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In the framework of Ω-sets, where Ω is a complete lattice, we introduce Ω-lattices, both as algebraic and as order structures. An Ω-poset is an Ω-set equipped with an Ω-valued order which is antisymmetric with respect to the corresponding Ω-valued equality. Using a cut technique, we prove that the quotient cut-substructures can be naturally ordered. Introducing notions of pseudo-infimum and pseudo-supremum, we obtain a definition of an Ω-lattice as an ordering structure. An Ω-lattice as an algebra is a bi-groupoid equipped with an Ω-valued equality, fulfilling particular lattice-theoretic formulas. On an Ω-lattice we introduce an Ω-valued order, and we prove that particular quotient substructures are classical lattices. Assuming Axiom of Choice, we prove that the two approaches are equivalent. [ABSTRACT FROM AUTHOR]

Omega rings (Ω -rings) (and other related structures) are lattice-valued structures (with Ω being the codomain lattice) defined on crisp algebras of the same type, with lattice-valued equality replacing the classical one. In this paper, Ω -ideals are introduced, and natural connections with Ω -congruences and homomorphisms are established. As an application, a framework of approximate solutions of systems of linear equations over Ω -fields is developed. [ABSTRACT FROM AUTHOR]

The topic of the paper are Ω-algebras, where Ω is a complete lattice. In this research we deal with congruences and homomorphisms. An Ω-algebra is a classical algebra which is not assumed to satisfy particular identities and it is equipped with an Ω-valued equality instead of the ordinary one. Identities are satisfied as lattice theoretic formulas. We introduce Ω-valued congruences, corresponding quotient Ω-algebras and Ω-homomorphisms and we investigate connections among these notions. We prove that there is an Ω-homomorphism from an Ω-algebra to the corresponding quotient Ω-algebra. The kernel of an Ω-homomorphism is an Ω-valued congruence. When dealing with cut structures, we prove that an Ω-homomorphism determines classical homomorphisms among the corresponding quotient structures over cut subalgebras. In addition, an Ω-congruence determines a closure system of classical congruences on cut subalgebras. Finally, identities are preserved under Ω-homomorphisms. [ABSTRACT FROM AUTHOR]

This paper lays down the axioms of Ω -valued normed spaces and shows their invariance under the so-called tilde-construction. Important examples are given by separated presheaves of normed spaces and by probabilistic normed spaces. Finally, there exists a natural Ω -valued topologization of Ω -valued normed spaces which makes it possible to study continuous linear operators between them. [ABSTRACT FROM AUTHOR]