. Lattices of coarse structures, Mat. Stud. 48 (2017), 115-123. We consider the lattice of coarse structures on a set X and study metrizable, locally finite and cellular coarse structures on X from the lattice point of view.To Michael Zarichnyi on 60th birthday 1. Introduction. Following [11], we say that a family E of subsets of X × X is a coarse structure on a set X ifEach ε ∈ E is called an entourage of the diagonal. We note that E is closed under finite unions (as ε ∪ δ ⊆ εδ), but E is not an ideal in the Boolean algebra of all subsets of X × X because E is not closed under formation of all subsets of its members.A subset E ′ ⊆ E is called a base for E if, for every ε ∈ E there exists ε ′ ∈ E ′ such that ε ⊆ ε ′ . The pair (X, E) is called a coarse space. For x ∈ X and ε ∈ E, we denote B(x, ε) = {y ∈ X : (x, y) ∈ ε} and say that B(x, ε) is a ball of radius ε around x. We note that a coarse space can be considered as an asymptotic counterpart of a uniform topological space and could be defined in terms of balls, see [7], [9]. In this case a coarse space is called a ballean. For categorical look at the balleans and coarse structures as two faces of the same coin, see [1].A coarse structure E on X is called connected if, for any x, y ∈ X, there is ε ∈ E such that y ∈ B(x, ε). A subset Y of X is called bounded if there exist x ∈ X and ε ∈ E such that Y ⊆ B(x, ε). A coarse structure E is called bounded if X is bounded, otherwise E is called unbounded. We note that on every set X, there exists the unique connected bounded coarse structure {ε ⊆ X × X : △ X ⊆ ε}, and if X is finite this structure is the unique connected coarse structure on X.In what follows, we consider only connected coarse structures on infinite sets. Given a set X, the family L X of all coarse structures on X is partially ordered by the inclusion, and L X can be considered as a lattice with the operations ∧ and ∨:• E ∧ E ′ is the strongest coarse structure such that E ∧ E ′ ⊆ E, E ∧ E ′ ⊆ E ′ ;2010 Mathematics Subject Classification: 06B05.
