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Compact topological semilattices Oleg Gutik National University of Lviv Uniwersytet Jana Kazimierza we Lwowie Tartu Ülikooli, 14 detsember, 2011 Oleg Gutik Compact topological semilattices General Topology Definition: A topological space X is called T4 -space, or a normal space, if X is: T1 -space (i.e., for every pairs of distinct points x1 , x2 ∈ X there exists an open set U in X such that x1 ∈ U and x2 ∈ / U ); and for every pairs of disjoint closed subsets A, B ⊂ X there exist open disjoint subsets U, V ∈ X such that A ⊆ U and B ⊆ V . Definition: A topological space X is called T3 1 -space, or a Tychonoff space, or a 2 completely regular space, if X is: T1 -space; and for every x ∈ X and every closed subset F ⊂ X such that x ∈ / F there exists a continuous function f : X → I such that f (x) = 0 and f (y) = 1 for all y ∈ F . Oleg Gutik Compact topological semilattices General Topology Definition: A topological space X is called T4 -space, or a normal space, if X is: T1 -space (i.e., for every pairs of distinct points x1 , x2 ∈ X there exists an open set U in X such that x1 ∈ U and x2 ∈ / U ); and for every pairs of disjoint closed subsets A, B ⊂ X there exist open disjoint subsets U, V ∈ X such that A ⊆ U and B ⊆ V . Definition: A topological space X is called T3 1 -space, or a Tychonoff space, or a 2 completely regular space, if X is: T1 -space; and for every x ∈ X and every closed subset F ⊂ X such that x ∈ / F there exists a continuous function f : X → I such that f (x) = 0 and f (y) = 1 for all y ∈ F . Oleg Gutik Compact topological semilattices General Topology Urysohn’s Lemma (1925): For every pairs A and B of disjoint closed subsets of a normal space X there exists a continuous function f : X → I such that f (x) = 0 for x ∈ A and f (y) = 1 for y ∈ B. Definition: For a topological space X the cardinal number w(X) = min {|B| : B is a base of X} is called the weight of X. Definition: The Tychonoff cube of weight m > ℵ0 is the space I m , i.e., the Cartesian Q product s∈S Is with Tychonoff topology, where Is = I and |S | = m. The Tychonoff cube I ℵ0 is called the Hilbert cube. Tychoff’s Theorem (1930): The Tychonoff cube I m is universal for all Tychonoff topological spaces of weight m > ℵ0 . Oleg Gutik Compact topological semilattices General Topology Urysohn’s Lemma (1925): For every pairs A and B of disjoint closed subsets of a normal space X there exists a continuous function f : X → I such that f (x) = 0 for x ∈ A and f (y) = 1 for y ∈ B. Definition: For a topological space X the cardinal number w(X) = min {|B| : B is a base of X} is called the weight of X. Definition: The Tychonoff cube of weight m > ℵ0 is the space I m , i.e., the Cartesian Q product s∈S Is with Tychonoff topology, where Is = I and |S | = m. The Tychonoff cube I ℵ0 is called the Hilbert cube. Tychoff’s Theorem (1930): The Tychonoff cube I m is universal for all Tychonoff topological spaces of weight m > ℵ0 . Oleg Gutik Compact topological semilattices General Topology Urysohn’s Lemma (1925): For every pairs A and B of disjoint closed subsets of a normal space X there exists a continuous function f : X → I such that f (x) = 0 for x ∈ A and f (y) = 1 for y ∈ B. Definition: For a topological space X the cardinal number w(X) = min {|B| : B is a base of X} is called the weight of X. Definition: The Tychonoff cube of weight m > ℵ0 is the space I m , i.e., the Cartesian Q product s∈S Is with Tychonoff topology, where Is = I and |S | = m. The Tychonoff cube I ℵ0 is called the Hilbert cube. Tychoff’s Theorem (1930): The Tychonoff cube I m is universal for all Tychonoff topological spaces of weight m > ℵ0 . Oleg Gutik Compact topological semilattices General Topology Urysohn’s Lemma (1925): For every pairs A and B of disjoint closed subsets of a normal space X there exists a continuous function f : X → I such that f (x) = 0 for x ∈ A and f (y) = 1 for y ∈ B. Definition: For a topological space X the cardinal number w(X) = min {|B| : B is a base of X} is called the weight of X. Definition: The Tychonoff cube of weight m > ℵ0 is the space I m , i.e., the Cartesian Q product s∈S Is with Tychonoff topology, where Is = I and |S | = m. The Tychonoff cube I ℵ0 is called the Hilbert cube. Tychoff’s Theorem (1930): The Tychonoff cube I m is universal for all Tychonoff topological spaces of weight m > ℵ0 . Oleg Gutik Compact topological semilattices Partial Order Definition: A binary relation 6 on a non-empty set X is called a partial order if the following conditions hold: x 6 x for all x ∈ X; if x 6 y and y 6 x, then x = y, for x, y ∈ X; if x 6 y and y 6 z, then x 6 z, for x, y, z ∈ X. A set X equipped with a partial order 6 is called a partially ordered set and it will be denoted by (X, 6). For a partially ordered set (X, 6), x ∈ X and A ⊆ X we denote: [ [ ↓x = {y ∈ X : y 6 x}, ↑x = {y ∈ X : x 6 y}, ↓A = ↓a and ↑A = ↑a. a∈A Definition: A partial order determined on a topological space X is called closed (or continuous) if the relation 6 is a closed subset of the product X × X. Oleg Gutik Compact topological semilattices a∈A Partial Order Definition: A binary relation 6 on a non-empty set X is called a partial order if the following conditions hold: x 6 x for all x ∈ X; if x 6 y and y 6 x, then x = y, for x, y ∈ X; if x 6 y and y 6 z, then x 6 z, for x, y, z ∈ X. A set X equipped with a partial order 6 is called a partially ordered set and it will be denoted by (X, 6). For a partially ordered set (X, 6), x ∈ X and A ⊆ X we denote: [ [ ↓x = {y ∈ X : y 6 x}, ↑x = {y ∈ X : x 6 y}, ↓A = ↓a and ↑A = ↑a. a∈A Definition: A partial order determined on a topological space X is called closed (or continuous) if the relation 6 is a closed subset of the product X × X. Oleg Gutik Compact topological semilattices a∈A Partial Order Definition: A binary relation 6 on a non-empty set X is called a partial order if the following conditions hold: x 6 x for all x ∈ X; if x 6 y and y 6 x, then x = y, for x, y ∈ X; if x 6 y and y 6 z, then x 6 z, for x, y, z ∈ X. A set X equipped with a partial order 6 is called a partially ordered set and it will be denoted by (X, 6). For a partially ordered set (X, 6), x ∈ X and A ⊆ X we denote: [ [ ↓x = {y ∈ X : y 6 x}, ↑x = {y ∈ X : x 6 y}, ↓A = ↓a and ↑A = ↑a. a∈A Definition: A partial order determined on a topological space X is called closed (or continuous) if the relation 6 is a closed subset of the product X × X. Oleg Gutik Compact topological semilattices a∈A Closed Partial Order Exercise: A partial order 6 on a topological space X is continuous if and only if for x, y ∈ X, x y implies there exist open neighbourhoods U (x) and U (y) of x and y in X such that a b for all a ∈ U (x) and b ∈ U (y). Definition: A topological space X equipped with a continuous partial order 6 is called a partially ordered topological space (or pospace). Exercise: Every partially ordered topological space is Hausdorff. Exercise: Q The Cartesian product s∈S Xs of an arbitrary family of partially ordered sets {(Xs , 6)}s∈S with the product order (i.e., (xs ) 6 (ys ) if and only if xs 6 ys for all s ∈ S ) is a partially ordered set. Oleg Gutik Compact topological semilattices Closed Partial Order Exercise: A partial order 6 on a topological space X is continuous if and only if for x, y ∈ X, x y implies there exist open neighbourhoods U (x) and U (y) of x and y in X such that a b for all a ∈ U (x) and b ∈ U (y). Definition: A topological space X equipped with a continuous partial order 6 is called a partially ordered topological space (or pospace). Exercise: Every partially ordered topological space is Hausdorff. Exercise: Q The Cartesian product s∈S Xs of an arbitrary family of partially ordered sets {(Xs , 6)}s∈S with the product order (i.e., (xs ) 6 (ys ) if and only if xs 6 ys for all s ∈ S ) is a partially ordered set. Oleg Gutik Compact topological semilattices Closed Partial Order Exercise: A partial order 6 on a topological space X is continuous if and only if for x, y ∈ X, x y implies there exist open neighbourhoods U (x) and U (y) of x and y in X such that a b for all a ∈ U (x) and b ∈ U (y). Definition: A topological space X equipped with a continuous partial order 6 is called a partially ordered topological space (or pospace). Exercise: Every partially ordered topological space is Hausdorff. Exercise: Q The Cartesian product s∈S Xs of an arbitrary family of partially ordered sets {(Xs , 6)}s∈S with the product order (i.e., (xs ) 6 (ys ) if and only if xs 6 ys for all s ∈ S ) is a partially ordered set. Oleg Gutik Compact topological semilattices Closed Partial Order Exercise: A partial order 6 on a topological space X is continuous if and only if for x, y ∈ X, x y implies there exist open neighbourhoods U (x) and U (y) of x and y in X such that a b for all a ∈ U (x) and b ∈ U (y). Definition: A topological space X equipped with a continuous partial order 6 is called a partially ordered topological space (or pospace). Exercise: Every partially ordered topological space is Hausdorff. Exercise: Q The Cartesian product s∈S Xs of an arbitrary family of partially ordered sets {(Xs , 6)}s∈S with the product order (i.e., (xs ) 6 (ys ) if and only if xs 6 ys for all s ∈ S ) is a partially