Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T13:07:14.304Z Has data issue: false hasContentIssue false
  • English
  • Français

Congruence quasi-orderability in subtractive varieties

Part of: Algebraic structures

Published online by Cambridge University Press:  09 April 2009

Paolo Agliano
Paolo Agliano
Affiliation:
Dipartimento di Matematica, Via del Captino 15, 53100, Siena, Italy e-mail: [email protected]
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 congruence quasi-orderable. Though this concept has its origin in abstract algebraic logic, it seems to be worth investigating in a purely algebraic fashion. Besides clarifying the algebraic meaning of this notion, we obtain several structure theorems about such varieties. Also several examples are provided to illustrate the theory.

Keywords

subtractive varietyquasi-orderability

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. 421 - 446
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Agliano, P., ‘Fregean subtractive varieties with definable congruences’, J. Austral. Math. Soc. 71 (2000), 353–366.Google Scholar
[2]Agliano, P. and Ursini, A., ‘Ideals and other generalizations of congruence classes’, J. Austral. Math. Soc. Ser A 53 (1992), 103–115.CrossRefGoogle Scholar
[3]Agliano, P. and Ursini, A., ‘On subtractive varieties II: general properties’, Algebra Universalis 36 (1996), 222–259.CrossRefGoogle Scholar
[4]Agliano, P. and Ursini, A., ‘On subtractive varieties III: From ideals to congruences’, Algebra Universalis 37 (1997), 296–333.Google Scholar
[5]Agliano, P. and Ursini, A., ‘On subtractive varieties IV: definability of principal ideals’, Algebra Universalis 38 (1997), 355–389.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.Google Scholar
[7]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
[8]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 Appl. Math. 203 (Marcel Dekker, New York, 1999) pp. 167–186.Google Scholar
[9]Büchi, J. R. and Owens, M. T., ‘Complemented monoids and hoops’, unpublished manuscript.Google Scholar
[10]Czelakowski, J. and Pigozzi, D., ‘Fregean algebraic logic’, preprint.Google Scholar
[11]Czelakowski, J. and Pigozzi, D., ‘Fregean logics with the multiterm deduction theorem and their algebrization’, preprint.Google Scholar
[12]Font, J., ‘On the Leibniz congruences’, in: Algebraic methods in logic an computer science (ed. Rauszer, C.), Banach Center Publ. 28 (Polish. Acad. Sci., Warszawa, 1993) pp. 17–36.Google Scholar
[13]Font, J. and Jansana, R., A general semantics for sentential logics, Lecture Notes in Logic 7 (Springer, New York, 1996).CrossRefGoogle Scholar
[14]Hecht, T. and Katrinak, T., ‘Equational classes of relative Stone algebras’, Notre Dame J. Formal Logic 13 (1972), 248–254.Google Scholar
[15]Idziak, P. M., Slomczyńska, K. and Wroński, A., ‘Equivalential algebras: a study of Fregean varieties’, preprint, 1996.Google Scholar
[16]Kabziński, J. K. and Wroński, A., ‘On equivalential algebras’, in: Proceedings of the 1975 international Symposium om Multiple-Valued Logic (Bloomington. Ind., 1975) (IEEE Comput. Soc., Long Beach, CA. 1975) pp. 419–428.Google Scholar
[17]Köhler, P. and Pigozzi, D., ‘Varieties with equationally definable principal congruences’, Algebra Universalis 11 (1980), 213–219.Google Scholar
[18]Nemitz, W. and Whaley, T., ‘Varieties of implicative semilattices’, Pacific J. Math. 37 (1971), 759–769.CrossRefGoogle Scholar
[19]Pigozzi, D., ‘Fregean algebraic logic’, in: Algebraic logic (eds. Andréka, H., Monk, J. D. and Németi, I.), Colloq. Math. Soc. János Bolyai 54 (North-Holland, Amsterdam, 1991) pp. 473–502.Google Scholar
[20]Ursini, A., ‘On subtractive varieties I’, Algebra Universalis 31 (1994), 204–222.CrossRefGoogle Scholar
[21]Wroński, A., ‘BCK-algebras do not form a variety’, Math. Japon. 28 (1983), 211–213.Google Scholar