  It is customary to expect from a logical system that it can be algebraizable,
in the sense that an algebraic companion of the deductive machinery can always
be found. Since the inception of da Costa's paraconsistent calculi $C_n$,
algebraic equivalents for such systems have been sought. It is known, however,
that these systems are not self-extensional (i.e., they do not satisfy the
replacement property). More than this, they are not algebraizable in the sense
of Blok-Pigozzi. The same negative results hold for several systems of the
hierarchy of paraconsistent logics known as Logics of Formal Inconsistency
(LFIs). Because of this, several systems belonging to this class of logics are
only characterizable by semantics of a non-deterministic nature. This paper
offers a solution for two open problems in the domain of paraconsistency, in
particular connected to algebraization of LFIs, by extending with rules several
LFIs weaker than $C_1$ , thus obtaining the replacement property (that is, such
LFIs turn out to be self-extensional). Moreover, these logics become
algebraizable in the standard Lindenbaum-Tarski's sense by a suitable variety
of Boolean algebras extended with additional operations. The weakest LFI
satisfying replacement presented here is called RmbC, which is obtained from
the basic LFI called mbC. Some axiomatic extensions of RmbC are also studied.
In addition, a neighborhood semantics is defined for such systems. It is shown
that RmbC can be defined within the minimal bimodal non-normal logic E+E
defined by the fusion of the non-normal modal logic E with itself. Finally, the
framework is extended to first-order languages. RQmbC, the quantified extension
of RmbC, is shown to be sound and complete w.r.t. the proposed algebraic
semantics.
