scholarly journals On maps between Stone-Cech compactifications induced by lattice homomorphisms

Filomat ◽  
2016 ◽  
Vol 30 (9) ◽  
pp. 2465-2474 ◽  
Author(s):  
Themba Dube
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Lattice Homomorphism ◽  
Completely Regular ◽  
Zero Sets ◽  
Continuous Map ◽  
Necessary And Sufficient ◽  
Lattice Homomorphisms ◽  
Tychonoff Spaces

Broverman has shown that if X and Y are Tychonoff spaces and t:Z(Y)?Z(X) is a lattice homomorphism between the lattices of their zero-sets, then there is a continuous map ?: ?X ? ?Y induced by t. In this note we expound this idea and supplement Broverman?s results by first showing that this phenomenon holds in the category of completely regular frames. Among results we obtain, which were not considered by Broverman, are necessary and sufficient conditions (in terms of properties of the map t) for the induced map ? to be (i) the inclusion of a subspace, (ii) surjective, and (iii) irreducible. We show that if X and Y are pseudocompact then t pulls back z-ultrafilters to z-ultrafilters if and only if cl?X t(Z) = ?? [cl?YZ] for every Z ? Z(Y) if and only if t is ?-homomorphism.

scholarly journals Homomorphisms Between Lattices of Zero-Sets

1978 ◽  
Vol 21 (1) ◽  
pp. 1-5 ◽  
Author(s):  
S. Broverman
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Lattice Homomorphism ◽  
Completely Regular ◽  
Zero Sets ◽  
Hausdorff Topological Space ◽  
Continuous Map ◽  
Lattice Homomorphisms

AbstractFor a completely regular Hausdorff topological space X, let Z(X) denote the lattice of zero-sets of X. If T is a continuous map from X to Y, then there is a lattice homomorphism T” from Z(Y) to Z(X) induced by T which is defined by τ‘(A) = τ←(A). A characterization is given of those lattice homomorphisms from Z(Y) to Z(X) which are induced in the above way by a continuous function from X to Y.

scholarly journals Unicyclic Graphs Whose Completely Regular Endomorphisms form a Monoid

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 240
Author(s):  
Rui Gu ◽  
Hailong Hou
Keyword(s):  
Sufficient Conditions ◽  
Unicyclic Graph ◽  
Necessary And Sufficient Conditions ◽  
Unicyclic Graphs ◽  
Completely Regular ◽  
Odd Cycle ◽  
Necessary And Sufficient

In this paper, completely regular endomorphisms of unicyclic graphs are explored. Let G be a unicyclic graph and let c E n d ( G ) be the set of all completely regular endomorphisms of G. The necessary and sufficient conditions under which c E n d ( G ) forms a monoid are given. It is shown that c E n d ( G ) forms a submonoid of E n d ( G ) if and only if G is an odd cycle or G = G ( n , m ) for some odd n ≥ 3 and integer m ≥ 1 .

scholarly journals Equicontinuity of iterates of circle maps

1998 ◽  
Vol 21 (3) ◽  
pp. 453-458 ◽  
Author(s):  
Antonios Valaristos
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Circle Maps ◽  
Continuous Map ◽  
The Family ◽  
Necessary And Sufficient

Letfbe a continuous map of the circle to itself. Necessary and sufficient conditions are given for the family ofiterates{fn}n=1∞to be equicontinuous.

Some semigroup classes and congruences on Bruck-Reilly and generalized Bruck-Reilly ∗-extensions of monoids

2015 ◽  
Vol 08 (04) ◽  
pp. 1550075
Author(s):  
Seda Oğuz ◽  
Eylem G. Karpuz
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Completely Regular ◽  
Necessary And Sufficient

In this paper, we determine necessary and sufficient conditions for Bruck–Reilly and generalized Bruck–Reilly ∗-extensions of monoids to be unit regular, completely regular and orthodox. In addition, we characterize the congruences on generalized Bruck–Reilly ∗-extension of a monoid.

scholarly journals Finiteness at infinity

1971 ◽  
Vol 17 (4) ◽  
pp. 299-304 ◽  
Author(s):  
P. A. Firby
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Locally Compact ◽  
Necessary And Sufficient ◽  
Tychonoff Spaces

If X is a Tychonoff topological space, and if βX is the Stone-Cech compactification of X, then βX\X will denote the complement of X in βX. If A is a subset of X, then cl [A: X] will denote the closure of A in X, and int [A: X] will denote the interior of A in X. In Isbell ((3), p. 119) a property of βX\X is called a property which X has at infinity, and it is the aim of this paper to give necessary and sufficient conditions for X to be finite at infinity. Since βX is T1 we can say that if X is finite at infinity, then βX\X is closed in βX. So we lose nothing by restricting our attention to locally compact, Tychonoff spaces, and for the remainder of the paper X will denote such a space.

scholarly journals Note on pointwise contractive projections

1993 ◽  
Vol 16 (4) ◽  
pp. 817-818
Author(s):  
L. M. Sanchez Ruiz ◽  
J. R. Ferrer Villanueva
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Necessary And Sufficient Conditions ◽  
Compact Open Topology ◽  
Completely Regular ◽  
Contractive Projection ◽  
Necessary And Sufficient ◽  
Contractive Projections

LetC(X)be the space of real-valued continuous functions on a Hausdorff completely regular topological spaceX. endowed with the compact-open topology. In this paper necessary and sufficient conditions are given for a subspace ofC(X)to be the range of a pointwise contractive projection inC(X).

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

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