ordered set. Oleg Gutik Compact topological semilattices Normally Ordered Space Definition: A map f from a partially ordered set X into a partially ordered set Y is said to be monotone if x 6 y implies f (x) 6 f (y), for x, y ∈ X. Definition: A subset A of a partially ordered set X is called order convex if a 6 b 6 c and a, b ∈ A imply c ∈ A. Definition (Nachbin, 1948): A pospace X said to be normally ordered (or monotone normal), if given two closed disjoint subset A = ↑A and B = ↓B in X, there exist open disjoint subsets U = ↑U and V = ↓V such that A ⊆ U and B ⊆ V . Notation: By I we denote the unit interval [0, 1] with the usual topology and the usual order 6. Oleg Gutik Compact topological semilattices Normally Ordered Space Definition: A map f from a partially ordered set X into a partially ordered set Y is said to be monotone if x 6 y implies f (x) 6 f (y), for x, y ∈ X. Definition: A subset A of a partially ordered set X is called order convex if a 6 b 6 c and a, b ∈ A imply c ∈ A. Definition (Nachbin, 1948): A pospace X said to be normally ordered (or monotone normal), if given two closed disjoint subset A = ↑A and B = ↓B in X, there exist open disjoint subsets U = ↑U and V = ↓V such that A ⊆ U and B ⊆ V . Notation: By I we denote the unit interval [0, 1] with the usual topology and the usual order 6. Oleg Gutik Compact topological semilattices Normally Ordered Space Definition: A map f from a partially ordered set X into a partially ordered set Y is said to be monotone if x 6 y implies f (x) 6 f (y), for x, y ∈ X. Definition: A subset A of a partially ordered set X is called order convex if a 6 b 6 c and a, b ∈ A imply c ∈ A. Definition (Nachbin, 1948): A pospace X said to be normally ordered (or monotone normal), if given two closed disjoint subset A = ↑A and B = ↓B in X, there exist open disjoint subsets U = ↑U and V = ↓V such that A ⊆ U and B ⊆ V . Notation: By I we denote the unit interval [0, 1] with the usual topology and the usual order 6. Oleg Gutik Compact topological semilattices Normally Ordered Space Definition: A map f from a partially ordered set X into a partially ordered set Y is said to be monotone if x 6 y implies f (x) 6 f (y), for x, y ∈ X. Definition: A subset A of a partially ordered set X is called order convex if a 6 b 6 c and a, b ∈ A imply c ∈ A. Definition (Nachbin, 1948): A pospace X said to be normally ordered (or monotone normal), if given two closed disjoint subset A = ↑A and B = ↓B in X, there exist open disjoint subsets U = ↑U and V = ↓V such that A ⊆ U and B ⊆ V . Notation: By I we denote the unit interval [0, 1] with the usual topology and the usual order 6. Oleg Gutik Compact topological semilattices Normally Ordered Space The Urysohn-Nachbin Lemma (Nachbin, 1948): Let X be a normally ordered pospace. If A = ↑A and B = ↓B are closed disjoint subsets in X, then there exists a monotone function f : X → I such that f (B) = 0 and f (A) = 1. Theorem (Nachbin, 1948): Every compact partially ordered space is normally ordered. Theorem (Nachbin, 1948): Every compact partially ordered space X is ordered convex (i.e., X has a base of the topology which consists of open ordered convex subsets). Oleg Gutik Compact topological semilattices Normally Ordered Space The Urysohn-Nachbin Lemma (Nachbin, 1948): Let X be a normally ordered pospace. If A = ↑A and B = ↓B are closed disjoint subsets in X, then there exists a monotone function f : X → I such that f (B) = 0 and f (A) = 1. Theorem (Nachbin, 1948): Every compact partially ordered space is normally ordered. Theorem (Nachbin, 1948): Every compact partially ordered space X is ordered convex (i.e., X has a base of the topology which consists of open ordered convex subsets). Oleg Gutik Compact topological semilattices Normally Ordered Space The Urysohn-Nachbin Lemma (Nachbin, 1948): Let X be a normally ordered pospace. If A = ↑A and B = ↓B are closed disjoint subsets in X, then there exists a monotone function f : X → I such that f (B) = 0 and f (A) = 1. Theorem (Nachbin, 1948): Every compact partially ordered space is normally ordered. Theorem (Nachbin, 1948): Every compact partially ordered space X is ordered convex (i.e., X has a base of the topology which consists of open ordered convex subsets). Oleg Gutik Compact topological semilattices Topological Semilattices Definition: A commutative semigroup of idempotents is called a semilattice. The Natural Partial Order on a Semilattice: Let E be a semilattice. We put e6f if and only if e · f = f · e = e, for e, f ∈ E. Definition: A Hausdorff topological space E with a continuous semilattice operation is called a topological