A partial order on a bounded lattice is called t-order if it is defined by means of the t-norm on . It is obtained that for a t-norm on a bounded lattice the relation iff for some is a partial order. The goal of the paper is to determine some conditions such that the new partial order induces a bounded lattice on the subset of all idempotent elements of and a complete lattice on the subset of all elements of which are the supremum of a subset of atoms.
We prove an analogue to Dordal’s result in P.L. Dordal, A model in which the base-matrix tree cannot have cofinal branches, J. Symbolic Logic 52 (1980), 651–664. He obtained a model of ZFC in which there is a tree -base for with no branches yet of height . We establish that this is also possible for using a natural modification of Mathias forcing.
In this paper, an equivalence on the class of nullnorms on a bounded lattice based on the equality of the orders induced by nullnorms is introduced. The set of all incomparable elements w.r.t. the order induced by nullnorms is investigated. Finally, the recently posed open problems have been solved.
We show that AC is equivalent to the assertion that every compact completely regular topology can be extended to a compact Tychonoff topology.
We prove in ZFC that there is a set and a surjective function H: A → ⟨0,1⟩ such that for every null additive set X ⊆ ⟨0,1), is null additive in . This settles in the affirmative a question of T. Bartoszyński.
We study pairs (V, V₁), V ⊆ V₁, of models of ZFC such that adding κ-many Cohen reals over V₁ adds λ-many Cohen reals over V for some λ > κ.