Construction of popo-groups with quasi-divisors theory

summary:A method is presented making it possible to construct $po$-groups with a strong theory of quasi-divisors of finite character and with some prescribed properties as subgroups of restricted Hahn groups $H(\Delta ,\mathbb{Z})$, where $\Delta$ are finitely atomic root systems. Some examples of these constructions are presented

Ordered groups with greatest common divisors theory

An embedding (called a GCD theory) of partly ordered abelian group G into abelian l-group Γ is investigated such that any element of Γ is an infimum of a subset (possible non-finite) from G. It is proved that a GCD theory need not be unique. A complete GCD theory is introduced and it is proved that G admits a complete GCD theory if and only if it admits a GCD theory G→Γ such that Γ is an Archimedean l-group. Finally, it is proved that a complete GCD theory is unique (up to o-isomorphisms) and that a po-group admits the complete GCD theory if and only if any v-ideal is v-invertible

Functors among Relational Variants of Categories Related to L-Fuzzy Partitions, L-Fuzzy Pretopological Spaces and L-Fuzzy Closure Spaces

Various types of topological and closure operators are significantly used in fuzzy theory and applications. Although they are different operators, in some cases it is possible to transform an operator of one type into another. This in turn makes it possible to transform results relating to an operator of one type into results relating to another operator. In the paper relationships among 15 categories of modifications of topological L-valued operators, including Čech closure or interior L-valued operators, L-fuzzy pretopological and L-fuzzy co-pretopological operators, L-valued fuzzy relations, upper and lower F-transforms and spaces with fuzzy partitions are investigated. The common feature of these categories is that their morphisms are various L-fuzzy relations and not only maps. We prove the existence of 23 functors among these categories, which represent transformation processes of one operator into another operator, and we show how these transformation processes can be mutually combined

Semiring-Valued Fuzzy Sets and F-Transform

The notion of a semiring-valued fuzzy set is introduced for special commutative partially pre-ordered semirings, including basic operations with these fuzzy structures. It is showed that many standard MV-algebra-valued fuzzy type structures with standard operations, such as hesitant, intuitionistic, neutrosophic or fuzzy soft sets are, for appropriate semirings, isomorphic to semiring-valued fuzzy sets with operations defined. F-transform and inverse F-transform are introduced for semiring-valued fuzzy sets and properties of these transformations are investigated. Using the transformation of MV-algebra-valued fuzzy type structures to semiring-valued fuzzy sets, the F-transforms for these fuzzy type structures is introduced. The advantage of this procedure is, among other things, that the properties of this F-transform are analogous to the properties of the classical F-transform and because these properties are proven for any semiring-valued fuzzy sets, it is not necessary to prove them for individual fuzzy type structures