For a closure space (P,φ) with φ(∅) = ∅, the closures of open subsets of P , called the regular closed subsets, form an ortholattice Reg(P,φ), extending the poset Clop(P,φ) of all clopen subsets. If (P,φ) is a finite convex geometry, then Reg(P, φ) is pseudocomplemented. The Dedekind-MacNeille completion of the poset of regions of any central hyperplane arrangement can be obtained in this way, ...

A convexity structure satisfies the separation property 5'4 if any two disjoint convex sets extend to complementary half-spaces. This property is investigated for alignment spaces, n-ary convexities, and graphs. In particular, it is proven that a) an n-ary convexity is $4 iff every pair of disjoint polytopes with at most n vertices can be separated by complementary half spaces, and b) an interv...

In this paper, we prove some splitting theorems for CAT(0) spaces on which some product group acts geometrically and show a splitting theorem for compact geodesic spaces of nonpositive curvature. A CAT(0) group Γ is said to be rigid, if Γ determines the boundary up to homeomorphism of a CAT(0) space on which Γ acts geometrically. Croke and Kleiner have constructed a non-rigid CAT(0) group. As a...

Topological spaces can be induced by various algebraic ordering relations such as, linear, partial and the inclusion-ordering of open sets forming chains chain complexes. In general, classifications covering are made using fundamental groups lifting. However, Riesz ordered n-spaces Urysohn interpretations real-valued continuous functions as provide new perspectives. This paper proposes formulat...

Starting with the underlying motivation of developing a general theory of L-fuzzy sets where L is a multilattice (a particular case of non-deterministic algebra), we study the relationship between the crisp notions of congruence, homomorphism and substructure on some non-deterministic algebras which have been used in the literature, i.e. hypergroups, and join spaces. Moreover, we provide suitab...

Let (rI , r, ,...) be a sequence of non-negative integers summing to n. We determine under what conditions there exists a finite distributive lattice L of rank n with ri join-irreducibles of rank i, for all i = 1,2,.. . . When I. exists, we give explicit expressions for the greatest number of elements L can have of any given rank, and for the greatest total number of elements L can have. The pr...