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Judit X. Madarász

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Interpolation in Algebraizable Logics;
Semantics for Non-normal Multi-modal Logic.

Abstract.
The two main directions pursued in the present paper are the following.

The first direction was (perhaps) started by Pigozzi in 1969. Maksimova [M91,
M79] proved that a normal modal logic (with a single unary modality) has the
Craig interpolation property iff the corresponding class of algebras has the
superamalgamation property. In this paper we extend Maksimova's theorem to
normal multi-modal logics with arbitrarily many, not necessarily unary
modalities, and to not necessarily normal multi-modal logics with modalities
of ranks smaller than 2.

To extend the characterization beyond multi-modal logics, we look at arbitrary
algebraizable logics. We will introduce an algebraic property equivalent with
the Craig interpolation property in algebraizable (and in strongly nice)
logics, and prove that the superamalgamation property implies the Craig
interpolation property. The problem of extending the characterization to
non-normal non-unary modal logics will be discussed, too.

On the second direction pursued herein: For non-normal modal logic with one
unary modality Lemmon [L66] gave a possible worlds semantics. Here we give a
more general possible worlds semantics for not necessarily normal multi-modal
logics with arbitrarily many not necessarily unary modalities. Strongly
related to the above is the theorem, proved e.g. in J\'onsson-Tarski~ [JT52]
and Henkin-Monk-Tarski~ [HMT71], that every normal Boolean algebra with
operators (bao) can be represented as a subalgebra of the complex algebra of
some relational structure. We extend this result to not necessarily normal
bao's as follows. We define partial relational structures and show that every
not necessarily normal bao is embeddable into the complex algebra of a partial
relational structure. This gives a possible worlds semantics for not
necessarily normal multi-modal logics (with arbitrarily many, not necessarily
unary modalities).

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