Skew residuated lattices

Chajda, Ivan; Krňávek, Jan

doi:10.1016/j.fss.2012.11.019

Cited by 7 publications

(3 citation statements)

References 3 publications

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“…Then the structure (R + , ≤, •, ÷, 0) is a commutative positive divisibility semiloop in the sense of Bosbach [2] (also see [10]), and hence, by [2,Prop. 2.11], it is the positive cone a linearly ordered commutative loop, for instance, of the linearly ordered commutative loop (R, ≤, •, ∼, 0) from Example 3.3.…”

Section: Lexicographic Productsmentioning

confidence: 99%

On non-associative generalizations of MV-algebras and lattice-ordered commutative loops

Krňávek

Kühr

2016

Fuzzy Sets and Systems

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Section: Lexicographic Productsmentioning

confidence: 99%

On non-associative generalizations of MV-algebras and lattice-ordered commutative loops

Krňávek

Kühr

2016

Fuzzy Sets and Systems

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“…Residuated lattices, introduced by Ward and Dilworth in [17], are a common structure among algebras associated with logical systems. Chajda et.al introduced a non-commutative generalization of the residuated lattice and called it skew residuated lattice [3]. Another non-commutative generalization of the residuated lattice is given by residuum on the skew lattice which is called residuated skew lattice [2,18].…”

Section: Introductionmentioning

confidence: 99%

(Skew) Filters in Residuated Skew Lattic

Koohnavard¹,

Saeid²

2018

SACS

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In this paper, we show the relationship between (skew) deductive system and (skew) filter in residuated skew lattices. It is shown that if a residuated skew lattice is conormal, then any skew deductive system is a skew filter under a condition and deductive system and skew deductive system are equivalent under some conditions too. It is investigated that in branchwise residuated skew lattice, filter, deductive system and skew deductive system are equivalent. We define some types of prime (skew) filters in residuated skew lattices and show the relationship between prime (skew) filters and residuated skew chains. It is proved that in prelinear residuated skew lattice any proper filter can be extended to a maximal, prime filter of type (I). The notion of the radical of a filter is defined and several characterizations of the radical of a filter are given. We show that in non conormal prelinear residuated skew lattice with element 0, infinitesimal elements are equal to intersection of all the maximal filters.

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“…A good candidate for this purpose could be the so-called skew residuated lattice introduced by the author and J. Krňávek in [6]. However, a certain modification is necessary.…”

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A representation of skew effect algebras

Chajda

2014

Mathematica Slovaca

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ABSTRACT. It was shown by the author and R. Halaš that every effect algebra can be organized into a conditionally residuated structure. Skew effect algebras were introduced as a non-associative modification of effect algebras. Hence, there is natural question if a similar characterization by a certain residuated structure is possible. For this we use the so-called skew residuated structure introduced recently by the author and J. Krňávek. It is shown that this is really a suitable tool for the representation. , and they form a useful tool for description of the domain of probabilities and of observables in the logic of quantum mechanics. Effect algebras form an important tool also for investigations of time dimension of quantum events as it was described in [5]. A certain modification of an effect algebra which does not demand associativity of addition was introduced by the author and H. Länger in [7] under the name skew effect algebra. The motivation for these algebras was justified by the fact that skew effect algebras are closely related to a non-associative fuzzy logic introduced by M. Botur and R. Halaš in [3]. This non-associative fuzzy logic has its algebraic counterpart in the variety of commutative basic algebras, see [1], [2] for details.Recently, it was shown by the author and R. Halaš in [4] that effect algebras can be represented by the so-called conditionally residuated structures. There is 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 03G12, 03G25, 06F99. K e y w o r d s: residuated structure, skew residuated structure, effect algebra, skew effect algebra.

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Skew residuated lattices

Cited by 7 publications

References 3 publications

On non-associative generalizations of MV-algebras and lattice-ordered commutative loops

On non-associative generalizations of MV-algebras and lattice-ordered commutative loops

(Skew) Filters in Residuated Skew Lattic

A representation of skew effect algebras

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