Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-10T20:05:56.309Z Has data issue: false hasContentIssue false
  • English
  • Français

A LOGICAL AND ALGEBRAIC CHARACTERIZATION OF ADJUNCTIONS BETWEEN GENERALIZED QUASI-VARIETIES

Published online by Cambridge University Press:  23 October 2018

TOMMASO MORASCHINI
TOMMASO MORASCHINI*
Affiliation:
DEPARTMENT OF THEORETICAL COMPUTER SCIENCEINSTITUTE OF COMPUTER SCIENCE CZECH ACADEMY OF SCIENCES PRAGUE, CZECH REPUBLIC E-mail: moraschini@cs.cas.cz
Get access Rights & Permissions [Opens in a new window]

Abstract

We present a logical and algebraic description of right adjoint functors between generalized quasi-varieties, inspired by the work of McKenzie on category equivalence. This result is achieved by developing a correspondence between the concept of adjunction and a new notion of translation between relative equational consequences.

Keywords

18A1518A4003C0503G27adjunctionadjoint functorcategory theoryuniversal algebracategory equivalencematrix powercontextual translationlocally presentable category
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 3 , September 2018 , pp. 899 - 919
Copyright
Copyright © The Association for Symbolic Logic 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Adámek, J., How many variables does a quasivariety need? Algebra Universalis, vol. 27 (1990), pp. 44–48.CrossRefGoogle Scholar
Adámek, J., Herrlich, H., and Strecker, G. E., Abstract and concrete categories: The joy of cats. Reprints in Theory and Applications of Categories, vol. 17 (2006), pp. 1–507.Google Scholar
Adámek, J. and Rosický, J., Locally Presentable and Accessible Categories, London Mathematical Society Lecture Note Series, vol. 189, Cambridge University Press, Cambridge, 1994.CrossRefGoogle Scholar
Adámek, J., Rosický, J., and Vitale, E. M., Algebraic Theories: A Categorical Introduction to General Algebra, Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2011.Google Scholar
Awodey, S., Category Theory, Oxford Logic Guides, vol. 49, The Clarendon Press Oxford University Press, New York, 2006.CrossRefGoogle Scholar
Bergman, C.. Universal Algebra: Fundamentals and Selected Topics, Pure and Applied Mathematics, Chapman and Hall/CRC, London, 2011.CrossRefGoogle Scholar
Blok, W. J. and Jónsson, B., Equivalence of consequence operations. Studia Logica, vol. 83 (2006), no. 1–3, pp. 91–110.CrossRefGoogle Scholar
Blok, W. J. and Pigozzi, D., Algebraizable Logics, Memoirs of the American Mathematical Society, vol. 396, American Mathematical Society, Providence, 1989.Google Scholar
Burris, S. and Sankappanavar, H. P., A Course in Universal Algebra. The millennium edition, 2012. Available at https://www.math.uwaterloo.ca/∼snburris/htdocs/ualg.html.Google Scholar
Cignoli, R., The class of Kleene algebras satisfying an interpolation property and Nelson algebras. Algebra Universalis, vol. 23 (1986), no. 3, pp. 262–292.CrossRefGoogle Scholar
Davey, B. A. and Priestley, H. A., Introduction to Lattices and Order, second ed., Cambridge University Press, New York, 2002.CrossRefGoogle Scholar
Dukarm, J. J., Morita equivalence of algebraic theories. Colloquium Mathematicum, vol. 55 (1988), pp. 11–17.CrossRefGoogle Scholar
Dummett, M. and Lemmon, E. J., Modal logics between S4 and S5. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 5 (1959), pp. 250–264.CrossRefGoogle Scholar
Freyd, P., Algebra valued functors in general and tensor products in particular. Colloquium Mathematicum, vol. 14 (1966), pp. 89–106.CrossRefGoogle Scholar
Gödel, K., Eine Interpretation des intuitionistischen Aussagenkalküls. Ergebnisse eines mathematisches Kolloquiums, vol. 4 (1933), pp. 39–40.Google Scholar
Gorbunov, V. A., Algebraic Theory of Quasivarieties, Siberian School of Algebra and Logic, Consultants Bureau, New York, 1998.Google Scholar
Kalman, J. A., Lattices with involution. Transactions of the Americal Mathematical Society, vol. 87 (1958), pp. 485–491.CrossRefGoogle Scholar
Kolmogorov, A. N., Sur le principe de tertium non datur. Matematicheskii Sbornik, vol. 32 (1925), pp. 646–667.Google Scholar
Kracht, M., Tools and Techniques in Modal Logic, Studies in Logic and the Foundations of Mathematics, vol. 142, North-Holland, Amsterdam, 1999.CrossRefGoogle Scholar
Kracht, M., Modal consequence relations, Handbook of Modal Logic, Chapter 8, Elsevier Science Inc., New York, NY, 2006.Google Scholar
Mac Lane, S., Categories for the Working Mathematician, second ed., Graduate Texts in Mathematics, vol. 5, Springer-Verlag, New York, 1998.Google Scholar
Maksimova, L. L. and Rybakov, V. V., A lattice of normal modal logics. Algebra and Logic, vol. 13 (1974), pp. 105–122.CrossRefGoogle Scholar
Mal’cev, A. I., The Metamathematics of Algebraic Systems, Collected Papers: 1936–1967, North-Holland, Amsterdam, 1971.Google Scholar
McKenzie, R., An algebraic version of categorical equivalence for varieties and more general algebraic categories, Logic and Algebra (Pontignano, 1994) (Aglianò, P. and Magari, R., editors), Lecture Notes in Pure and Applied Mathematics, vol. 180, Dekker, New York, 1996, pp. 211–243.Google Scholar
McKenzie, R. N., McNulty, G. F., and Taylor, W. F., Algebras, Lattices, Varieties, vol. I, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1987.Google Scholar
McKinsey, J. C. C. and Tarski, A., Some theorems about the sentential calculi of Lewis and Heyting, this JOURNAL, vol. 13 (1948), pp. 1–15.Google Scholar
Porst, H.-E., Equivalence for varieties in general and for ${\cal B}{\cal O}{\cal O}{\cal L}$ in particular . Algebra Universalis , vol. 43 (2000), pp. 157–186.CrossRefGoogle Scholar
Porst, H.-E., Generalized morita theories. Notices of the South African Mathematical Society, vol. 32 (2001), pp. 4–16.Google Scholar