semilattice (i.e., for any e, f ∈ E and for every open neighbourhood U (e · f ) of the point e · f in E there exist open neighbourhoods V (e) and V (f ) of e and f in E such that V (e) · V (f ) ⊆ U (e · f )). Example 1: Imin = ([0, 1], min). Example 2: (exp∞ (X), ∪) with the Vietories topology. Oleg Gutik Compact topological semilattices Topological Semilattices Definition: A commutative semigroup of idempotents is called a semilattice. The Natural Partial Order on a Semilattice: Let E be a semilattice. We put e6f if and only if e · f = f · e = e, for e, f ∈ E. Definition: A Hausdorff topological space E with a continuous semilattice operation is called a topological semilattice (i.e., for any e, f ∈ E and for every open neighbourhood U (e · f ) of the point e · f in E there exist open neighbourhoods V (e) and V (f ) of e and f in E such that V (e) · V (f ) ⊆ U (e · f )). Example 1: Imin = ([0, 1], min). Example 2: (exp∞ (X), ∪) with the Vietories topology. Oleg Gutik Compact topological semilattices Topological Semilattices Definition: A commutative semigroup of idempotents is called a semilattice. The Natural Partial Order on a Semilattice: Let E be a semilattice. We put e6f if and only if e · f = f · e = e, for e, f ∈ E. Definition: A Hausdorff topological space E with a continuous semilattice operation is called a topological semilattice (i.e., for any e, f ∈ E and for every open neighbourhood U (e · f ) of the point e · f in E there exist open neighbourhoods V (e) and V (f ) of e and f in E such that V (e) · V (f ) ⊆ U (e · f )). Example 1: Imin = ([0, 1], min). Example 2: (exp∞ (X), ∪) with the Vietories topology. Oleg Gutik Compact topological semilattices Topological Semilattices Definition: A commutative semigroup of idempotents is called a semilattice. The Natural Partial Order on a Semilattice: Let E be a semilattice. We put e6f if and only if e · f = f · e = e, for e, f ∈ E. Definition: A Hausdorff topological space E with a continuous semilattice operation is called a topological semilattice (i.e., for any e, f ∈ E and for every open neighbourhood U (e · f ) of the point e · f in E there exist open neighbourhoods V (e) and V (f ) of e and f in E such that V (e) · V (f ) ⊆ U (e · f )). Example 1: Imin = ([0, 1], min). Example 2: (exp∞ (X), ∪) with the Vietories topology. Oleg Gutik Compact topological semilattices Topological Semilattices Definition: A commutative semigroup of idempotents is called a semilattice. The Natural Partial Order on a Semilattice: Let E be a semilattice. We put e6f if and only if e · f = f · e = e, for e, f ∈ E. Definition: A Hausdorff topological space E with a continuous semilattice operation is called a topological semilattice (i.e., for any e, f ∈ E and for every open neighbourhood U (e · f ) of the point e · f in E there exist open neighbourhoods V (e) and V (f ) of e and f in E such that V (e) · V (f ) ⊆ U (e · f )). Example 1: Imin = ([0, 1], min). Example 2: (exp∞ (X), ∪) with the Vietories topology. Oleg Gutik Compact topological semilattices Topological Semilattices Theorem: The natural partial order on a Hausdorff topological semilattice is closed, and hence every Hausdorff topological semilattice is a partially ordered topological space. Definition: Let E be a topological semilattice. By Hom(E, Imin ) we denote the set of all continuous homomorphisms from E into Imin . We shall say that Hom(E, Imin ) separates points of E if for any distinct e, f ∈ E with e 6 f there exists h ∈ Hom(E, Imin ) such that h(e) = 0 and h(f ) = 1. Anderson Problem (1953): Let E be a compact topological semilattice. Does Hom(E, Imin ) separate points of E? Oleg Gutik Compact topological semilattices Topological Semilattices Theorem: The natural partial order on a Hausdorff topological semilattice is closed, and hence every Hausdorff topological semilattice is a partially ordered topological space. Definition: Let E be a topological semilattice. By Hom(E, Imin ) we denote the set of all continuous homomorphisms from E into Imin . We shall say that Hom(E, Imin ) separates points of E if for any distinct e, f ∈ E with e 6 f there exists h ∈ Hom(E, Imin ) such that h(e) = 0 and h(f ) = 1. Anderson Problem (1953): Let E be a compact topological semilattice. Does Hom(E, Imin ) separate points of E? Oleg Gutik Compact topological semilattices Topological Semilattices Theorem: The natural partial order on a Hausdorff topological semilattice is closed, and hence every Hausdorff topological semilattice is a partially ordered topological space. Definition: Let E be a topological semilattice. By Hom(E, Imin ) we denote the set of all