The Birkhoff-Maltsev problem asks for a characterization of those lattices each of which is isomorphic to the lattice L(K) of all subquasivarieties for some quasivariety K of algebraic systems. The current status of this problem, which is still open, is discussed. Various unsolved questions that are related to the Birkhoff-Maltsev problem are also considered, including ones that stem from the theory of propositional logics.Having whetted the reader's appetite in the preface to this special issue by claiming that much remains to be done in the theory of quasivarieties, we feel some responsibility to justify our claim. To do so, we will discuss the current status of the Birkhoff-Maltsev problem and consider open questions related to it.

Modes are idempotent and entropic algebras. Although it had been established many years ago that groupoid modes embed as subreducts of semimodules over commutative semirings, the general embeddability question remained open until Stronkowski and Stanovský's recent constructions of isolated examples of modes without such an embedding. The current paper now presents a broad class of modes that are not embeddable into semimodules, including structural investigations and an analysis of the lattice of varieties.

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Combination of sliding window method with physical properties scale of amino acids is a classical approach for linear B-cell epitope prediction. But it was shown that accuracy of these methods is poor. We reviewed classical and new algorithms of epitope prediction and present own implementation of one of them. The AAPPred software is available online at http://www.bioinf.ru/aappred/.

We present some equivalent conditions for a quasivariety K of structures to be generated by a single structure. The first such condition, called the embedding property was found by A.I. Mal tsev in [6]. It says that if A, B ∈ K are nontrivial, then there exists C ∈ K such that A and B are embeddable into C. One of our equivalent conditions states that the set of quasi-identities valid in K is closed under a certain Gentzen type rule which is due to J. Loś and R. Suszko [5].