of Claim 3.3.42: "powerset of F^P" should be "powerset of A". Two lines later: h_r instead of h.
 p. 213, Theorem 4.4.12: First order logic with n variables L_n does NOT have the weak interpolation property (for n>1). The proof is similar to the one in Example 4.4.4 on p. 210.
RELATED
 Here are some more details about Definition 3.3.27(6). This item of the definition is used slightly informally in Claim 3.3.42, here is what to do to make it precise (a general lemma that can be used everywhere when 3.3.27(6) is referred to).
 Preprints of the Algebraic Logic Group at the Renyi Institute; Papers in Algebra, Logic and Algebraic Logic by Hajnal Andréka and István Németi.

A recent paper around Theorem 4.4.18 and Problem 4.4.20:
Amalgamation and Robinson property in Universal Algebraic Logic 
Papers on local and modelwise interpolation property. See subsection 4.4.1. Theorem 4.4.12 is stated in the correct version. Problem 4.4.30 is touched upon.
The Modelwise Interpolation Property of Semantic Logics
Algebraic characterization of the local Craig interpolation property