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Recently in [17, 18], new algebras, called weak pseudo EMV-algebras, wPEMV-algebras for short, were introduced generalizing pseudo MV-algebras, generalized Boolean algebras and pseudo EMV-algebras. For these algebras a top element is not assumed a priori. For this class of algebras, we define a state as a finitely additive mapping from a wPEMV-algebra into the real interval [0, 1] which preserves a partial addition of two non-interactive elements and attaining the value 1 in some element. It can happen that some commutative wPEMV-algebras are stateless, e.g. cancellative ones. The paper is divided into two parts. Part I deals with basic properties of states and state-morphisms which are wPEMVhomomorphisms from a wPEMV-algebra into the real interval [0, 1] endowed as a commutative wPEMV-algebra. We show that there is a one-to-one correspondence between the set of state-morphisms and the set of maximal and normal ideals having a special property. In Part II, we present an analogue of the Krein-Mil’man theorem applied to the set of states. We characterize the space of the state-morphisms of a wPEMV-algebra without top element as a Hausdorff locally compact space in the weak topology of states and we present its Alexandroff’s one-point compactification. Moreover, we give an integral representation of any (finitely additive) state by a unique regular Borel σ-additive probability measure. Recently in [17, 18], new algebras, called weak pseudo EMV-algebras, wPEMV-algebras for short, were introduced generalizing pseudo MV-algebras, generalized Boolean algebras and pseudo EMV-algebras. For these algebras a top element is not assumed a priori. For this class of algebras, we define a state as a finitely additive mapping from a wPEMV-algebra into the real interval [0, 1] which preserves a partial addition of two non-interactive elements and attaining the value 1 in some element. It can happen that some commutative wPEMV-algebras are stateless, e.g. cancellative ones. The paper is divided into two parts. Part I deals with basic properties of states and state-morphisms which are wPEMVhomomorphisms from a wPEMV-algebra into the real interval [0, 1] endowed as a commutative wPEMV-algebra. We show that there is a one-to-one correspondence between the set of state-morphisms and the set of maximal and normal ideals having a special property. In Part II, we present an analogue of the Krein-Mil’man theorem applied to the set of states. We characterize the space of the state-morphisms of a wPEMV-algebra without top element as a Hausdorff locally compact space in the weak topology of states and we present its Alexandroff’s one-point compactification. Moreover, we give an integral representation of any (finitely additive) state by a unique regular Borel σ-additive probability measure. [ABSTRACT FROM AUTHOR]

Standard completeness, completeness on the real unit interval [0, 1], is one of important research areas in mathematical fuzzy logic. Recently, standard completeness for semilinear logics with knotted axioms has been investigated prooftheoretically by introducing and eliminating density rule. This paper introduces model-theoretic completeness for such logics. To this end, it is first shown that knotted axioms can be divided into left and right ones and then proved that mianorm-based logic systems with left and right knotted axioms are standard complete. This completeness is provided by embedding linearly ordered algebras into densely ordered ones and these algebras again into [0, 1]. More exactly, mianorm-based systems with left and right knotted axioms and their algebraic structures are first discussed. After some examples of mianorms satisfying left and right knotted properties are introduced, standard completeness for those logics is established model-theoretically using the above construction. Finally, this investigation is extended to their corresponding involutive fixpointed systems. [ABSTRACT FROM AUTHOR]

This paper further studies topological structures induced by L-fuzzifying approximation operators, where L denotes a completely distributive De Morgan algebra. Firstly, the Alexandrov topologies induced by L-fuzzy relations are investigated with respect to L-fuzzifying approximation operators. Especially, the relationships among those Alexandrov topologies are discussed. Secondly, pseudo-similarity sets of L-fuzzy relations are proposed based on the Alexandrov topologies induced by upper and lower L-fuzzifying approximation operators. Meanwhile, the properties of pseudosimilarity sets are discussed, where some examples are presented to show the differences between similarity set of fuzzy relations and pseudo-similarity set of L-fuzzy relations. [ABSTRACT FROM AUTHOR]

In this article, with the aim of further investigating BL-algebras, the concepts of Unity Co-annihilator-filters (UC-filters) is introduced and discussed. Also, for the faster study of UC-filters in BL-algebras, some equivalent conditions are obtained, and (with some examples) it is shown that these filters have differences. In addition, we consider several additional conditions imposed on UC-filters and prime filters and establish links between them. So, we get relationships between these types of filters and prime filters in BL-algebras. Finally, the form of all UC-saturated ∨-closed subsets of a G- algebras (by the concept of UC-saturated ∨-closed subsets) is stated. [ABSTRACT FROM AUTHOR]

We introduce the notions of Sasaki mapping and of sharp elements on a d0-algebra. We investigate the relationships of sharp elements with Sasaki mappings and with central elements, thus generalizing some results known for D-lattices. Namely, we give a characterization of sharp elements, by means of Sasaki mappings, which extends a result of Bennett and Foulis; we also prove that an element is central if and only if it is a sharp element and every element is compatible with it: This generalizes a result of Riečanová. [ABSTRACT FROM AUTHOR]