{"title":"Topological S-act congruence","authors":"S. Maity, Monika Paul","doi":"10.56415/qrs.v30.23","DOIUrl":null,"url":null,"abstract":"In this paper, we establish the necessary and sufficient condition for an equivalence relation ρ on an S-act A endowed with a topology such that A/ρ becomes a Hausdorff topological S-act. Also, we show that if A1 and A2 be two topological S-acts, then for any homomorphism ϕ : A1 → A2, A1/ ker ϕ is a topological S-act if and only if ϕ is ϕ-saturated continuous. Moreover, we establish for any two congruences θ1 and θ2 on an S-act A endowed with a topology, θ1 ∩ θ2 is a topological S-act congruence on A if and only if the mapping ϕ : A → A/θ1 × A/θ2, defined by ϕ(a) = (aθ1, aθ2), for all a ∈ A, is ϕ-saturated continuous, where S is a topological semigroup.","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Quasigroups and Related Systems","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.56415/qrs.v30.23","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we establish the necessary and sufficient condition for an equivalence relation ρ on an S-act A endowed with a topology such that A/ρ becomes a Hausdorff topological S-act. Also, we show that if A1 and A2 be two topological S-acts, then for any homomorphism ϕ : A1 → A2, A1/ ker ϕ is a topological S-act if and only if ϕ is ϕ-saturated continuous. Moreover, we establish for any two congruences θ1 and θ2 on an S-act A endowed with a topology, θ1 ∩ θ2 is a topological S-act congruence on A if and only if the mapping ϕ : A → A/θ1 × A/θ2, defined by ϕ(a) = (aθ1, aθ2), for all a ∈ A, is ϕ-saturated continuous, where S is a topological semigroup.