continuous homomorphisms from E into Imin . We shall say that Hom(E, Imin ) separates points of E if for any distinct e, f ∈ E with e 6 f there exists h ∈ Hom(E, Imin ) such that h(e) = 0 and h(f ) = 1. Anderson Problem (1953): Let E be a compact topological semilattice. Does Hom(E, Imin ) separate points of E? Oleg Gutik Compact topological semilattices Lawson Semilattices Definition (Lawson, 1969) & (McWaters, 1969): A topological semilattice S is said to have small semilattices at x ∈ S if the point x has a basis of neighborhoods which are subsemilattices of S. The topological semilattice S has small semilattices (or is a Lawson semilattice) iff it has small semilattices at every point. Proposition (Lawson, 1969) & (McWaters, 1969): (i) Let S be a Lawson topological semilattice and let T be a subsemilattice (equipped with the relative topology). Then T is a Lawson semilattice. (ii) Let Q {Sj : j ∈ J } be a collection of Lawson topological semilattices. Then {Sj : j ∈ J } endowed with coordinatewise operations and the product topology is a Lawson semilattice. Oleg Gutik Compact topological semilattices Lawson Semilattices Definition (Lawson, 1969) & (McWaters, 1969): A topological semilattice S is said to have small semilattices at x ∈ S if the point x has a basis of neighborhoods which are subsemilattices of S. The topological semilattice S has small semilattices (or is a Lawson semilattice) iff it has small semilattices at every point. Proposition (Lawson, 1969) & (McWaters, 1969): (i) Let S be a Lawson topological semilattice and let T be a subsemilattice (equipped with the relative topology). Then T is a Lawson semilattice. (ii) Let Q {Sj : j ∈ J } be a collection of Lawson topological semilattices. Then {Sj : j ∈ J } endowed with coordinatewise operations and the product topology is a Lawson semilattice. Oleg Gutik Compact topological semilattices Lawson Semilattices Theorem (Lawson, 1969): Let S be a locally compact Hausdorff topological semilattice. For x ∈ S the following statements are equivalent: (i) S has small semilattices at x; (ii) the semilattice ↓x has small semilattices at x; (iii) if V is open in S and x ∈ V , then there exists an element y ∈ V such that x ∈ Int(↑y). Theorem (Lawson, 1969): Let S and T be compact topological semilattices and let f be a continuous homomorphism from S onto T . If S is Lawson, then T is Lawson, too. Oleg Gutik Compact topological semilattices Lawson Semilattices Theorem (Lawson, 1969): Let S be a locally compact Hausdorff topological semilattice. For x ∈ S the following statements are equivalent: (i) S has small semilattices at x; (ii) the semilattice ↓x has small semilattices at x; (iii) if V is open in S and x ∈ V , then there exists an element y ∈ V such that x ∈ Int(↑y). Theorem (Lawson, 1969): Let S and T be compact topological semilattices and let f be a continuous homomorphism from S onto T . If S is Lawson, then T is Lawson, too. Oleg Gutik Compact topological semilattices The Fundamental Theorem of Compact Semilattices The Fundamental Theorem of Compact Semilattices (Lawson, 1969): Let S be a compact Hausdorff topological semilattice. Then the following statements are equivalent: (i) S is Lawson; (ii) Hom(S, Imin ) separates points of S; (iii) S is topologically isomorphic to a closed subsemilattice of a product of copies of Imin . Examples of Lawson semilattices: (i) Every linearly ordered topological semilattice. (ii) Every compact 0-dimensional topological semilattice (A topological space X is called 0-dimensional if X has a base which consists of closed-and-open subsets of X). (iii) (exp∞ (X), ∪) with the Vietories topology: the free Lawson semilattice over a topological space S. Oleg Gutik Compact topological semilattices The Fundamental Theorem of Compact Semilattices The Fundamental Theorem of Compact Semilattices (Lawson, 1969): Let S be a compact Hausdorff topological semilattice. Then the following statements are equivalent: (i) S is Lawson; (ii) Hom(S, Imin ) separates points of S; (iii) S is topologically isomorphic to a closed subsemilattice of a product of copies of Imin . Examples of Lawson semilattices: (i) Every linearly ordered topological semilattice. (ii) Every compact 0-dimensional topological semilattice (A topological space X is called 0-dimensional if X has a base which consists of closed-and-open subsets of X). (iii) (exp∞ (X), ∪) with the Vietories topology: the free Lawson semilattice over a topological space S. Oleg Gutik Compact topological semilattices Examples of non-Lawson Semilattices Jimmie D. Lawson. Lattices with no interval homomorphisms. Pacific Journal of