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A note on derivations on basic algebras

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Abstract

Generalizing derivations on MV-algebras, we introduce derivations on the so-called “basic algebras” which are a common abstraction of MV-algebras and orthomodular lattices. We prove that derivations coincide with projections onto certain intervals and in some particular cases, such as in MV-algebras, they correspond to certain direct product decompositions.

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Notes

  1. In Krňávek and Kühr (2011), this result was proved for basic algebras satisfying the identity \(x\oplus (\lnot x\wedge y)=x\oplus y\), and in Botur and Kühr (2014), for those satisfying the identity \(x\le x\oplus y\). Note that the former identity holds in all linearly ordered basic algebras.

  2. In a non-commutative basic algebra \(A\), the conditions (4) and (5) are not equivalent without additivity of \(d\), but we are going to show that if \(d:A\rightarrow A\) is additive, then it satisfies (4) iff it satisfies (5). Thus there is only one type of derivations on \(A\), in symbols, \(\mathcal {D}({A}) = \mathcal {D}_1({A}) = \mathcal {D}_2({A})\).

  3. Though it is clear, we emphasize that (i) entails \(d(x)=x\) for \(x\le d(1)\), and in particular, \(d(d(x))=d(x)\) for every \(x\in A\). Thus \(d\) is an interior operator on \(A\).

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Authors and Affiliations

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

    Jan Krňávek & Jan Kühr

Authors
  1. Jan Krňávek
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  2. Jan Kühr
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Corresponding author

Correspondence to Jan Kühr.

Additional information

Communicated by L. Spada.

Supported by the ESF project “Algebraic Methods in Quantum Logic”, CZ.1.07/2.3.00/20.0051, and by the Palacký University project “Mathematical Structures”, IGA PrF 2014016.

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Krňávek, J., Kühr, J. A note on derivations on basic algebras. Soft Comput 19, 1765–1771 (2015). https://doi.org/10.1007/s00500-014-1586-0

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  • DOI: https://doi.org/10.1007/s00500-014-1586-0

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