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The structure of completely meet irreducible congruences in strongly Fregean algebras

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Abstract

A strongly Fregean algebra is an algebra such that the class of its homomorphic images is Fregean and the variety generated by this algebra is congruence modular. To understand the structure of these algebras we study the prime intervals projectivity relation in the posets of their completely meet irreducible congruences and show that its cosets have the natural structure of a Boolean group. In particular, this approach allows us to represent congruences and elements of such algebras as the subsets of upward closed subsets of these posets with some special properties.

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References

  1. Aichinger, E.: Congruence lattices forcing nilpotency. J. Algebra Appl. 17, 1850033 (2018)

    Article  MathSciNet  Google Scholar 

  2. Czelakowski, J., Pigozzi, D.: Fregean logics. Ann. Pure Appl. Logic 127, 17–76 (2004)

    Article  MathSciNet  Google Scholar 

  3. Czelakowski, J., Pigozzi, D.: Fregean logics with the multiterm deduction theorem and their algebraization. Studia Logica 78, 171–212 (2004)

    Article  MathSciNet  Google Scholar 

  4. Font, J.M., Jansana, R., Pigozzi, D.: A survey of abstract algebraic logic. Studia Logica 74, 13–97 (2003)

    Article  MathSciNet  Google Scholar 

  5. Freese, R., McKenzie, R.: Commutator theory for congruence modular varieties. London Math. Soc. Lecture Notes, vol. 125. Cambridge University Press, Cambridge (1987)

  6. Grätzer, G.: Lattice Theory: Foundation. Birkhäuser, Basel (2011)

    Book  Google Scholar 

  7. Hagemann, J.: On regular and weakly regular congruences (1973) (preprint)

  8. Idziak, P.M., Słomczyńska, K.: Polynomially rich algebras. J. Pure Appl. Algebra 156, 33–68 (2001)

    Article  MathSciNet  Google Scholar 

  9. Idziak, P.M., Słomczyńska, K., Wroński, A.: Equivalential algebras: a study of Fregean varieties. In: Font, J., Jansana, R., Pigozzi, D. (eds.) Proceedings of the Workshop on Abstract Algebraic Logic (1997). CRM Quaderns, vol. 10, pp. 95–100. Centre de Recerca Matemàtica, Barcelona (1998)

  10. Idziak, P.M., Słomczyńska, K., Wroński, A.: Fregean varieties. Int. J. Algebra Comput. 19, 595–645 (2009)

    Article  MathSciNet  Google Scholar 

  11. Kabziński, J., Wroński, A.: On equivalential algebras. In: Epstein, G. (ed.) Proceedings of the 1975 International Symposium on Multiple-Valued Logic, pp. 419–428. Indiana University, Bloomington (1975)

  12. Pigozzi, D.: Fregean algebraic logic. In: Andréka, H., Monk, J.D., Németi, I. (eds.) Algebraic Logic, Colloq. Math. Soc. János Bolyai, vol. 54, pp. 473–502. North-Holland, Amsterdam (1991)

  13. Przybyło, S., Słomczyńska, K.: Free finitely generated linear Hilbert algebras with supremum. J. Mult.-Valued Logic Soft Comput. 29, 135–156 (2017)

  14. Słomczyńska, K.: Equivalential algebras. Part I: Representation. Algebra Universalis 35, 524–547 (1996)

    Article  MathSciNet  Google Scholar 

  15. Słomczyńska, K.: Free spectra of linear equivalential algebras. J. Symb. Log. 70, 1341–1358 (2005)

    Article  MathSciNet  Google Scholar 

  16. Słomczyńska, K.: Free equivalential algebras. Ann. Pure Appl. Logic 155, 86–96 (2008)

    Article  MathSciNet  Google Scholar 

  17. Słomczyńska, K.: Algebraic semantics for the (\(\leftrightarrow ,\lnot \))-fragment of IPC and its properties. Math. Log. Q. 63, 202–210 (2017)

    Article  MathSciNet  Google Scholar 

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  1. Institute of Mathematics, Pedagogical University, Podchora̧żych 2, 30-084, Kraków, Poland

    Katarzyna Słomczyńska

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  1. Katarzyna Słomczyńska
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Correspondence to Katarzyna Słomczyńska.

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Presented by J. G. Raftery.

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Słomczyńska, K. The structure of completely meet irreducible congruences in strongly Fregean algebras. Algebra Univers. 83, 31 (2022). https://doi.org/10.1007/s00012-022-00787-0

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