Mathematics 32 (1970), 459—465. Gerhard Gierz. A compact semilattice on the Hilbert cube with no interval homomorphism. Proceedings of the American Mathematical Society 101 (1987), 592–594. Oleg Gutik Compact topological semilattices Examples of non-Lawson Semilattices Jimmie D. Lawson. Lattices with no interval homomorphisms. Pacific Journal of Mathematics 32 (1970), 459—465. Gerhard Gierz. A compact semilattice on the Hilbert cube with no interval homomorphism. Proceedings of the American Mathematical Society 101 (1987), 592–594. Oleg Gutik Compact topological semilattices The Fundamental Theorem of Compact 0-dimensional Semilattices The Fundamental Theorem of Compact 0-dimensional Semilattices (Hofmann, Mislove, Stralka, 1974): Let S be a compact Hausdorff topological semilattice. Then the following statements are equivalent: (i) S is 0-dimensional; (ii) Hom(S, {0, 1}min ) separates points of S; (iii) S is topologically isomorphic to a closed subsemilattice of a product of copies of {0, 1}min . Oleg Gutik Compact topological semilattices Compact Topological Semilattices with Open Translations Definition: A continuous map f : X → Y of topological spaces X and Y is called open if f (U ) is an open subset in Y for any open subset U ⊆ X. Definition: A topological semigroup S is called a semigroup with open translations if all left λs : S → S : x 7→ s · x and right ρs : S → S : x 7→ x · s translations in S are open maps, for all s ∈ S. Guran’s Problem (1991): To describe compact topological semilattices with open translations. Oleg Gutik Compact topological semilattices Compact Topological Semilattices with Open Translations Definition: A continuous map f : X → Y of topological spaces X and Y is called open if f (U ) is an open subset in Y for any open subset U ⊆ X. Definition: A topological semigroup S is called a semigroup with open translations if all left λs : S → S : x 7→ s · x and right ρs : S → S : x 7→ x · s translations in S are open maps, for all s ∈ S. Guran’s Problem (1991): To describe compact topological semilattices with open translations. Oleg Gutik Compact topological semilattices Compact Topological Semilattices with Open Translations Definition: A continuous map f : X → Y of topological spaces X and Y is called open if f (U ) is an open subset in Y for any open subset U ⊆ X. Definition: A topological semigroup S is called a semigroup with open translations if all left λs : S → S : x 7→ s · x and right ρs : S → S : x 7→ x · s translations in S are open maps, for all s ∈ S. Guran’s Problem (1991): To describe compact topological semilattices with open translations. Oleg Gutik Compact topological semilattices Compact Topological Semilattices with Open Translations Theorem: Let E be a compact topological semilattice with open translations. Then the following assertions hold: (i) the semilattice E is 0-dimensional, and hence is Lawson; (ii) the topological space E is scattered (i.e., every subset A in E has an isolated point in itself); (iii) E contains a dense subset K(E) of isolated points of E; (iv) K(E) is a subsemilattice of E; (v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable; (vi) every maximal antichain (i.e., a subset with point-wise incomparable elements) in K(E) is finite; (vii) K(E) is countable; (viii) E is a first countable space; (ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the topology at the point e ∈ E; (x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and χ(e, E) = min {|B(e)| : B(e) is a base at e}; (xi) χ(E) 6 ℵ0 , and hence E is countable. Oleg Gutik Compact topological semilattices Compact Topological Semilattices with Open Translations Theorem: Let E be a compact topological semilattice with open translations. Then the following assertions hold: (i) the semilattice E is 0-dimensional, and hence is Lawson; (ii) the topological space E is scattered (i.e., every subset A in E has an isolated point in itself); (iii) E contains a dense subset K(E) of isolated points of E; (iv) K(E) is a subsemilattice of E; (v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable; (vi) every maximal antichain (i.e., a subset with point-wise incomparable elements) in K(E) is finite; (vii) K(E) is countable; (viii) E is a first countable space; (ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the topology at the point e ∈ E; (x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and χ(e, E) = min {|B(e)| : B(e) is a base at e}; (xi) χ(E) 6 ℵ0 , and hence E is countable. Oleg Gutik Compact topological semilattices Compact Topological Semilattices with Open Translations Theorem: Let E be a compact topological