MV-algebras with internal states and probabilistic fuzzy logics

Flaminio, Tommaso; Montagna, Franco

doi:10.1016/j.ijar.2008.07.006

Cited by 107 publications

(62 citation statements)

References 15 publications

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“…Flaminio and Montagna ( [33]) have endowed the MV-algebras with a unary operation called an internal state or a state operator satisfying some basic properties of states and the new structures are called state MV-algebras. In fact, they developed a unified treatment of states and probabilistic many-valued logic in a logic and algebraic setting.…”

Section: State Pseudo-hoopsmentioning

confidence: 99%

Bounded pseudo-hoops with internal states

Ciungu

2013

Mathematica Slovaca

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ABSTRACT. State MV-algebras were introduced by Flaminio and Montagna as MV-algebras with internal states. Di Nola and Dvurečenskij presented the notion of state-morphism MV-algebra which is a stronger variation of a state MV-algebra. Rachůnek andŠalounová introduced state GMV-algebras (pseudo-MV algebras) and state-morphism GMV-algebras, while the state BL-algebras and state-morphism BL-algebras were defined by Ciungu, Dvurečenskij and Hyčko. Recently, Dvurečenskij, Rachůnek andŠalounová presented state R -monoids and state-morphism R -monoids. In this paper we study these concepts for more general fuzzy structures, namely pseudo-hoops and we present state pseudo-hoops and state-morphism pseudo-hoops.

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Section: State Pseudo-hoopsmentioning

confidence: 99%

Bounded pseudo-hoops with internal states

Ciungu

2013

Mathematica Slovaca

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“…For this reason it is interesting to de ne a probability with values in an abstract algebra. In this case, Flaminio and Montagna [16] were the rst to present a uni ed approach to states and probabilistic many-valued logic in a logical and algebraic setting. They added a unary operation, called internal state (or state operator) to the language of MV-algebras which preserves the usual properties of states.…”

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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“…State MV-algebras generalize, for example, Hájek's approach [14] to fuzzy logic with modality P r (interpreted as probably with the following semantic interpretation: The probability of an event a is presented as the truth value of P r(a). For a more detailed motivation of state MV-algebras and their relation to logic, see [15].In [1], the notion of a state operator was extended from MV-algebras to the more general frame of effect algebras. A state operator is there defined as an additive, unital and idempotent operator on E. A state operator on E is called strong, if it satisfies the additional condition…”

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confidence: 99%

“…Another approach to the state theory on MV-algebras has been presented recently in [15]. Namely, a new unary operator was added to the MV-algebras structure as an internal state (or so-called state operator).…”

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Effect algebras with state operator

Pulmannová¹

EPiC Series in Computing

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This paper is a joint work with Anna Jenčová.Effect algebras have been introduced by Foulis and Bennett [2] (see also [3,4] for equivalent definitions) for modeling unsharp measurements in quantum mechanical systems [5]. They are a generalization of many structures which arise in the axiomatization of quantum mechanics (Hilbert space effects [7]), noncommutative measure theory and probability (orthomodular lattices and posets, [6]), fuzzy measure theory and many-valued logic 9]).A state, as an analogue of a probability measure, is a basic notion in algebraic structures used in the quantum theories (see e.g., [8]), and properties of states have been deeply studied by many authors.In MV-algebras, states as averaging the truth value were first studied in [11]. In the last few years, the notion of a state has been studied by many experts in MV-algebras, e.g, [13,12].Another approach to the state theory on MV-algebras has been presented recently in [15]. Namely, a new unary operator was added to the MV-algebras structure as an internal state (or so-called state operator). MV-algebras with the added state operator are called state MValgebras. The idea is that an internal state has some properties reminiscent of states, but, while a state is a map from an MV-algebra into [0, 1], an internal state is an operator of the algebra. State MV-algebras generalize, for example, Hájek's approach [14] to fuzzy logic with modality P r (interpreted as probably with the following semantic interpretation: The probability of an event a is presented as the truth value of P r(a). For a more detailed motivation of state MV-algebras and their relation to logic, see [15].In [1], the notion of a state operator was extended from MV-algebras to the more general frame of effect algebras. A state operator is there defined as an additive, unital and idempotent operator on E. A state operator on E is called strong, if it satisfies the additional condition(1)Since MV-algebras form a special subclass of effect algebras, so-called MV-effect algebras, it was shown that the definition of a state operator on effect algebras coincides with the original definition on MV-algebras if and only if the state operator is strong. Moreover, if τ is faithful, i.e., τ (a) = 0 implies a = 0, then property (1) is automatically satisfied.In the present paper, we show that state operators on an effect algebra E are related with states on E in the following way: (1) every state on E induces a state operator on the tensor product [0, 1] ⊗ E. (2) If E admits an ordering set of states, then every state operator on E induces a state on E. We study state operators mainly on convex effect algebras. Convex effect algebras as effect algebras with an additional convexity structure were introduced and studied in [17,16]. It was proved in [17], that every convex effect algebra is isomorphic with the interval [0, u] in an ordered real linear space (V, V + ), where u is an order unit. Moreover, (V, u) is an order unit space, i.e. u is an archimedean order unit, if and only if E ad...

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MV-algebras with internal states and probabilistic fuzzy logics

Cited by 107 publications

References 15 publications

Bounded pseudo-hoops with internal states

Bounded pseudo-hoops with internal states

Generalized state maps and states on pseudo equality algebras

Effect algebras with state operator

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