We (i.e. I) present a simplified version of Shelah’s “preserving a little implies preserving much”: If I is the ideal generated by a Suslin ccc forcing (e.g. Lebesque– null or meager), and P is a Suslin forcing, and P is I–preserving (i.e. it doesn’t make any positive Borel–set small), then P preserves generics over candidates and therefore is strongly I–preserving (i.e. doesn’t make any positi...

Morphisms of some categories of sets with similarity relations (Ω-sets) are investigated, where Ω is a complete residuated lattice. Namely a category SetF(Ω) with morphisms (A, δ) → (B, γ) defined as special maps A → B and a category SetR(Ω) with morphisms defined as a special relations A × B → Ω. It is proved that arbitrary maps A → Ω and A × B → Ω can be extended onto morphisms (A, δ) → (Ω,↔)...

In this paper we consider fuzzy relations compatible with algebraic operations, which are called fuzzy relational morphisms. In particular, we aim our attention to those fuzzy relational morphisms which are uniform fuzzy relations, called uniform fuzzy relational morphisms, and those which are partially uniform F -functions, called fuzzy homomorphisms. Both uniform fuzzy relations and partially...

We (i.e. I) present a simplified version of Shelah’s “preserving a little implies preserving much”: If I is the ideal generated by a Suslin ccc forcing (e.g. Lesbeque– null or meager), and P is a Suslin forcing, and P is I–preserving (i.e. it doesn’t make any positive Borel–set small), then P preserves generics over candidates and therefore is strongly I–preserving (i.e. doesn’t make any positi...

In a recent article Facchini and Finocchiaro considered natural pretorsion theory in the category of preordered sets inducing corresponding stable category. present work we propose an alternative construction $\mathsf{PreOrd} (\mathbb C)$ internal preorders any coherent $\mathbb C$, that enlightens categorical nature this notion. When C$ is pretopos prove quotient functor from to associated pre...

In the study of pre-Lie algebras, concept pre-morphism arises naturally as a generalization standard notion morphism. Pre-morphisms can be defined for arbitrary (not-necessarily associative) algebras over any commutative ring k with identity, and dualized in various ways to generalized morphisms (related pre-Jordan algebras) anti-pre-morphisms anti-pre-Lie algebras). We consider idempotent pre-...