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Soft Computing
Published in: Soft Computing 14/2018

20-11-2017 | Foundations

On special elements and pseudocomplementation in lattices with antitone involutions

Authors: Petr Emanovský, Jan Kühr

Published in: Soft Computing | Issue 14/2018

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Abstract

The so-called basic algebras correspond in a natural way to lattices with antitone involutions and hence generalize both MV-algebras and orthomodular lattices. The paper deals with several types of special elements of basic algebras and with pseudocomplemented basic algebras.
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Footnotes
1
This identity is one of the aforementioned additional conditions which lead to algebras similar to MV-algebras; see Krňávek and Kühr (2011) and Botur et al. (2014).
 
2
A note on terminology: when speaking of lattices with antitone involution(s), we omit the adjective “bounded”.
 
3
In fact, the quasi-identity \((x\le \lnot y\) & \(x\oplus y\le \lnot z)\) \(\Rightarrow \) \((x\oplus y)\oplus z=x\oplus (z\oplus y)\) was used in Chajda et al. (2009a, b), but it is possible to show that it is equivalent to (2.10).
 
4
In the variety generated by linearly ordered basic algebras, (2.13) is equivalent to the quasi-identity \(x\le y\) \(\Rightarrow \) \(z\oplus x\le z\oplus y\), but we do not know whether this is true in general.
 
5
Given \(B\subseteq A\), this is not equivalent to saying that \((B,\vee ,\wedge ,\lnot ,0,1)\) is a Boolean algebra. It can easily happen that \((B,\vee ,\wedge ,\lnot ,0,1)\) is a Boolean algebra, but \((B,\oplus ,\lnot ,0,1)\) is not a Boolean subalgebra of \((A,\oplus ,\lnot ,0,1)\), because B need not be closed under \(\oplus \).
 
6
As in the case of Boolean subalgebras, this is stronger than saying that \((B,\vee ,\wedge ,\lnot ,0,1)\) is an orthomodular lattice.
 
7
This means that the relative complementation in [a, 1], which is the natural antitone involution in [a, 1], is replaced with another antitone involution. Of course, this is possible, provided that the interval has more than two elements. For a concrete example, see Chajda and Kühr (2013b), Example 3.1 or Krňávek and Kühr (2015), Example 14.
 
8
Namely, the identity \(x\oplus (\lnot x\wedge y)=x\oplus y\) in Krňávek and Kühr (2011), and the identity \(x\le x\oplus y\) in Botur and Kühr (2014).
 
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Metadata
Title
On special elements and pseudocomplementation in lattices with antitone involutions
Authors
Petr Emanovský
Jan Kühr
Publication date
20-11-2017
Publisher
Springer Berlin Heidelberg
Published in
Soft Computing / Issue 14/2018
Print ISSN: 1432-7643
Electronic ISSN: 1433-7479
DOI
https://doi.org/10.1007/s00500-017-2926-7

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