Lavinia Corina Ciungu

Generalized Bosbach states: Part II

We continue the investigation of generalized Bosbach states that we began in Part I, restricting our research to the commutative case and treating further aspects related to these states. Part II is concerned with similarity convergences, continuity of states and the construction of the s-completion of a commutative residuated lattice, where s is a generalized Bosbach state.

Bounded pseudo-hoops with internal states

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 and Šalounová 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 and Šalounová presented state Rℓ-monoids and state-morphism Rℓ-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.

Internal states on equality algebras

This paper investigates properties of equality algebras introduced by Jenei as a possible algebraic semantic for fuzzy type theory. We define and study the pointed equality algebras and its subclass of compatible pointed equality algebras. We introduce and investigate the internal states and the state-morphism operators on equality algebras and on their corresponding BCK-meet-semilattices. We prove that any internal state (state-morphism) on an equality algebra is also an internal state (state-morphism) on its corresponding BCK-meet-semilattice, and we prove the converse for the case of linearly ordered equality algebras. Another main result consists of proving that any state-morphism on a linearly ordered equality algebra is an internal state on it. We show that any internal state on a linearly ordered BCK-meet-semilattice satisfying the distributivity condition is also an internal state on its corresponding equality algebra and a state-morphism on a BCK-meet-semilattice satisfying the distributivity condition is also a state-morphism on its corresponding equality algebra.

On pseudo-equality algebras

Recently, a new algebraic structure called pseudo-equality algebra has been defined by Jenei and Kóródi as a generalization of the equality algebra previously introduced by Jenei. As a main result, it was proved that the pseudo-equality algebras are term equivalent with pseudo-BCK meet-semilattices. We found a gap in the proof of this result and we present a counterexample and a correct version of the theorem. The correct version of the corresponding result for equality algebras is also given.

Local pseudo-BCK algebras with pseudo-product

Pseudo-BCK algebras were introduced by G. Georgescu and A. Iorgulescu as a generalization of BCK algebras in order to give a corresponding structure to pseudo-MV algebras, since the bounded commutative BCK algebras correspond to MV algebras. Properties of pseudo-BCK algebras and their connections with other fuzzy structures were established by A. Iorgulescu and J. Kühr. The aim of this paper is to define and study the local pseudo-BCK algebras with pseudo-product. We will also introduce the notion of perfect pseudo-BCK algebras with pseudo-product and we will study their properties. We define the radical of a bounded pseudo-BCK algebra with pseudo-product and we prove that it is a normal deductive system. Another result consists of proving that every strongly simple pseudo-hoop is a local bounded pseudo-BCK algebra with pseudo-product.

Relative negations in non-commutative fuzzy structures

In this paper we investigate the properties of the relative negations in non-commutative residuated lattices and their applications. We define the notion of a relative involutive FL-algebra and we generalize to relative negations some results proved for involutive pseudo-BCK algebras. The relative locally finite IFL-algebra is defined and it is proved that an interval algebra of a relative locally finite divisible IFL-algebra is relative involutive. Starting from the observation that in the definition of states, the standard MV-algebra structure of [0, 1] intervenes, there were introduced the states on bounded pseudo-BCK algebras, pseudo-hoops and residuated lattices with values in the same kind of structures and they were studied under the name of generalized states. For the case of commutative residuated lattices the generalized states were studied in the sense of relative negation. We define and study the relative generalized states on non-commutative residuated lattices. One of the main results consists of proving that every order-preserving generalized Bosbach state is a relative generalized Riečan state. Some conditions are given for a relative generalized Riečan state to be a generalized Bosbach state. Finally, we develop a concept of states on IFL-algebras.

