Boolean Algebra Retracts

1971 ◽  
Vol 23 (2) ◽  
pp. 339-344
Author(s):  
Timothy Cramer
Keyword(s):  
Boolean Algebra ◽  
Dual Problem ◽  
Sufficient Conditions ◽  
Boolean Algebras ◽  
Necessary And Sufficient Conditions ◽  
Infinite Cardinal ◽  
Necessary And Sufficient

A Boolean algebra B is a retract of an algebra A if there exist homomorphisms ƒ: B → A and g: A → B such that gƒ is the identity map B. Some important properties of retracts of Boolean algebras are stated in [3, §§ 30, 31, 32]. If A and B are a-complete, and A is α-generated by B, Dwinger [1, p. 145, Theorem 2.4] proved necessary and sufficient conditions for the existence of an α-homomorphism g: A → B such that g is the identity map on B. Note that if a is not an infinite cardinal, B must be equal to A. The dual problem was treated by Wright [6]; he assumed that A and B are σ-algebras, and that g: A → B is a σ-homomorphism, and gave conditions for the existence of a homomorphism ƒ:B → A such that gƒ is the identity map.

On Birkhoff's Problem 73 For Monoids

1960 ◽  
Vol 3 (3) ◽  
pp. 217-220
Author(s):  
Barron Brainerd
Keyword(s):  
Boolean Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Congruence Relations ◽  
Necessary And Sufficient ◽  
Generalized Boolean Algebra

Birkhoff in [2] poses the following problem:“Problem 73. Find necessary and sufficient conditions in order that the correspondence between the congruence relations and the (neutral) ideals of a lattice be one-one”.This problem has been solved by Areškin [l] and Hashimoto [3]. Essentially the conditions reduce to the r e quirement that the lattice be a generalized Boolean algebra.

scholarly journals Integral cayley graphs over semi-dihedral groups

2021 ◽  
pp. 1-1
Author(s):  
Tao Cheng ◽  
Lihua Feng ◽  
Guihai Yu ◽  
Chi Zhang
Keyword(s):  
Boolean Algebra ◽  
Dihedral Group ◽  
Cayley Graphs ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Character Table ◽  
Hard Problem ◽  
Dihedral Groups ◽  
Necessary And Sufficient ◽  
Integral Graphs

Classifying integral graphs is a hard problem that initiated by Harary and Schwenk in 1974. In this paper, with the help of character table, we treat the corresponding problem for Cayley graphs over the semi-dihedral group SD8n = ?a,b | a4n = b2 = 1; bab = a2n-1?, n ? 2. We present several necessary and sufficient conditions for the integrality of Cayley graphs over SD8n, we also obtain some simple sufficient conditions for the integrality of Cayley graphs over SD8n in terms of the Boolean algebra of hai. In particular, we give the sufficient conditions for the integrality of Cayley graphs over semi-dihedral groups SD2n (n?4) and SD8p for a prime p, from which we determine several infinite classes of integral Cayley graphs over SD2n and SD8p.

XIV.—On Eigenvectors of Boolean Matrices

1967 ◽  
Vol 67 (3) ◽  
pp. 196-204
Author(s):  
T. S. Blyth
Keyword(s):  
Boolean Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Matrix ◽  
Necessary And Sufficient

SynopsisThroughout this paper we shall be concerned with n × n matrices X=[xij] whose elements xij belong to a given Boolean algebra B(≤, ∩, ∪, ′). In tne first part of the paper we show that the set S(λ, x) of matrices with a given eigenvector x and eigenvalue λ is a subgerbier (i.e.), γ-semi-reticulated sub-semigroup of Mn(B), the Boolean algebra of all n × n matrices over B. We also determine the greatest element M(λ, x) of S(λ, x) and consider some of its properties. In the second part of the paper we consider the structure of matrices which possess a given primitive eigenvector x and show in particular that, for any given λ ∈ B, there is a matrix, namely M(λ, x), having x as the maximum primitive eigenvector associated with the eigenvalue λ. We also determine necessary and sufficient conditions under which the matrix M(λ, x) is the same for all λ in any given sublattice of B.

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.

On the domain of Riesz mean in the space Ls

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 925-940 ◽  
Author(s):  
Medine Yeşilkayagil ◽  
Feyzi Başar
Keyword(s):  
Banach Space ◽  
Sequence Space ◽  
Double Sequence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Double Sequences ◽  
Riesz Mean ◽  
Double Sequence Space ◽  
Necessary And Sufficient ◽  
Barrelled Space

Let 0 < s < ?. In this study, we introduce the double sequence space Rqt(Ls) as the domain of four dimensional Riesz mean Rqt in the space Ls of absolutely s-summable double sequences. Furthermore, we show that Rqt(Ls) is a Banach space and a barrelled space for 1 ? s < 1 and is not a barrelled space for 0 < s < 1. We determine the ?- and ?(?)-duals of the space Ls for 0 < s ? 1 and ?(bp)-dual of the space Rqt(Ls) for 1 < s < 1, where ? ? {p, bp, r}. Finally, we characterize the classes (Ls:Mu), (Ls:Cbp), (Rqt(Ls) : Mu) and (Rqt(Ls):Cbp) of four dimensional matrices in the cases both 0 < s < 1 and 1 ? s < 1 together with corollaries some of them give the necessary and sufficient conditions on a four dimensional matrix in order to transform a Riesz double sequence space into another Riesz double sequence space.

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