This paper gives algebraic characterizations of Halld\'{e}n completeness, and of Maksimova's variable separation property and its deductive form. Though algebraic characterizations of these properties have been already studied for modal and superintuitionistic logics, e.g. in Wro\'{n}ski [12], pp.126--129), Maksimova [7], pp.168--184), [9], pp.99--112), a deeper analysis of these properties and non-trivial modifications of these results are needed to extend them to those for substructural logics, because of the lack of some structural rules in them. The first attempt in this direction (...) was made in the dissertation \cite{Kih06} of the first author. Results of this paper are partly announced also in Chapter 5 of the book [2]. (shrink)

In this paper we find an algebraic equivalent of the Hallden property in modal logics, namely, we prove that the Hallden-completeness in any normal modal logic is equivalent to the so-called super-embedding property of a suitable class of modal algebras. The joint embedding property of a class of algebras is equivalent to the Pseudo-Relevance Property. We consider connections of the above-mentioned properties with interpolation and amalgamation. Also an algebraic equivalent of of the principle of variable separation in superintuitionistic logics will (...) be found. (shrink)