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Link to original content: https://link.springer.com/doi/10.1007/s00500-012-0915-4
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Ideals and congruences of basic algebras

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Abstract

MV-algebras as well as orthomodular lattices can be seen as a particular case of so-called “basic algebras” which are an alter ego of bounded lattices whose sections are equipped with fixed antitone involutions. The class of basic algebras is an ideal variety. In the paper, we give an internal characterization of congruence kernels (ideals) and find a finite basis of ideal terms, with focus on monotone and effect basic algebras. We also axiomatize basic algebras that are subdirect products of linearly ordered ones.

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Notes

  1. Almost nothing—in fact, the intersection of basic algebras and BL-algebras are MV-algebras. Indeed, if \(\langle A;\vee ,\wedge ,\odot ,\rightarrow ,0,1\rangle \) is a BL-algebra, then for each \(a\in A\), the map \(\varphi _a:[a,1]\rightarrow [a,1]\) defined by \(\varphi _a(x)=x\rightarrow a\) is antitone, but it need not be an involution. The \(\varphi _a\)’s are involutions only if the BL-algebra satisfies the law of double negation, i.e., if it is (equivalent to) an MV-algebra.

  2. At first glance, this might seem to be a misprint, but we do mean “p-ideals”, not pre-ideals. It can happen that a pre-ideal of an orthomodular basic algebra is not a p-ideal (or even a lattice ideal) of the corresponding orthomodular lattice (see Krňávek and Kühr 2011).

  3. The original result is due to Ravindran (1996); in Dvurečenskij and Vetterlein (2001), a more general result is proved for pseudo-effect algebras and pseudo-MV-algebras that are non-commutative counterparts of effect algebras and MV-algebras, respectively.

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Acknowledgments

This research has been supported by the ESF project no. CZ.1.07/2.3.00/20.0051 “Algebraic Methods in Quantum Logic”.

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Authors and Affiliations

  1. Department of Algebra and Geometry, Faculty of Science, Palacký University in Olomouc, 17. listopadu 12, 77146, Olomouc, Czech Republic

    Ivan Chajda & Jan Kühr

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  1. Ivan Chajda
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  2. Jan Kühr
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Correspondence to Jan Kühr.

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Chajda, I., Kühr, J. Ideals and congruences of basic algebras. Soft Comput 17, 401–410 (2013). https://doi.org/10.1007/s00500-012-0915-4

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