semilattice with open translations. Then the following assertions hold: (i) the semilattice E is 0-dimensional, and hence is Lawson; (ii) the topological space E is scattered (i.e., every subset A in E has an isolated point in itself); (iii) E contains a dense subset K(E) of isolated points of E; (iv) K(E) is a subsemilattice of E; (v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable; (vi) every maximal antichain (i.e., a subset with point-wise incomparable elements) in K(E) is finite; (vii) K(E) is countable; (viii) E is a first countable space; (ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the topology at the point e ∈ E; (x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and χ(e, E) = min {|B(e)| : B(e) is a base at e}; (xi) χ(E) 6 ℵ0 , and hence E is countable. Oleg Gutik Compact topological semilattices Compact Topological Semilattices with Open Translations Theorem: Let E be a compact topological semilattice with open translations. Then the following assertions hold: (i) the semilattice E is 0-dimensional, and hence is Lawson; (ii) the topological space E is scattered (i.e., every subset A in E has an isolated point in itself); (iii) E contains a dense subset K(E) of isolated points of E; (iv) K(E) is a subsemilattice of E; (v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable; (vi) every maximal antichain (i.e., a subset with point-wise incomparable elements) in K(E) is finite; (vii) K(E) is countable; (viii) E is a first countable space; (ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the topology at the point e ∈ E; (x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and χ(e, E) = min {|B(e)| : B(e) is a base at e}; (xi) χ(E) 6 ℵ0 , and hence E is countable. Oleg Gutik Compact topological semilattices Compact Topological Semilattices with Open Translations Theorem: Let E be a compact topological semilattice with open translations. Then the following assertions hold: (i) the semilattice E is 0-dimensional, and hence is Lawson; (ii) the topological space E is scattered (i.e., every subset A in E has an isolated point in itself); (iii) E contains a dense subset K(E) of isolated points of E; (iv) K(E) is a subsemilattice of E; (v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable; (vi) every maximal antichain (i.e., a subset with point-wise incomparable elements) in K(E) is finite; (vii) K(E) is countable; (viii) E is a first countable space; (ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the topology at the point e ∈ E; (x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and χ(e, E) = min {|B(e)| : B(e) is a base at e}; (xi) χ(E) 6 ℵ0 , and hence E is countable. Oleg Gutik Compact topological semilattices Compact Topological Semilattices with Open Translations Theorem: Let E be a compact topological semilattice with open translations. Then the following assertions hold: (i) the semilattice E is 0-dimensional, and hence is Lawson; (ii) the topological space E is scattered (i.e., every subset A in E has an isolated point in itself); (iii) E contains a dense subset K(E) of isolated points of E; (iv) K(E) is a subsemilattice of E; (v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable; (vi) every maximal antichain (i.e., a subset with point-wise incomparable elements) in K(E) is finite; (vii) K(E) is countable; (viii) E is a first countable space; (ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the topology at the point e ∈ E; (x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and χ(e, E) = min {|B(e)| : B(e) is a base at e}; (xi) χ(E) 6 ℵ0 , and hence E is countable. Oleg Gutik Compact topological semilattices Compact Topological Semilattices with Open Translations Theorem: Let E be a compact topological semilattice with open translations. Then the following assertions hold: (i) the semilattice E is 0-dimensional, and hence is Lawson; (ii) the topological space E is scattered (i.e., every subset A in E has an isolated point in itself); (iii) E contains a dense subset K(E) of isolated points of E; (iv) K(E) is a subsemilattice of E; (v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable; (vi) every maximal antichain (i.e., a subset with point-wise incomparable elements) in K(E) is finite; (vii) K(E) is countable; (viii) E is a first countable space; (ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the topology at the point e ∈ E; (x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and χ(e, E) = min {|B(e)| : B(e) is a base at e}; (xi) χ(E) 6 ℵ0 , and hence E is countable. Oleg Gutik Compact topological semilattices Compact Topological Semilattices with Open Translations Theorem: Let E be a compact topological semilattice with open translations. Then the following assertions hold: (i) the semilattice E is 0-dimensional, and hence is Lawson; (ii) the topological space E is scattered (i.e., every subset A in E has an isolated point in itself); (iii) E contains a dense subset K(E) of isolated points of E; (iv) K(E) is a subsemilattice of E; (v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable; (vi) every maximal antichain (i.e., a subset with point-wise incomparable elements) in K(E) is finite; (vii) K(E) is countable; (viii) E is a first countable space; (ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the topology at the point e ∈ E; (x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and χ(e, E) = min {|B(e)| : B(e) is a base at e}; (xi) χ(E) 6 ℵ0 , and hence E is countable. Oleg Gutik Compact topological semilattices Compact Topological Semilattices with Open Translations Theorem: Let E be a compact topological semilattice with open translations. Then the following assertions hold: (i) the semilattice E is 0-dimensional, and hence is Lawson; (ii) the topological space E is scattered (i.e., every subset A in E has an isolated point in itself); (iii) E contains a dense subset K(E) of isolated points of E; (iv) K(E) is a subsemilattice of E; (v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable; (vi) every maximal antichain (i.e., a subset with point-wise incomparable elements) in K(E) is finite; (vii) K(E) is countable; (viii) E is a first countable space; (ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the topology at the point e ∈ E; (x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and χ(e, E) = min {|B(e)| : B(e) is a base at e}; (xi) χ(E) 6 ℵ0 , and hence E is countable. Oleg Gutik Compact topological semilattices Compact Topological Semilattices with Open Translations Theorem: Let E be a compact topological semilattice with open translations. Then the following assertions hold: (i) the semilattice E is 0-dimensional, and hence is Lawson; (ii) the topological space E is scattered (i.e., every subset A in E has an isolated point in itself); (iii) E contains a dense subset K(E) of isolated points of E; (iv) K(E) is a subsemilattice of E; (v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable; (vi) every maximal antichain (i.e., a subset with point-wise incomparable elements) in K(E) is finite; (vii) K(E) is countable; (viii) E is a first countable space; (ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the topology at the point e ∈ E; (x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and χ(e, E) = min {|B(e)| : B(e) is a base at e}; (xi) χ(E) 6 ℵ0 , and hence E is countable. Oleg Gutik Compact topological semilattices Compact Topological Semilattices with Open Translations Theorem: Let E be a compact topological semilattice with open translations. Then the following assertions hold: (i) the semilattice E is 0-dimensional, and hence is Lawson; (ii) the topological space E is scattered (i.e., every subset A in E has an isolated point in itself); (iii) E contains a dense subset K(E) of isolated points of E; (iv) K(E) is a subsemilattice of E; (v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable; (vi) every maximal antichain (i.e., a subset with point-wise incomparable elements) in K(E) is finite; (vii) K(E) is countable; (viii) E is a first countable space; (ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the topology at the point e ∈ E; (x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and χ(e, E) = min {|B(e)| : B(e) is a base at e}; (xi) χ(E) 6 ℵ0 , and hence E is countable. Oleg Gutik Compact topological semilattices Compact Topological Semilattices with Open Translations Theorem: Let E be a compact topological semilattice with open translations. Then the following assertions hold: (i) the semilattice E is 0-dimensional, and hence is Lawson; (ii) the topological space E is scattered (i.e., every subset A in E has an isolated point in itself); (iii) E contains a dense subset K(E) of isolated points of E; (iv) K(E) is a subsemilattice of E; (v) every maximal chain (i.e., a linearly ordered subset) in K(E) is countable; (vi) every maximal antichain (i.e., a subset with point-wise incomparable elements) in K(E) is finite; (vii) K(E) is countable; (viii) E is a first countable space; (ix) the family B(e) = {Ux (e) = ↓e ∩ ↑x : x ∈ ↓e ∩ K(E)} is a base of the topology at the point e ∈ E; (x) χ(E) = |E|, where χ(E) = supe∈E χ(e, E) and χ(e, E) = min {|B(e)| : B(e) is a base at e}; (xi) χ(E) 6 ℵ0 , and hence E is countable. Oleg Gutik Compact topological semilattices