Representing Upper Probability Measures over Rational Lukasiewicz Logic.
Riemann average truth-value of Łukasiewicz formulas
Sequential convergences on pseudo MV-algebras
Sharp and fuzzy elements of an RC-group
Simple games in Łukasiewicz calculus and their cores
We propose a generalization of simple coalition games in the context of games with fuzzy coalitions. Mimicking the correspondence of simple games with non-constant monotone formulas of classical logic, we introduce simple Łukasiewicz games using monotone formulas of Łukasiewicz logic, one of the most prominent fuzzy logics. We study the core solution on the class of simple Łukasiewicz games and show that cores of such games are determined by finitely-many linear constraints only. The non-emptiness...
Spectral topologies of dually residuated lattice-ordered monoids
Dually residuated lattice-ordered monoids (-monoids for short) generalize lattice-ordered groups and include for instance also -algebras (pseudo -algebras), a non-commutative extension of -algebras. In the present paper, the spectral topology of proper prime ideals is introduced and studied.
State-homomorphisms on -algebras
Riečan [12] and Chovanec [1] investigated states in -algebras. Earlier, Riečan [11] had dealt with analogous ideas in -posets. In the monograph of Riečan and Neubrunn [13] (Chapter 9) the notion of state is applied in the theory of probability on -algebras. We remark that a different definition of a state in an -algebra has been applied by Mundici [9], [10] (namely, the condition (iii) from Definition 1.1 above was not included in his definition of a state; in other words, only finite additivity...
States on basic algebras
States on commutative basic algebras were considered in the literature as generalizations of states on MV-algebras. It was a natural question if states exist also on basic algebras which are not commutative. We answer this question in the positive and give several examples of such basic algebras and their states. We prove elementary properties of states on basic algebras. Moreover, we introduce the concept of a state-morphism and characterize it among states. For basic algebras which are the certain...
States on pseudo-effect algebras with general comparability
Strong subdirect products of MV-algebras
Subdirect product decompositions of -algebras
Subdirectly irreducible MV-algebras
In this note we characterize the one-generated subdirectly irreducible MV-algebras and use this characterization to prove that a quasivariety of MV-algebras has the relative congruence extension property if and only if it is a variety.
The existence of states on every Archimedean atomic lattice effect algebra with at most five blocks
Effect algebras are very natural logical structures as carriers of probabilities and states. They were introduced for modeling of sets of propositions, properties, questions, or events with fuzziness, uncertainty or unsharpness. Nevertheless, there are effect algebras without any state, and questions about the existence (for non-modular) are still unanswered. We show that every Archimedean atomic lattice effect algebra with at most five blocks (maximal MV-subalgebras) has at least one state, which...
The prime and maximal spectra and the reticulation of BL-algebras
In this paper we study the prime and maximal spectra of a BL-algebra, proving that the prime spectrum is a compact T 0 topological space and that the maximal spectrum is a compact Hausdorff topological space. We also define and study the reticulation of a BL-algebra.
The σ-complete MV-algebras which have enough states
We characterize Łukasiewicz tribes, i.e., collections of fuzzy sets that are closed under the standard fuzzy complementation and the Łukasiewicz t-norm with countably many arguments. As a tool, we introduce σ-McNaughton functions as the closure of McNaughton functions under countable MV-algebraic operations. We give a measure-theoretical characterization of σ-complete MV-algebras which are isomorphic to Łukasiewicz tribes.
Topological locally finite MV-algebras and Riemann surfaces.
Values and minimal spectrum of an algebraic lattice
Very true operators on MTL-algebras
The main goal of this paper is to investigate very true MTL-algebras and prove the completeness of the very true MTL-logic. In this paper, the concept of very true operators on MTL-algebras is introduced and some related properties are investigated. Also, conditions for an MTL-algebra to be an MV-algebra and a Gödel algebra are given via this operator. Moreover, very true filters on very true MTL-algebras are studied. In particular, subdirectly irreducible very true MTL-algebras are characterized...
Weak Boolean products of bounded dually residuated -monoids
In the paper we deal with weak Boolean products of bounded dually residuated -monoids (DR-monoids). Since bounded DRl-monoids are a generalization of pseudo MV-algebras and pseudo BL-algebras, the results can be immediately applied to these algebras.
Weak homogeneity and Pierce’s theorem for -algebras
In this paper we prove a theorem on weak homogeneity of -algebras which generalizes a known result on weak homogeneity of Boolean algebras. Further, we consider a homogeneity condition for -algebras which is defined by means of an increasing cardinal property.