In analogy with effect algebras, we introduce the test spaces and $MV$-test spaces. A test corresponds to a hypothesis on the propositional system, or, equivalently, to a partition of unity. We show that there is a close correspondence between $MV$-algebras and $MV$-test spaces.
A version of Dieudonné theorem is proved for lattice group-valued modular measures on lattice ordered effect algebras. In this way we generalize some results proved in the real-valued case.
Effect basic algebras (which correspond to lattice ordered effect algebras) are studied. Their ideals are characterized (in the language of basic algebras) and one-to-one correspondence between ideals and congruences is shown. Conditions under which the quotients are OMLs or MV-algebras are found.
We present a categorical approach to the extension of probabilities, i.e. normed σ-additive measures. J. Novák showed that each bounded σ-additive measure on a ring of sets A is sequentially continuous and pointed out the topological aspects of the extension of such measures on A over the generated σ-ring σ(A): it is of a similar nature as the extension of bounded continuous functions on a completely regular topological space X over its Čech-Stone compactification βX (or as the extension of continuous functions on X over its Hewitt realcompactification υX). He developed a theory of sequential envelopes and (exploiting the Measure Extension Theorem) he proved that σ(A) is the sequential envelope of A with respect to the probabilities. However, the sequential continuity does not capture other properties (e.g. additivity) of probability measures. We show that in the category ID of D-posets of fuzzy sets (such D-posets generalize both fields of sets and bold algebras) probabilities are morphisms and the extension of probabilities on A over σ(A) is a completely categorical construction (an epireflection). We mention applications to the foundations of probability and formulate some open problems.
In this paper, considering L being a completely distributive lattice, we first introduce the concept of L -fuzzy ideal degrees in an effect algebra E , in symbol Dei . Further, we characterize L -fuzzy ideal degrees by cut sets. Then it is shown that an L -fuzzy subset A in E is an L -fuzzy ideal if and only if Dei(A)=⊤, which can be seen as a generalization of fuzzy ideals. Later, we discuss the relations between L -fuzzy ideals and cut sets (Lβ -nested sets and Lα -nested sets). Finally, we obtain that the L -fuzzy ideal degree is an (L,L) -fuzzy convexity. The morphism between two effect algebras is an (L,L) -fuzzy convexity-preserving mapping.
Several open problems posed during FSTA 2006 (Liptovský Ján, Slovakia) are presented. These problems concern the classification of strict triangular norms, Lipschitz t-norms, interval semigroups, copulas, semicopulas and quasi-copulas, fuzzy implications, means, fuzzy relations, MV-algebras and effect algebras.
Eighteen open problems posed during FSTA 2010 (Liptovský Ján, Slovakia) are presented. These problems concern copulas, triangular norms and related aggregation functions. Some open problems concerning effect algebras are also included.
We show that an effect tribe of fuzzy sets T⊆[0,1]X with the property that every f∈T is B0(T)-measurable, where B0(T) is the family of subsets of X whose characteristic functions are central elements in T, is a tribe. Moreover, a monotone σ-complete effect algebra with RDP with a Loomis-Sikorski representation (X,T,h), where the tribe T has the property that every f∈T is B0(T)-measurable, is a σ-MV-algebra.