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It was noticed that there is a misprint in the initial of the name of one of the authors in the original online version of the paper.

A meet semilattice with a partial join operation satisfying some axioms is a JP-semilattice. A PJP-semilattice is a pseudocomplemented JP-semilattice. We characterize Stone JP-semilattices, give some characterizations of kernel ideals of a Stone JP-semilattice, and describe the join of two kernel ideals of a Stone JP-semilattice.

We introduce an alternative definition of fibrewise uniformity and discuss consequences deduced from new axioms. By modifying James’ definition of fibrewise uniform structure, which is a slightly strengthened one, we define a new fibrewise uniformity which is symmetric in global and realizes 1-1 correspondence between fibrewise entourage uniformities and fibrewise covering uniformities. Moreover, we obtain a characterization of the fibrewise completion of fibrewise generalized uniform space as a fibrewise extension of a fibrewise space. As an application of the fibrewise completion theory, we show that there exists a fibrewise Shanin compactification of a fibrewise space.

Monotone normality is usually defined in the class of T1 spaces. In this paper new characterizations of monotone normality, free of T1 axiom, are provided and it is shown that in this context it is not a hereditary property. Also, a Tietze-type extension theorem for lattice-valued functions for this class of spaces is given.

We prove that axiom CPAgameprism, which follows from the Covering Property Axiom CPA and holds in the iterated perfect set model, implies that there exists a Hamel basis which is a union of less than continuum many pairwise disjoint perfect sets. We will also give two consequences of this last fact.

A Riemannian manifold satisfies the axiom of 2-planes if at each point, there are suficiently many totally geodesic surfaces passing through that point. Real hypersurfaces in quaternionic space forms admit nice families of tangent planes, namely, totally real, half-quaternionic and quaternionic. Several definitions of axiom of planes arise naturally when we consider such families of tangent planes. We are able to classify real hypersurfaces in quaternionic space forms satisfying these definitions.

Let (M, E) be a Finsler manifold. A triplet (¯D, ¯h, α) is said to be a Wagner connection on M if (¯D, ¯h) is a Finsler connection, α ∈ C ∞ (M) and the axioms (W1)–(W4), formulated originally by M. Hashiguchi, are satisfied. Then ¯h is called a Wagner endomorphism on M. We establish an explicit relation between the (canonical) Barthel endomorphism of (M, E) and a Wagner endomorphism ¯h. We show that the second Cartan tensors ¯C′, ¯C′ b belonging to ¯h are symmetric and totally symmetric, respectively. An explicit relation between the "canonical" tensors C′, C′ b and the "Wagnerian" ones is also derived. We can conclude that the rules of calculation with respect to a Wagner connection are formally the same as those with respect to the classical Cartan connection. We establish some basic curvature identities concerning a Wagner connection, including Bianchi identities. Finally, we present a new, intrinsic definition as well as several tensorial characterizations of Wagner manifolds.