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Lattice-Like Total Perfect Codes

100%
Araujo C., Dejter I.
Discussiones Mathematicae Graph Theory
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2014
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tom 34
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nr 1
57-74
EN
A contribution is made to the classification of lattice-like total perfect codes in integer lattices Λn via pairs (G, Φ) formed by abelian groups G and homomorphisms Φ: Zn → G. A conjecture is posed that the cited contribution covers all possible cases. A related conjecture on the unfinished work on open problems on lattice-like perfect dominating sets in Λn with induced components that are parallel paths of length > 1 is posed as well.
2
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Four-Valued Logics BD and DM4: Expansions

100%
Karpenko A. S.
Bulletin of the Section of Logic
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2017
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tom 46
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nr 1/2
EN
The paper discusses functional properties of some four-valued logics which are the expansions of four-valued Belnap’s logic DM4. At first, we consider the logics with two designated values, and then logics defined by matrices having the same underlying algebra, but with a different choice of designated values, i.e. with one designated value. In the preceding literature both approaches were developed independently. Moreover, we present the lattices of the functional expansions of DM4.
3
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The structure of atoms (hereditarily indecomposable continua)

88%
Ball R. N., Hagler J. N., Sternfeld Y.
Fundamenta Mathematicae
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1998
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tom 156
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nr 3
261-278
EN
Let X be an atom (= hereditarily indecomposable continuum). Define a metric ϱ on X by letting $ϱ(x,y) = W(A_{xy})$ where $A_{x,y}$ is the (unique) minimal subcontinuum of X which contains x and y and W is a Whitney map on the set of subcontinua of X with W(X) = 1. We prove that ϱ is an ultrametric and the topology of (X,ϱ) is stronger than the original topology of X. The ϱ-closed balls C(x,r) = {y ∈ X:ϱ ( x,y) ≤ r} coincide with the subcontinua of X. (C(x,r) is the unique subcontinuum of X which contains x and has Whitney value r.) It is proved that for any two (nontrivial) atoms and any Whitney maps on them, the corresponding ultrametric spaces are isometric. This implies in particular that the combinatorial structure of subcontinua is identical in all atoms. The set M(X) of all monotone upper semicontinuous decompositions of X is a lattice when ordered by refinement. It is proved that for two atoms X and Y, M(X) is lattice isomorphic to M(Y) if and only if X is homeomorphic to Y.
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The partially pre-ordered set of compactifications of Cp(X, Y)

75%
Dorantes-Aldama A., Rojas-Hernández R., Tamariz-Mascarúa Á.
Topological Algebra and its Applications
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2015
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tom 3
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nr 1
EN
In the set of compactifications of X we consider the partial pre-order defined by (W, h) ≤X (Z, g) if there is a continuous function f : Z ⇢ W, such that (f ∘ g)(x) = h(x) for every x ∈ X. Two elements (W, h) and (Z, g) of K(X) are equivalent, (W, h) ≡X (Z, g), if there is a homeomorphism h : W ! Z such that (f ∘ g)(x) = h(x) for every x ∈ X. We denote by K(X) the upper semilattice of classes of equivalence of compactifications of X defined by ≤X and ≡X. We analyze in this article K(Cp(X, Y)) where Cp(X, Y) is the space of continuous functions from X to Y with the topology inherited from the Tychonoff product space YX. We write Cp(X) instead of Cp(X, R). We prove that for a first countable space Y, K(Cp(X, Y)) is not a lattice if any of the following cases happen: (a) Y is not locally compact, (b) X has only one non isolated point and Y is not compact. Furthermore, K(Cp(X)) is not a lattice when X satisfies one of the following properties: (i) X has a non-isolated point with countable character, (ii) X is not pseudocompact, (iii) X is infinite, pseudocompact and Cp(X) is normal, (iv) X is an infinite generalized ordered space. Moreover, K(Cp(X)) is not a lattice when X is an infinite Corson compact space, and for every space X, K(Cp(Cp(X))) is not a lattice. Finally, we list some unsolved problems.
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