Pure Hilbert Algebras with Infimum
In [6], iH-algebras were introduced in order to indicate an equational version of the class of Hilbert algebras where each pair of elements has infimum. These authors also proved that this variety hasthe class of Curry's implicative semilattices ([9]) as a proper subvariety. On the other hand, in [4] a special class of Hilbert algebras associated with ordered sets, which they called order algebras, were investigated. These algebras were also studied in [1] under the name of pure Hilbert algebras.Bearing in mind the above results, in this paper we introduce the notion of pure Hilbert algebras with infimum (or ipH-algebras, for short). Furthermore, we characterize the lattice of ipH-congruences and we determine the subdirectly irreducible ipH-algebras. Besides, we prove that subdirectly irreducible ipH-algebras are alsosubdirectly irreducible iH-algebras.
Prior’s OIC nonconservativity example revisited
In his 1964 note, ‘Two Additions to Positive Implication’, A. N. Prior showed that standard axioms governing conjunction yield a nonconservative extension of the pure implicational intermediate logic OIC (order implicational calculus) of R. A. Bull. Here, after reviewing the situation with the aid of an adapted form of the Kripke semantics for intuitionistic and intermediate logics, we proceed to illuminate this example by transposing it to the setting of modal logic, and then relate it to the propositional logic of what have been called ‘Hilbert algebras with infimum’.
Struktur- und anzahlformeln für topologien auf endlichen mengen
Every topology τ on the finite set X={x1,...,xN} has a minimal base Μ(τ); separation axioms and connectivity are simply characterized; further we obtain a recursive construction of finite topologies.In the second section we give some formulae connecting A(N) (the number of topologies on X), Z(N) (the number of connected topologies on X) and associated numbers. The theory of representation matrices discussed in the third section leads to an easy description of many topological notions as closure operator, induced topology, connectivity; it also yields some more combinatorial formulae.In the last part we improve the remainder term R(N)=Id A(N)−1/4(N)2=0(N3/2ld N) given by Kleitman and Rothschild to 0(N·ld2N). Then, among many other asymptotical results, we show that A(N) and Z(N) are asymptotically equal.