Ω-groups in the language of Ω-groupoids

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.