A note on derivations on basic algebras

Krňávek, Jan; Kühr, Jan

doi:10.1007/s00500-014-1586-0

Cited by 13 publications

(7 citation statements)

References 13 publications

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“…Although Łukasiewicz semirings are in a one-to-one correspondence with MValgebras, we do not define derivations like in [1] or [10] since there the MV-operation is used instead of C. Hence also our results differ essentially from those obtained in the mentioned papers. Now we can adopt the definition of a derivation from [9].…”

Section: Derivationscontrasting

confidence: 66%

“…It was shown in [10], Lemma 5, that basic algebras have only principal derivations. We are going to show that this is not the case for Łukasiewicz semirings.…”

Section: A B/mentioning

confidence: 99%

“…Starting with 2010, several authors extended the study of derivations to certain algebras forming an algebraic semantic for non-classical logics including the logic of quantum mechanics. In this context let us mention the papers [1] and [10]. It was shown in [5] that every MV-algebra can be converted into a socalled Łukasiewicz semiring.…”

Section: Introductionmentioning

confidence: 99%

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Derivations in Lukasiewicz semirings

Chajda¹,

Länger²

2018

MMN

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An axiomatization of classical propositional logic is provided by means of Boolean algebras which are term equivalent to Boolean rings. This is important because rings form a classical part of algebra whose tools can be used for the investigations. The Łukasiewicz manyvalued logic was axiomatized via so-called MV-algebras by C. C. Chang in 1950's. MV-algebras are successfully applied in the logic of quantum mechanics and hence they are considered as quantum structures. It is a natural question if also MV-algebras have their alter ego among classical structures. For this reason the so-called Łukasiewicz semirings were introduced by the first author and his collaborators in [3]-[4]. As shown, Łukasiewicz semirings are term equivalent to MV-algebras and we can use with advantage several developed tools for their study. In particular, we investigate derivations in semirings which were introduced formerly but here these semirings are enriched by an involution.

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Section: Derivationscontrasting

confidence: 66%

“…It was shown in [10], Lemma 5, that basic algebras have only principal derivations. We are going to show that this is not the case for Łukasiewicz semirings.…”

Section: A B/mentioning

confidence: 99%

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Derivations in Lukasiewicz semirings

Chajda¹,

Länger²

2018

MMN

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“…They studied properties of derivations on lattices and characterized modular lattices and distributive lattices by isotone derivations. For subsequent work, see [1,17,19,22,28,37]. There are also studies on generalizations of derivations on lattices, such as generalized derivations [2], higher derivations [7], f -derivations [9], n-derivations and (n, m)-derivations [8].…”

Section: Introductionmentioning

confidence: 99%

On differential lattices

Gan¹,

Guo²

2021

Preprint

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This paper studies the differential lattice, defined to be a lattice L equipped with a map d : L → L that satisfies a lattice analog of the Leibniz rule for a derivation. Isomorphic differential lattices are studied and classifications of differential lattices are obtained for some basic lattices. Several families of derivations on a lattice are explicitly constructed, giving realizations of the lattice as lattices of derivations. Derivations on a finite distributive lattice are shown to have a natural structure of lattice. Moreover, derivations on a complete infinitely distributive lattice form a complete lattice. For a general lattice, it is conjectured that its poset of derivations is a lattice that uniquely determines the given lattice. Contents 1. Introduction 1 2. Differential lattices and basic properties 3 3. Isomorphic classes of differential lattices 6 3.1. Isomorphic differential lattices 6 3.2. Classification of differential lattices 9 4. The lattices of derivations 13 4.1. Lattice structures for derivations on distributive lattices 13 4.2. Lattice structures on inner and other special derivations 16 4.3. Lattice structures for derivations on specific lattices 18 References 21

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“…The notion of derivations from the analytic theory was introduced in 1957 by Posner ([41]) to a prime ring (R, +, •) as a map d : R −→ R satisfying the conditions d(x + y) = d(x) + d(y) and d(x • y) = d(x) • y + x • d(y), for all x, y ∈ R. Since the derivation proved to be helpful for studying the properties of algebraic systems, this notion has been defined and studied by many authors for the cases of lattices ( [44], [23], [45], [46]) and algebras of fuzzy logic: MValgebras ( [3]), [47], [22]), BCI-algebras ( [30], [1], [2]), commutative residuated lattice ( [25]), BCC-algebras ( [42], [4]), BE-algebras ( [32])), basic algebras ( [34]) and pseudo-MV algebras ( [43]).…”

Section: Introductionmentioning

confidence: 99%

Pseudo BCI-algebras with derivations

Ciungu

2019

Preprint

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In this paper we define two types of implicative derivations on pseudo-BCI algebras, we investigate their properties and we give a characterization of regular implicative derivations of type II. We also define the notion of a d-invariant deductive system of a pseudo-BCI algebra A proving that d is a regular derivation of type II if and only if every deductive system on A is d-invariant. It is proved that a pseudo-BCI algebra is p-semisimple if and only if the only regular derivation of type II is the identity map. Another main result consists of proving that the set of all implicative derivations of a p-semisimple pseudo-BCI algebra forms a commutative monoid with respect to function composition. Two types of symmetric derivations on pseudo-BCI algebras are also introduced and it is proved that in the case of p-semisimple pseudo-BCI algebras the sets of type II implicative derivations and type II symmetric derivations are equal.

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A note on derivations on basic algebras

Cited by 13 publications

References 13 publications

Derivations in Lukasiewicz semirings

Derivations in Lukasiewicz semirings

On differential lattices

Pseudo BCI-algebras with derivations

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