Commutative pseudo-equality algebras

Pseudo-equality algebras were initially introduced by Jenei and Kóródi as a possible algebraic semantic for fuzzy-type theory, and they have been revised by Dvurečenskij and Zahiri under the name of JK-algebras. In this paper, we define and study the commutative pseudo-equality algebras. We give a characterization of commutative pseudo-equality algebras, and we prove that an invariant pseudo-equality algebra is commutative if and only if its corresponding pseudo-BCK(pC)-meet-semilattice is commutative. Other results consist of proving that every commutative pseudo-equality algebra is a distributive lattice and every finite invariant commutative pseudo-equality algebra is a symmetric pseudo-equality algebra. We introduce the notion of a commutative deductive system of a pseudo-equality algebra, and we give equivalent conditions for this notion. As applications of these notions and results, we define and study the measures and measure morphisms on pseudo-equality algebras, we prove new properties of state pseudo-equality algebras, and we introduce and investigate the pseudo-valuations on pseudo-equality algebras. We prove that any measure morphism on a pseudo-equality algebra is a measure on it, and the kernel of a measure is a commutative deductive system. We show that the quotient pseudo-equality algebra over the kernel of a measure is a commutative pseudo-equality algebra. It is also proved that a pseudo-equality algebra possessing an order-determining system is commutative. We prove that the two types of internal states on a pseudo-equality algebra coincide if and only if it is a commutative pseudo-equality algebra. Given a pseudo-equality algebra A, it is proved that the kernel of a commutative pseudo-valuation on A is a commutative deductive system of A. If, moreover, A is commutative, then we prove that any pseudo-valuation on A is commutative.

On state pseudo-hoops

Abstract Recently, the bounded pseudo-hoops with internal states have been defined and studied. Some of results are based on the hypothesis that a bounded pseudo-hoop has the Glivenko and (mN) properties. In this paper we show that this hypothesis is superfluous, namely every good pseudo-hoop satisfies the Glivenko and (mN) properties and those results are reformulated without that hypothesis. The pointed pseudo-hoops are introduced and studied and the notion of state operator has been generalized for the case of unbounded pseudo-hoops. We define the Bosbach and Riečan states on pointed pseudo-hoops and we make some considerations regarding the relationship between the state operators and states on these structures.

Pseudo-BCK Algebras

BCK algebras were introduced originally by K. Iseki with a binary operation ∗ modeling the set-theoretical difference and with a constant element 0, that is, a least element. Another motivation is from classical and non-classical prepositional calculi modeling logical implications. Such algebras contain as a special subfamily the family of MV-algebras where some important fuzzy structures can be studied. Pseudo-BCK algebras were introduced by G. Georgescu and A. lorgulescu as algebras with “two differences”, a left- and right-difference, instead of one ∗ and with a constant element 0 as the least element. Nowadays pseudo-BCK algebras are used in a dual form, with two implications, → and ⇝ and with one constant element 1, that is, the greatest element. Thus such pseudo-BCK algebras are in the “negative cone” and are also called “left-ones”. More properties of pseudo-BCK algebras and their connection with other fuzzy structures were established by A. lorgulescu. In this chapter we prove new properties of pseudo-BCK algebras with pseudo-product and pseudo-BCK algebras with pseudo-double negation and we show that every pseudo-BCK algebra can be extended to a good one. Examples of proper pseudo-BCK algebras, good pseudo-BCK algebras and pseudo-BCK lattices are given and the orthogonal elements in a pseudo-BCK algebra are characterized. Finally, we define the maximal and normal deductive systems of a pseudo-BCK algebra with pseudo-product and we study their properties.

Commutative deductive systems in probability theory on generalizations of fuzzy structures

Abstract The aim of this paper is to introduce the notion of commutative deductive systems on generalizations of fuzzy structures, and to emphasize their role in the probability theory on these algebras. We give a characterization of commutative pseudo-BE algebras and we generalize an axiom system consisting of four identities to the case of commutative pseudo-BE algebras. We define the commutative deductive systems of pseudo-BE algebras and we investigate their properties. It is proved that, if a pseudo-BE(A) algebra A is commutative, then all deductive systems of A are commutative. Moreover, we generalize the notions of measures, state-measures and measure-morphisms to the case of pseudo-BE algebras and we also prove that there is a one-to-one correspondence between the set of all Bosbach states on a bounded pseudo-BE algebra and the set of its state-measures. The notions of internal states and state-morphism operators on pseudo-BCK algebras are extended to the case of pseudo-BE algebras and we also prove that any type II state operator on a pseudo-BE algebra is a state-morphism operator on it. The notions of pseudo-valuation and commutative pseudo-valuation on pseudo-BE algebras are defined and investigated. For the case of commutative pseudo-BE algebras we prove that the two kind of pseudo-valuations coincide. Characterizations of pseudo-valuations and commutative pseudo-valuations are given. We show that the kernel of a Bosbach state (state-morphism, measure, type II state operator, pseudo-valuation) is a commutative deductive system.