Strongly s-dense injective hull and Banaschewski’s theorems for acts
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Hasan Barzegar
Abstract
For a class 𝓜 of monomorphisms of a category, mathematicians usually use different types of essentiality. Essentiality is an important notion closely related to injectivity. Banaschewski defines and gives sufficient conditions on a category 𝓐 and a subclass 𝓜 of its monomorphisms under which 𝓜-injectivity well-behaves with respect to the notions such as 𝓜-absolute retract and 𝓜-essentialness.
In this paper, 𝓐 is taken to be the category of acts over a semigroup S and 𝓜sd to be the class of strongly s-dense monomorphisms. We study essentiality with respect to strongly s-dense monomorphisms of acts. Depending on a class 𝓜 of morphisms of a category 𝓐, In some literatures, three different types of essentialness are considered. Each has its own benefits in regards with the behavior of 𝓜-injectivity. We will show that these three different definitions of essentiality with respect to the class of strongly s-dense monomorphisms are equivalent. Also, the existence and the explicit description of a strongly s-dense injective hull for any given act which is equivalent to the maximal such essential extension and minimal strongly s-dense injective extension with respect to strongly s-dense monomorphism is investigated. At last we conclude that strongly s-dense injectivity well behaves in the category Act-S.
Communicated by Aleš Pultr
References
[1] Banaschewski, B.: Injectivity and essential extensions in equational classes of algebras. Queens Papers in Pure and Appl. Math. 25, 1970, pp. 131–147.Suche in Google Scholar
[2] Banaschewski, B.–-Nelson, E.: Equational compactness in equational classes of algebras, Algebra Universalis 2 (1972), 152–165.10.1007/BF02945023Suche in Google Scholar
[3] Barzegar, H.: Strongly s-dense monomorphism, J. Hyperstructures 1(1) (2012), 14–26.Suche in Google Scholar
[4] Barzegar, H.: Sequentially compactS-act, Journal of Algebraic Systems 5(2) (2017), 111–125.Suche in Google Scholar
[5] Barzegar, H.–-Ebrahimi, M. M.–-Mahmoudi, M.: Essentiality and injectivity, Appl. Categ. Structures 18(1) (2010), 73–83.10.1007/s10485-008-9165-0Suche in Google Scholar
[6] Berthiaume, P.: The injective envelope ofS-Sets, Canad. Math. Bull. 10(2) (1967), 261–273.10.4153/CMB-1967-026-1Suche in Google Scholar
[7] Kilp, M.–-Knauer U.–-Mikhalev, A.: Monoids, Acts and Categories, Walter de Gruyter, Berlin, New York, 2000.10.1515/9783110812909Suche in Google Scholar
[8] Tholen, W.: Injective objects and cogenerating sets, J. Algebra 73(1) (1981), 139–155.10.1016/0021-8693(81)90351-3Suche in Google Scholar
[9] Tholen, W.: Injectivity versus exponentiability, Cah. Topol. Géom. Différ. Catég. 49(3) (2008), 228–240.Suche in Google Scholar
[10] Warfield, R. B.: Purity and algebraic compactness for modules, Pacific J. Math. 28 (1969), 699–719.10.2140/pjm.1969.28.699Suche in Google Scholar
© 2020 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- Regular papers
- Relatively residuated lattices and posets
- Strongly s-dense injective hull and Banaschewski’s theorems for acts
- Wild sets in global function fields
- Generators and integral points on elliptic curves associated with simplest quartic fields
- New Filbert and Lilbert matrices with asymmetric entries
- Returning functions with closed graph are continuous
- On sets of points of approximate continuity and ϱ-upper continuity
- Investigation of the fifth Hankel determinant for a family of functions with bounded turnings
- On solvability of some nonlocal boundary value problems for biharmonic equation
- The study of piecewise pseudo almost periodic solutions for impulsive Lasota-Wazewska model with discontinuous coefficients
- Strongly increasing solutions of higher-order quasilinear ordinary differential equations
- Oscillation of second order half-linear neutral differential equations with weaker restrictions on shifted arguments
- Filippov solutions of vector Dirichlet problems
- Sequences of positive homoclinic solutions to difference equations with variable exponent
- Modified Lupaş-Jain operators
- New fixed point results in bv(s)-metric spaces
- Improved Young and Heinz operator inequalities for unitarily invariant norms
- Scrutiny of some fixed point results by S-operators without triangular inequality
- The lattices of families of regular sets in topological spaces
- Iterated partial summations applied to finite-support discrete distributions
- Hamiltonicity of a class of toroidal graphs
Artikel in diesem Heft
- Regular papers
- Relatively residuated lattices and posets
- Strongly s-dense injective hull and Banaschewski’s theorems for acts
- Wild sets in global function fields
- Generators and integral points on elliptic curves associated with simplest quartic fields
- New Filbert and Lilbert matrices with asymmetric entries
- Returning functions with closed graph are continuous
- On sets of points of approximate continuity and ϱ-upper continuity
- Investigation of the fifth Hankel determinant for a family of functions with bounded turnings
- On solvability of some nonlocal boundary value problems for biharmonic equation
- The study of piecewise pseudo almost periodic solutions for impulsive Lasota-Wazewska model with discontinuous coefficients
- Strongly increasing solutions of higher-order quasilinear ordinary differential equations
- Oscillation of second order half-linear neutral differential equations with weaker restrictions on shifted arguments
- Filippov solutions of vector Dirichlet problems
- Sequences of positive homoclinic solutions to difference equations with variable exponent
- Modified Lupaş-Jain operators
- New fixed point results in bv(s)-metric spaces
- Improved Young and Heinz operator inequalities for unitarily invariant norms
- Scrutiny of some fixed point results by S-operators without triangular inequality
- The lattices of families of regular sets in topological spaces
- Iterated partial summations applied to finite-support discrete distributions
- Hamiltonicity of a class of toroidal graphs
