Authors: Šešelja, Branimir
Tepavčević, Andreja 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Ω-groups in the language of Ω-groupoids
Journal: Fuzzy Sets and Systems
Volume: 397
First page: 152
Last page: 167
Issue Date: 15-Oct-2020
Rank: M21a
ISSN: 0165-0114
DOI: 10.1016/j.fss.2019.08.007
Abstract: 
We introduce Ω-groups as particular Ω-groupoids, a structure with a single binary operation and an Ω-equality replacing the classical one. The membership values belong to a complete lattice Ω. We analyze and compare languages in which such structures can be introduced. We prove the equivalence of approaches to Ω-groups as algebras with three operations and those in the language of Ω-groupoids. We also introduce a wider class of Ω-groups, so called weak Ω-groups for which different neutral elements and different inverses of the same member are equal up to the Ω-equality. For all these, quotient structures with respect to cuts of the Ω-equality are classical groups. We present basic features of Ω-groups in the language with one binary operation. As an application, we show that linear equations can be uniquely (up to Ω-equality) solved in these structures.
Keywords: Group | Lattice-valued groupoid | Ω-algebra | Ω-group | Ω-set
Publisher: Elsevier
Project: Development of methods of computation and information processing: theory and applications 
Provincial Secretariat for Higher Education and Scientific Research AP Vojvodina, grant “Computational intelligence and relational equations in forensics”, Grant No. 142-451-3642/2017-01

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