{"title":"内射和射影偏置作用","authors":"L. Shahbaz","doi":"10.56415/qrs.v31.11","DOIUrl":null,"url":null,"abstract":"In this paper, after recalling the category {\\bf PosAct}-$S$ of all poset acts over a pomonoid $S$; an $S$-act in the category {\\bf Pos} of all posets, with action preserving monotone maps between them, some categorical properties of the category {\\bf PosAct}-$S$ are considered. In particular, we describe limits and colimits such as products, coproducts, equalizers, coequalizers and etc. in this category. Also, several kinds of epimorphisms and monomorphisms are characterized in {\\bf PosAct}-$S$. Finally, we study injectivity and projectivity in {\\bf PosAct}-$S$ with respect to (regular) monomorphisms and (regular) epimorphisms, respectively, and see that although there is no non-trivial injective poset act with respect to monomorphisms, {\\bf PosAct}-$S$ has enough regular injectives with respect to regular monomorphisms. Also, it is proved that regular injective poset acts are exactly retracts of cofree poset acts over complete posets.","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Injective and projective poset acts\",\"authors\":\"L. Shahbaz\",\"doi\":\"10.56415/qrs.v31.11\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, after recalling the category {\\\\bf PosAct}-$S$ of all poset acts over a pomonoid $S$; an $S$-act in the category {\\\\bf Pos} of all posets, with action preserving monotone maps between them, some categorical properties of the category {\\\\bf PosAct}-$S$ are considered. In particular, we describe limits and colimits such as products, coproducts, equalizers, coequalizers and etc. in this category. Also, several kinds of epimorphisms and monomorphisms are characterized in {\\\\bf PosAct}-$S$. Finally, we study injectivity and projectivity in {\\\\bf PosAct}-$S$ with respect to (regular) monomorphisms and (regular) epimorphisms, respectively, and see that although there is no non-trivial injective poset act with respect to monomorphisms, {\\\\bf PosAct}-$S$ has enough regular injectives with respect to regular monomorphisms. Also, it is proved that regular injective poset acts are exactly retracts of cofree poset acts over complete posets.\",\"PeriodicalId\":38681,\"journal\":{\"name\":\"Quasigroups and Related Systems\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-07-01\",\"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.v31.11\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Quasigroups and Related Systems","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.56415/qrs.v31.11","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
In this paper, after recalling the category {\bf PosAct}-$S$ of all poset acts over a pomonoid $S$; an $S$-act in the category {\bf Pos} of all posets, with action preserving monotone maps between them, some categorical properties of the category {\bf PosAct}-$S$ are considered. In particular, we describe limits and colimits such as products, coproducts, equalizers, coequalizers and etc. in this category. Also, several kinds of epimorphisms and monomorphisms are characterized in {\bf PosAct}-$S$. Finally, we study injectivity and projectivity in {\bf PosAct}-$S$ with respect to (regular) monomorphisms and (regular) epimorphisms, respectively, and see that although there is no non-trivial injective poset act with respect to monomorphisms, {\bf PosAct}-$S$ has enough regular injectives with respect to regular monomorphisms. Also, it is proved that regular injective poset acts are exactly retracts of cofree poset acts over complete posets.