Order dense injectivity of $S$-posets

Document Type : Research Paper


Department of Mathematics, University of Maragheh


‎‎‎In this paper‎, ‎the‎ notion of injectivity with respect to order dense embeddings in ‎‎the category of $S$-posets‎, ‎posets with a monotone action of a‎ pomonoid $S$ on them‎, ‎is studied‎. ‎We give a criterion‎, ‎like the Baer condition for injectivity of modules‎, ‎or Skornjakov criterion for injectivity of $S$-sets‎, ‎for the order dense injectivity‎. ‎Also‎, ‎we consider such injectivity for $S$ itself‎, and its order dense ideals‎. ‎Further‎, ‎we define and study some kinds of weak injectivity with respect to order dense embeddings‎, ‎consider their relations with order dense injectivity‎. ‎Also investigate if these kinds of injectivity are preserved or reflected by products‎, ‎coproducts‎, ‎and direct sums of‎‎$S$-posets‎.


[1] B‎. ‎Banaschewski‎, ‎ Injectivity and essential extensions in‎  equational classes of algebras, ‎Queen's Papers Pure Appl‎. ‎Math‎. ‎25 (1970)‎, ‎131-147‎.
[2] S‎. ‎Bulman-Fleming, ‎Subpullback flat $S$-posets‎ need not be subequlizer flat, ‎Semigroup Forum, 78(1) (2009)‎, ‎27-33‎.
[3] S‎. ‎Bulman-Fleming‎, ‎D‎. ‎Gutermuth‎, ‎A‎. ‎Gilmour‎, ‎and M‎. ‎Kilp, Flatness properties of $S$-posets, ‎Comm‎. ‎Algebra {34}(4)‎ ‎ (2006)‎, ‎1291-1317‎.
[4] S‎. ‎Bulman-Fleming and V‎. ‎Laan‎, ‎Lazard's theorem for $S$-posets, Math‎. ‎Nachr‎. ‎{278}(15) (2005)‎, 1743-1755‎.
[5] S‎. ‎Bulman-Fleming and M‎. ‎Mahmoudi, ‎ The category of S-posets, Semigroup Forum {71}(3) (2005)‎, ‎443-461‎.
[6] D‎. ‎Dikranjan and W‎. ‎Tholen‎, ‎Categorical structure of‎ closure operators‎, ‎with applications to topology‎, ‎algebra‎, ‎and‎‎ discrete mathematics"‎, ‎Mathematics and Its Applications‎, ‎Kluwer‎ Academic Publ.‎, ‎1995‎.
‎[7] M.M‎. ‎Ebrahimi‎, ‎M‎. ‎Haddadi‎, ‎and M‎. ‎Mahmoudi‎, ‎ Injectivity in a category‎: ‎an overview of well behaviour theorems‎, ‎Algebra‎, ‎Groups‎, ‎and Geometries 26 (2009)‎, ‎451-472‎.
‎[7] M.M‎. ‎Ebrahimi‎, ‎M‎. ‎Haddadi‎, ‎and M‎. ‎Mahmoudi‎, ‎‎ Injectivity in a category‎: ‎an overview on smallness conditions,‎ Categ‎. ‎General Alg‎. ‎Struct‎. ‎Appl‎. ‎2(1) (2014)‎, ‎83-112‎.
‎[8] M‎. ‎Mehdi Ebrahimi and M‎. ‎Mahmoudi‎, ‎ The category of $M$-sets‎, ‎Ital‎. ‎J‎. ‎Pure Appl‎. ‎Math. {9} (2001)‎, ‎123-132‎.
‎[9] M‎. ‎Mehdi Ebrahimi‎, ‎M‎. ‎Mahmoudi‎, ‎ and H‎. ‎Rasouli‎, ‎Banaschewski's theorem for S-posets‎: ‎regular‎ injectivity and completeness‎, ‎{Semigroup Forum} {80}(2)‎‎ (2010)‎, ‎313-324‎.
‎[10] S.M.-Fakhruddin‎, ‎On the category of S-posets, ‎Acta Sci‎. ‎Math‎.‎ (Szeged)} {52} (1988)‎, ‎85-92‎.
‎[11] M‎. ‎Kilp‎, ‎U‎. ‎Knauer‎, ‎and A‎. ‎Mikhalev‎, ‎Monoids‎, ‎Acts‎‎ and Categories‎, ‎Walter de Gruyter‎, ‎Berlin‎, ‎New York‎, ‎2000‎.
‎[12] H‎. ‎Rasouli, ‎Categorical properties of regular monomorphisms of $S$-posets, ‎Europ‎. ‎J‎. ‎Pure Appl‎. ‎Math‎. ‎{7}(2) (2014)‎, ‎166-178‎.
‎[13] L‎. ‎Shahbaz and M‎. ‎Mahmoudi,  Various kinds of regular injectivity for $S$-posets‎, ‎Bull‎. ‎Iranian Math‎. ‎Soc‎. ‎{40}(1) (2014)‎, ‎243-261‎.
‎[14] L‎. ‎Shahbaz and M‎. ‎Mahmoudi, Injectivity of $S$-posets with respect to down closed‎‎ regular monomorphisms, ‎Semigroup Forum‎, ‎Published online (2015)‎ DOI:10.1007/s00233-014-9676-y‎.
‎[15] X‎. ‎Shi, ‎ On flatness properties of cyclic $S$-posets‎,‎ {Semigroup Forum} {77}(2) (2008)‎, ‎248-266‎.
‎[16] X.P‎. ‎Shi‎, ‎Z‎. ‎Liu‎, ‎F.G‎. ‎Wang‎, ‎and S‎. ‎Bulman-Fleming‎, Indecomposable‎, ‎projective and flat $S$-posets‎, ‎Comm‎. ‎Algebra {33}(1) (2005)‎, ‎235-251‎.
‎[17] L.A.-Skornjakov‎, ‎On the injectivity of ordered left acts over monoids, ‎Vestnik Moskov‎. ‎Univ‎. ‎Ser‎. ‎I Math‎. ‎Mekh. (1986)‎, ‎17-19 (in Russian)‎.
‎[18] W‎. ‎Tholen‎, ‎Injective objects and‎‎ cogenerating sets, ‎J‎. ‎Algebra {73}(1) (1981)‎, ‎139-155‎.
‎[19] X‎. ‎Zhang‎, Regular injectivity of S-posets over Clifford pomonoids‎, ‎Southeast Asian Bull‎. ‎Math. {32} (2007)‎, ‎1007-1015‎.
‎[20] X‎. ‎Zhang and V‎. ‎Laan, ‎On homological classification of pomonoids by regular weak injectivity properties of S-posets, ‎Cent‎. ‎Eur‎.‎ J.‎‎Math. {5}(1) (2007)‎, ‎181-200‎.