Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-24T03:00:26.625Z Has data issue: false hasContentIssue false
  • English
  • Français

Free spectra of linear equivalential algebras

Published online by Cambridge University Press:  12 March 2014

Katarzyna Slomczyńska
Katarzyna Slomczyńska*
Affiliation:
Institute of Mathematics, Pedagogical University, UL. Podchorażych 2, 30-084 Kraków, Poland, E-mail: kslomcz@ap.krakow.pl
Get access Rights & Permissions [Opens in a new window]

Abstract

We construct the finitely generated free algebras and determine the free spectra of varieties of linear equivalential algebras and linear equivalential algebras of finite height corresponding, respectively, to the equivalential fragments of intermediate Gödel-Dummett logic and intermediate finite-valued logics of Gödel. Thus we compute the number of purely equivalential propositional formulas in these logics in n variables for an arbitrary n ∈ ℕ.

Keywords

Gödel-Dummett logicintuitionistic equivalencefree algebrasfree spectraequivalential algebras
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 4 , December 2005 , pp. 1341 - 1358
Copyright
Copyright © Association for Symbolic Logic 2005

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

[1]Agliano, P., Fregean subtractive varieties with definable congruence. Journal of the Australian Mathematical Society, vol. 71 (2001), pp. 353–366.CrossRefGoogle Scholar
[2]Berman, J. and Idziak, P. M., Generative complexity in algebra. Memoirs of the American Mathematical Society, vol. 175 (2005), no. 828.CrossRefGoogle Scholar
[3]Blok, W. J. and Pigozzi, D., Algebraizable logics. Memoirs of the American Mathematical Society, vol. 77 (1989), no. 396.CrossRefGoogle Scholar
[4]Czelakowski, J., Protoalgebraic logics, Kluwer, Amsterdam. 2000.Google Scholar
[5]Czelakowski, J. and Pigozzi, D., Fregean logics, Annals of Pure and Applied Logic, vol. 127 (2004), pp. 17–76.CrossRefGoogle Scholar
[6]Czelakowski, J., Fregean logics with the multiterm deduction theorem and their algebraization, Studia Logica, vol. 78 (2004), pp. 171–212.CrossRefGoogle Scholar
[7]Diego, A., Sur les algèbres de Hilbert, Gauthier-Villars, Paris, 1966.Google Scholar
[8]Dummett, M., A prepositional calculus with denumerable matrix, this Journal, vol. 24 (1959), pp. 97–106.Google Scholar
[9]Font, J. M., Jansana, R., and Pigozzi, D., A survey of abstract algebraic logic, Studia Logica, vol. 74 (2003), pp. 13–97.CrossRefGoogle Scholar
[10]Frege, G., Über Sinn und Bedeutung, Zeitschrift für Philosophie und philosophische Kritik, vol. 100 (1892), pp. 25–50, for English translation see: On sense and reference, Translations from the Philosophical Writings of Gottlob Frege (Geach, P. and Black, M., editors), Basil Blackwell, Oxford, 1952, pp. 56-78.Google Scholar
[11]Gödel, K., Zum intuitionistischen Aussagenkalkül, Anzeiger der Akademie der Wissenschaften in Wien, vol. 69 (1932), pp. 65–66.Google Scholar
[12]Grätzer, G. and Kisielewicz, A., A survey of some open problems on pn-sequences and free spectra of algebras and varieties. Universal algebra and quasigroup theory (Romanowska, A. and Smith, J.D.H., editors). Heldermann, Berlin, 1992, pp. 57–88.Google Scholar
[13]Hobby, D. and McKenzie, R., The structure of finite algebras. Contemporary Mathematics, vol. 76, American Mathematical Society, Providence, RI, 1988.Google Scholar
[14]Horn, A., Free L-algebras, this Journal, vol. 34 (1969), pp. 475–480.Google Scholar
[15]Idziak, P. M., Słomczyńska, K., and Wroński, A., Equivalential algebras: A study of fregean varieties. Proceedings of the workshop on abstract algebraic logic, Spain, July 1–5,1997 (Font, J., Jansana, R., and Pigozzi, D., editors), CRM Quaderns 10, Barcelona, 1998, pp. 95–100.Google Scholar
[16]Idziak, P. M. and Słomczyńska, K., Polynomially rich algebras. Journal of Pure and Applied Algebra, vol. 156 (2001), pp. 33–68.CrossRefGoogle Scholar
[17]Kabziński, J. K. and Wroński, A., On equivalential algebras, Proceedings of the 1975 international symposium on multiple-valued logic (Indiana University, Bloomington, Indianapolis, May 13-16, 1975), IEEE Computer Society, Long Beach, California, 1975, pp. 419–428.Google Scholar
[18]Pigozzi, D., Fregean algebraic logic, Algebraic logic (Budapest, 1988), Colloquia Mathematica Societatis János Bolyai, vol. 54, North-Holland, Amsterdam, 1991, pp. 473–502.Google Scholar
[19]Sloane, N. J. A., The on-line encyclopedia of integer sequences, published electronically at www.research.att.com/~njas/sequences/,2005.Google Scholar
[20]Słomczyńska, K., Linear equivalential algebras, Reports on Mathematical Logic, (1995), no. 29, pp. 41–58.Google Scholar
[21]Słomczyńska, K., Equivalential algebras. Part I: Representation, Algebra Universalis, vol. 35 (1996), pp. 524–547.CrossRefGoogle Scholar
[22]Suszko, R., Non-Fregean logic and theories, Analele Universităţii Bucuresţi Matemátică, Acta Logica, vol. 9 (1968), pp. 105–125.Google Scholar
[23]Suszko, R., Abolition of the Fregean axiom, Logic colloquium (Boston, MA, 1972-1973) (Parikh, R., editor), Lecture Notes in Mathematics, vol. 453, Springer, Berlin, 1975, pp. 169–239.Google Scholar
[24]Tax, R. E., On the intuitionistic equivalential calculus, Notre Dame Journal of Formal Logic, vol. 14 (1973), pp. 448–456.CrossRefGoogle Scholar
[25]Thomas, I., Finite limitations on Dummett's LC, Notre Dame Journal of Formal Logic, vol. 3 (1962), pp. 170–174.CrossRefGoogle Scholar
[26]Wilf, H. S., Generating functionology, Academic Press Inc., San Diego, 1994.Google Scholar
[27]Wojtylak, P. and Wroński, A., On the problem of R. E. Tax, Reports on Mathematical Logic, (2001), no. 35, pp. 87–101.Google Scholar
[28]Wroński, A., On the free equivalential algebras with three generators. Bulletin of the Section of Logic of the University of Lódź, vol. 22 (1993), pp. 37–39.Google Scholar