Semicorings and Semicomodules

REFERENCES

• Abuhlail , J. Y. Some remarks on tensor products and flatness of semimodules. DOI:10.1007/s00233-014-9566-3.
   Google Scholar
• Abuhlail , J. Y. Exact sequences of commutative monoids and semimodules. Homology, Homotopy, and Applications, in press.
   Google Scholar
• Abuhlail , J. Y. Exact Sequences in Non-Exact Categories. arXiv:1111.0330.
   Google Scholar
• Abuhlail , J. Y. ( 2003 ). Rational modules for corings . Commun. Algebra 31 ( 12 ): 5793 – 5840 .
   Web of Science ®Google Scholar
• Abuhlail , J. Y. ( 2013 ). Semiunital semimonoidal categories (Applications to semirings and semicorings) . Theor. Appl. Categories 28 ( 4 ): 123 – 149 .
   Google Scholar
• Abuhlail , J. Y. , Gómez-Torrecillas , J. , Lobillo , F. ( 2001 ). Duality and rational semimodules for Hopf algebras over commutative rings . J. Algebra 240 : 165 – 184 .
   Web of Science ®Google Scholar
• Adámek , J. , Herrlich , H. , Strecker , G. E. ( 2009 ). Abstract and Concrete Categories . Dover Publications Edition (available at: http://katmat.math.uni-bremen.de/acc).
   Google Scholar
• Agore , A. L. , Caenepeel , S. , Militaru , G. Braidings on the category of bimodules, Azumaya algebras and epimorphisms of rings . Appl. Categor. Structures (to appear).
   Web of Science ®Google Scholar
• Al-Thani , H. ( 1996 ). k-Projective semimodules . Kobe J. Math. 13 : 49 – 59 .
   Google Scholar
• Adámek , J. , Porst , H. ( 1994 ). Locally Presentable and Accessible Categories . Cambridge : Cambridge University Press .
   Google Scholar
• Adámek , J. , Porst , H. (2003). On varieties and covarieties in a category. Math. Struc. Comp. Science 13:201–232.
   Google Scholar
• Barr , M. ( 1971 ). Exact categories . In: Exact Categories and Categories of Sheaves . Lec. Not. Math. Vol. 236, 1–120, Springer .
   Google Scholar
• Borceux , F. , Bourn , D. ( 2004 ). Mal'cev, protomodular, homological and semi-abelian categories. Mathematics and Its Applications. Vol. 566. Dordrecht: Kluwer Academic Publishers .
   Google Scholar
• Böhm , G. , Brzeziński , T. , Wisbauer , R. ( 2009 ). Monads and comonads on module categories . J. Algebra 322 ( 5 ): 1719 – 1747 .
   Web of Science ®Google Scholar
• Brzeziński , T. , Majid , S. ( 1998 ). Coalgebra bundles . Comm. Math. Phys. 191 : 467 – 492 .
   Web of Science ®Google Scholar
• Borceux , F. ( 1994 ). Handbook of Categorical Algebra. I, Basic Category Theory. Cambridge Univ. Press .
   Google Scholar
• Borceux , F. , Rosický , J. ( 2004 ). On von Neumann varieties . Theory Appl. Categ. 13 ( 1 ): 5 – 26 .
   Google Scholar
• Bourbaki , N. ( 1974 ). Elements of Mathematics, Algebra I. Chapters 1–3. Hermann .
   Google Scholar
• Brzeziński , T. ( 2009 ). Comodules and corings. Handbook of Algebra Vol. 6. Handb. Algebr. 6. Amsterdam: Elsevier/North-Holland, pp. 237–318 .
   Google Scholar
• Brzezinski , T. , Wisbauer , R. ( 2003 ). Corings and Comodules. Lond. Math. Soc. Lec. Not. Ser. 309. Cambridge University Press .
   Google Scholar
• Barr , M. , Wells , C. ( 2005 ). Toposes, Triples and Theories; Corrected Reprint of the 1985 Original, Reprints in Theory Appl. Categ. 12 .
