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We present a logical and algebraic description of right adjoint functors between generalized quasi-varieties, inspired by the work of McKenzie on category equivalence. This result is achieved by developing a correspondence between the concept of adjunction and a new notion of translation between relative equational consequences.
Abstract algebraic logic is a theory that provides general tools for the algebraic study of arbitrary propositional logics. According to this theory, every logic
is associated with a matrix semantics
. This article is a contribution to the systematic study of the so-called truth sets of the matrices in
. In particular, we show that the fact that the truth sets of
can be defined by means of equations with universally quantified parameters is captured by an order-theoretic property of the Leibniz operator restricted to deductive filters of
. This result was previously known for equational definability without parameters. Similarly, it was known that the truth sets of
are implicitly definable if and only if the Leibniz operator is injective on deductive filters of
over every algebra. However, it was an open problem whether the injectivity of the Leibniz operator transfers from the theories of
to its deductive filters over arbitrary algebras. We show that this is the case for logics expressed in a countable language, and that it need not be true in general. Finally we consider an intermediate condition on the truth sets in
that corresponds to the order-reflection of the Leibniz operator.
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