Abstract. We present short single axioms for ortholattices, orthomodular lattices, and modular ortholattices, all in terms of the Sheffer stroke. The ortholattice axiom is the shortest possible. We also give multiequation bases in terms of the Sheffer stroke and in terms of join, meet, and complementation. Proofs are omitted but are available in an associated technical report and on the Web. We used computers extensively to find candidates, reject candidates, and search for proofs that candidates are single axioms. 1.

Short single axioms for ortholattices, orthomodular lattices, and modular ortholattices are presented, all in terms of the Sheffer stroke. The ortholattice axiom is the shortest possible. Other equational bases in terms of the Sheffer stroke and in terms of join, meet, and complement are presented. Proofs are omitted but are available in an associated technical report. Computers were used extensively to find candidates, reject candidates, and search for proofs that candidates are single axioms. The notion of computer proof is addressed. 1

Abstract. We give three examples of cofinal intervals in the lattice of (local) clones on an infinite set X, whose structure is on the one hand non-trivial but on the other hand reasonably well understood. Specifically, we will exhibit clones C1, C2, C3 such that: (1) the interval [C1, O] in the lattice of local clones is (as a lattice) isomorphic to {0, 1, 2,...} under the divisibility relation, (2) the interval [C2, O] in the lattice of local clones is isomorphic to the congruence lattice of an arbitrary semilattice, (3) the interval [C3, O] in the lattice of all clones is isomorphic to the lattice of all filters on X. 747 revision:2009-04-30 modified:2009-04-30 1.

Abstract. We give an overview of decidability results for modal logics having a binary modality. We put an emphasis on the demonstration of proof-techniques, and hope that this will also help in finding the borderlines between decidable and undecidable fragments of usual first-order logic. Key words: poly-modal and multi-modal logics, decision problems, Arrow logics, algebraic logic, relation algebra, associativity, dynamic logics, action algebras, Boolean algebras with operators We investigate here decidability problems concerning logics having an extra binary connective "o " beside the Boolean ones. These logics are strongly related to ordinary first-order logic, see Henkin et al. (1985, ch. 5.3) on this connection in an algebraic setting. Our most important aims are to give a transparent overview of the results and to stress the crucial points and ideas of the proofs, especially when the extra binary connective is associative 9 The emphasis is on those parts of the proof methods which have been well known for the specialists. (The reason for this is the didactic character of the present paper 9 For the details of those ideas which are our own contributions we give references to more technical papers. Since associativity of"o " corresponds to commutativity of first-order existential quantifiers in some sense (cf. op cit), our results and techniques can help to find decidable fragments of first-order logic as well (for such results see N6meti (1985, 1987, 1992)) 9