States on semi-divisible residuated lattices

Turunen, Esko; Mertanen, Janne

doi:10.1007/s00500-007-0182-y

Cited by 44 publications

(14 citation statements)

References 17 publications

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“…It is clear that the set of double complemented elements is the MV -center MV (L) of a residuated lattice L (thus of a BL-algebra, too), see [15].…”

Section: The Set Of Double Complemented Elementsmentioning

confidence: 99%

Some results in BL ‐algebras

Saeid¹,

Motamed

2009

Mathematical Logic Qtrly

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“…It is clear that the set of double complemented elements is the MV -center MV (L) of a residuated lattice L (thus of a BL-algebra, too), see [15].…”

Section: The Set Of Double Complemented Elementsmentioning

confidence: 99%

Some results in BL ‐algebras

Saeid¹,

Motamed

2009

Mathematical Logic Qtrly

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“…It was proved in [23] that condition (11) is equivalent to the condition that, for all x, y ∈ L holds…”

Section: Mathematical Preliminariesmentioning

confidence: 99%

“…Semi-divisible residuated lattices are not necessary strong; in [23] we introduced a semidivisible residuated lattice L sD on the unit real interval [0, 1] such that…”

Section: Proposition Any Divisible Residuated Lattice Is Strongmentioning

confidence: 99%

“…We got an impression that such generalization reduce extensively in probability theory on MValgebras introduced by Chovanec and Mundici (see [3] and [15], respectively) and published two papers where we showed that this indeed is the case (cf. [14,23]). More precisely, if the regular elements of a residuated lattice L fulfill a certain extra condition, then the MV-center becomes an MV-algebra and states (i.e.…”

Section: Introductionmentioning

confidence: 99%

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Axiomatic Extensions of Hohle's Monoidal Logic

Turunen¹

2011

Proceedings of the 7th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-2011)

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We introduce an axiomatic extension of Höhle's Monoidal Logic called Semi-divisible Monoidal Logic, and prove that it is complete by showing that semi-divisibility is preserved in MacNeille completion. Moreover, we introduce Strong semidivisible Monoidal Logic and conjecture that a predicate formula α is derivable in Strong Semi-divisible Monadic logic if, and only if its double negation ¬¬α is derivable in Łukasiewicz * logic.

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“…Di erent approaches to the generalization mainly gave rise to two di erent notions, namely, Bosbach states and Riečan states. Hence it is meaningful to extend the notion of states to other algebraic structures and their noncommutative cases [4,[10][11][12][13][14]. For example, Liu Lianzhen studied the existence of Bosbach states and Riečan states on nite monoidal t-norm based algebras (MTL-algebra for short) in [11].…”

Section: Introductionmentioning

confidence: 99%

Generalized state maps and states on pseudo equality algebras

Cheng

Xin

2018

Open Mathematics

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Abstract:In this paper, we attempt to cope with states in a universal algebraic setting, that is, introduce a notion of generalized state map from a pseudo equality algebra X to an arbitrary pseudo equality algebra Y. We give two types of special generalized state maps, namely, generalized states and generalized internal states. Also, we study two types of states, namely, Bosbach states and Riečan states. Finally, we discuss the relations among generalized state maps, states and internal states (or state operators) on pseudo equality algebras. We verify the results that generalized internal states are the generalization of internal states, and generalized states are the generalization of state-morphisms on pseudo equality algebras. Furthermore, we obtain that generalized states are the generalization of Bosbach states and Riečan states on linearly ordered and involutive pseudo equality algebras, respectively. Hence we can come to the conclusion that, in a sense, generalized state maps can be viewed as a possible united framework of the states and the internal states, the state-morphisms and the internal state-morphisms on pseudo equality algebras.

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States on semi-divisible residuated lattices

Cited by 44 publications

References 17 publications

Some results in BL ‐algebras

Some results in BL ‐algebras

Axiomatic Extensions of Hohle's Monoidal Logic

Generalized state maps and states on pseudo equality algebras

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