Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-28T01:15:26.448Z Has data issue: false hasContentIssue false
  • English
  • Français

Fregean subtractive varieties with definable congruence

Part of: Algebraic structures

Published online by Cambridge University Press:  09 April 2009

Paolo Agliano
Paolo Agliano
Affiliation:
Dipartimento di Matematica, Via del Capitano 15, 53100, Siena, Italy e-mail: agliano@unisi.it
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we investigate subtractive varieties of algebras that are Fregean in order to get structure theorems about them. For instance it turns out that a subtractive variety is Fregean and has equationally definable principal congruences if and only if it is termwise equivalent to a variety of Hilbert algebras with compatible operations. Several examples are provided to illustrate the theory.

Keywords

Fregeansubtractive varietydefinable congruences

MSC classification

Secondary: 08A99: None of the above, but in this section 08A30: Subalgebras, congruence relations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 71 , Issue 3 , December 2001 , pp. 353 - 366
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Agliano, P. and Ursini, A., ‘Ideals and other generalizations of congruence classes’, J. Austral. Math. Soc. Ser A 53 (1992), 103–115.CrossRefGoogle Scholar
[2]Agliano, P. and Ursini, A., ‘On subtractive varieties II: general properties’, Algebra Universalis 36 (1996), 222–259.CrossRefGoogle Scholar
[3]Agliano, P. and Ursini, A., ‘On subtractive varieties III: From ideals to congruences’, Algebra Universalis 37 (1997), 296–333.CrossRefGoogle Scholar
[4]Agliano, P. and Ursini, A., ‘On subtractive varieties IV: definability of principal ideals’, Algebra Universalis 38 (1997), 355–389.CrossRefGoogle Scholar
[5]Barbour, G. D. and Raftery, J. G., ‘On quasivarieties of logic, regularity conditions and parametrized algebrizations’, Preprint, 1997.Google Scholar
[6]Blok, W. J., Köhler, P. and Pigozzi, D., ‘On the structure of varieties with equationally definable principal congruences II’, Algebra Universalis 18 (1984), 334–379.CrossRefGoogle Scholar
[7]Blok, W. J. and Pigozzi, D., ‘On the structure of varieties with equationally definable principal congruences I’, Algebra Universalis 15 (1982), 195–227.CrossRefGoogle Scholar
[8]Blok, W. J. and Pigozzi, D., ‘The deduction theorem in algebraic logic’, unpublished manuscript, 1989.Google Scholar
[9]Blok, W. J. and Pigozzi, D., ‘Algebraic semantics for universal horn logic without equality’, in: Universal algebra and quasi-group theory (eds. Romanowska, A. and Smith, J. H. D.) (Heldermann, Berlin, 1992) pp. 1–56.Google Scholar
[10]Blok, W. J. and Raftery, J. G., ‘Ideals in quasivarieties of algebras’, in: Models, algebras and proofs (eds. Caceido, X. and Montenegro, C. H.), Lecture Notes in Pure and Appi. Math. 203 (Marcel Dekker, New York, 1999) pp. 167–186.Google Scholar
[11]Büchi, J. R. and Owens, M. T., ‘Skolem rings and their varieties’, in: The collected works of J. Richard Büchi (eds. MacLane, S. and Siefkes, D.) (Springer, New York, 1990) pp. 161–221.CrossRefGoogle Scholar
[12]Czelakowski, J. and Pigozzi, D., ‘Fregean logics with the multiterm deduction theorem and their algebrization’, Preprint.Google Scholar
[13]Diego, A., Sur les algèbres de Hubert, Collection de Logique Mathematique, Series A, no. 21 (Gauthier-Villars, Paris, 1966).Google Scholar
[14]Font, J. M. and Rius, M., ‘Tetravalent modal alebtas and logics’, Mathematicspreprint series 191, (Universitat de Barcelona, Barcelona, 1995).Google Scholar
[15]Gumm, H. P. and Ursini, A., ‘Ideals in universal algebra’, Algebra Universalis 19 (1984), 45–54.CrossRefGoogle Scholar
[16]Hagemann, J., ‘On regular and weakly regular congruences’, Preprint 75, (TH, Darmstadt, 1975).Google Scholar
[17]Hobby, D. and McKenzie, R., The structure of finite algebras (tame congruence theory), Contemp. Math. (American Mathematical Society, Providence, RI, 1988).CrossRefGoogle Scholar
[18]Idziak, P. M., Slomczyńska, K. and Wroński, A., ‘Equivalential algebras: a study of Fregean varieties’, Preprint, 1996.Google Scholar
[19]Köhler, P., ‘Brouwerian semilattices’, Trans. Amer Math. Soc. 68 (1981), 103–126.CrossRefGoogle Scholar
[20]Pigozzi, D., ‘Fregean algebraic logic’, in: Algebraic logic (eds. Andréka, H., Monk, J. D. and Németi, I.), Colloq. Math. Soc. J´nos Bolyai 54 (North-Holland, Amsterdam, 1991) pp. 473–502.Google Scholar
[21]Ursini, A., ‘On subtractive varieties I’, Algebra Universalis 31 (1994), 204–222.CrossRefGoogle Scholar