We compute the expected numbers of both formal concepts and proper premises in a formal context that is chosen uniformly at random among all formal contexts of given dimensions.

This report discusses clones with nullary operations and the corresponding relational clones, both defined on arbitrary non-empty sets. The relationship between such clones and clones in the usual sense, i.e. without nullary operations, is investigated, and in particular the latter type of clones is located in the lattice of all clones. By means of two pairs of kernel and closure operators, a framework is developed that allows to transfer statements about usual clones to statements about clones with nullary constants. In this respect, familiar operators and constructions from clone theory, like the operators Pol and Inv, the closure operators belonging to the clone lattices, and the different variants of local closure operators on sets of relations and operations, respectively, are translated from the usual setting to the more general one and vice versa. The applicability of the presented machinery is demonstrated using the example of the theorem characterising Galois closed sets w.r.t. Pol-Inv as local closures of clones and relational clones, respectively.

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In this article we establish a bridge between dynamical systems, including topological and measurable dynamical systems as well as continuous skew product flows and nonautonomous dynamical systems; and coalgebras in categories having all finite products. We introduce a straightforward unifying definition of abstract dynamical system on finite product categories. Furthermore, we prove that such systems are in a unique correspondence with monadic algebras whose signature functor takes products with the time space. We substantiate that the categories of topological spaces, metrisable and uniformisable spaces have exponential objects w.r.t. locally compact Hausdorff, σ-compact or arbitrary time spaces as exponents, respectively. Exploiting the adjunction between taking products and exponential objects, we demonstrate a one-to-one correspondence between monadic algebras (given by dynamical systems) for the left-adjoint functor and comonadic coalgebras for the other. This, finally, provides a new, alternative perspective on dynamical systems.

A restricted version of the Galois connection between polymorphisms and invariants, called Pol−CInv, is studied, where the invariant relations are restricted to so-called clausal relations. In this context, the relationship of maximal C-clones and maximal clones is investigated. It is shown that, with the exception of one special case occurring for Boolean domains, maximal C-clones are never maximal clones. / Wir untersuchen eine eingeschränkte Variante der Galoisverbindung zwischen Polymorphismen und invarianten Relationen, bezeichnet mit Pol−CInv, wobei die invarianten Relationen auf sogenannte klausale Relationen beschränkt werden. In diesem Zusammenhang wird die Beziehung zwischen maximalen C-Klonen und maximalen Klonen betrachtet. Es wird gezeigt, daß, mit Ausnahme eines Spezialfalles für Boolesche Grundmengen, maximale C-Klone niemals maximale Klone sind.