   Google Scholar
• Cuninghame-Green , R. A. ( 1979 ). Minimax Algebra . Lecture Notes in Economics and Mathematical Systems #166 . Berlin : Springer-Verlag .
   Google Scholar
• Caenepeel , S. , Militaru , G. , Zhu , S. ( 2002 ). Frobenius and Separable Functors for Generalized Module Categories and Nonlinear Equations . Vol. 1787. Lect. Not. Math. , Berlin : Springer-Verlag .
   Google Scholar
• Dedekind , R. ( 1894 ). Über die Theorie der ganzen algebraischen Zahlen . Supplement XI to P. G. Dirichlet, L.: Vorlesung über Zahlentheorie. 4 Aufl., Braunschweig: Druck und Verlag.
   Google Scholar
• Eilenberg , S. ( 1974 >). Automata, languages, and machines: Vol. A. Pure and Applied Mathematics. Vol. 58. New York: Academic Press .
   Google Scholar
• Eilenberg , S. ( 1976 ). Automata, Languages, and Machines: Vol. B, with two chapters “Depth decomposition theorem” and “Complexity of semigroups and morphisms” by Bret Tilson. Pure and Applied Mathematics. Vol. 59. New York-London: Academic Press .
   Google Scholar
• Faith , C. ( 1973 ). Algebra: Rings, Modules and Categories. I . Die Grundlehren der mathematischen Wissenschaften. Band 190 . Berlin-Heidelberg-New York : Springer-Verlag .
   Google Scholar
• Golan , J. ( 1999 ). Semirings and Their Applications . Dordrecht : Kluwer .
   Google Scholar
• Gómez-Torrecillas , J. ( 1998 ). Coalgebras and comodules over a commutative ring . Rom. J. Pure Appl. Math. 43 : 591 – 603 .
   Google Scholar
• Gómez-Torrecillas , J. ( 2006 ). Comonads and Galois corings . Appl. Categ. Structures 14 ( 5–6 ): 579 – 598 .
   Web of Science ®Google Scholar
• Gum , P. ( 1999 ). Elements of the General Theory of Coalgebras . Johannesburg , South Africa : LUATCS’99, Rand Afrikaans University .
   Google Scholar
• Gunawardena , J. (ed.). ( 1998 ). Idempotency. Papers from the Workshop Held in Bristol, October 3–7, 1994. Publications of the Newton Institute, 11. Cambridge: Cambridge University Press .
   Google Scholar
• Katsov , Y. (1997). Tensor products and injective envelopes of semimodules over additively regular semirings. Algebra Colloq. 4(2):121–131.
   Web of Science ®Google Scholar
• Katsov , Y. ( 2004 ). On flat semimodules over semirings . Algebra Universalis 51 ( 2–3 ): 287 – 299 .
   Web of Science ®Google Scholar
• Kelly , G. M. Basic Concepts of Enriched Category Theory. Repr. Theory Appl. Categ. 10, Reprint of the 1982 Original, Cambridge: Cambridge Univ. Press.
   Google Scholar
• Kolokoltsov , V. N. , Maslov , V. P. ( 1997 ). Idempotent analysis and its applications. Translation of “Idempotent Analysis and Its Application in Optimal Control” (Russian). Nauka, Moscow, 1994 (With an appendix by Pierre Del Moral); Mathematics and Its Applications. Vol. 401. Dordrecht: Kluwer Academic Publishers Group .
   Google Scholar
• Katsov , Y. , Nam , T. G. ( 2011 ). Morita equivalence and homological characterization of rings . J. Alg. Appl. 10 ( 3 ): 445 – 473 .
   Web of Science ®Google Scholar
• Litvinov , G. L. , Maslov , V. P. , Shpiz , G. B. ( 1999 ). Tensor products of idempotent semimodules. An algebraic approach . Math. Notes 65 ( 3-4 ): 479 – 489 .
   Web of Science ®Google Scholar
• Mesablishvili , B. , Wisbauer , R. ( 2011 ). Bimonads and Hopf monads on categories . J. K-Theory 7 ( 2 ): 349 – 388 .
   Web of Science ®Google Scholar
• Mesablishvili , B. , Wisbauer , R. On rational pairings of functors. Appl. Cat. Struct. DOI 10.1007/s10485-011-9264-1.
   Google Scholar
• Porst , H.-E. ( 2006 ). On corings and comodules. Arch. Math. (Brno) 42(4): 419–425 .
   Google Scholar
• Porst , H.-E. ( 2008 ). Fundamental constructions for coalgebras, corings, and comodules . Appl. Categ. Struct. 16 ( 1–2 ): 223 – 238 .
   Web of Science ®Google Scholar
• Porst , H.-E. ( 2008 ). On categories of monoids, comonoids, and bimonoids . Quaest. Math. 31 ( 2 ): 127 – 139 .
   Web of Science ®Google Scholar
• Puppe , D. ( 1962 ). Korrespondenzen in Abelschen Kategorien . Math. Ann. 148 : 1 – 30 .
   Web of Science ®Google Scholar
• Richter-Gebert , J. , Sturmfels , B. , Theobald , T. ( 2005 ). First steps in tropical geometry . In: Idempotent Mathematics and Mathematical Physics, volume 377 of Contemp. Math. . Providence , RI : Amer. Math. Soc. , pp. 289 – 317 .
   Google Scholar
• Settele , W. ( 1974 ). Über die Eigenschaften der Katergorie der Comoduln über einer Coalgebra, Diplomarbeit, München .
   Google Scholar
• Sweedler , M. ( 1969 ). Hopf Algebras . Benjamin : New York .
   Google Scholar
• Sweedler , M. ( 1975 ). The predual theorem to the Jacobson-Bourbaki theorem . Trans. Amer. Math. Soc. 213 : 391 – 406 .
   Web of Science ®Google Scholar
• Takahashi , M. ( 1981 ). On the bordism categories. II. Elementary properties of semimodules . Math. Sem. Notes Kobe Univ. 9 ( 2 ): 495 – 530 .
   Google Scholar
• Takahashi , M. ( 1982 ). On the bordism categories. III. Functors Hom and for semimodules. Math. Sem. Notes Kobe Univ. 10(2):551–562 .
   Google Scholar
• Takahashi , M. ( 1983 ). Extensions of semimodules, II . Math. Sem. Notes Kobe Univ. 11 ( 1 ): 83 – 118 .
   Google Scholar
• Vercruysse , J. ( 2013 ). Hopf algebras. Variant notions and reconstruction theorems. In: Quantum Physics and Linguistics, A Compositional, Diagrammatic Discourse, Heunen, C., Sadrzadeh, M., Grefenstette E. (eds.). Oxford University Press .
   Google Scholar
• Wisbauer , R. ( 1991 ). Foundations of module and ring theory. A Handbook for Study and Research, Algebra, Logic and Applications. Vol. 3. Philadelphia: Gordon and Breach Science Publishers .
   Google Scholar
• Wisbauer , R. ( 1996 ). Modules and algebras: Bimodule structure and group actions on algebras. Pitman Monographs and Surveys in Pure and Applied Mathematics. Vol. 81. Longman, Harlow .
   Google Scholar
• Wisbauer , R. ( 2012 ). Coalgebraic structures in module theory . Lin. Mult. Alg. 60 : 829 – 853 .
   Web of Science ®Google Scholar
• Worthington , J. (2012). A bialgebraic approach to automata and formal language theory. Ann. Pure Appl. Logic 163(7):745–762.
   Web of Science ®Google Scholar
• Wischnewsky , M. ( 1975 ). On linear representations of affine groups. I . Pac. J. Math. 61 : 551 – 572 .
   Web of Science ®Google Scholar
• Zimmermann-Huisgen , B. ( 1976 ). Pure submodules of direct products of free modules . Math. Ann. 224 : 233 – 245 .
   Web of Science ®Google Scholar
• Communicated by V. A. Artamonov.
   Google Scholar