On Semilattice Structure of Mizar Types

Let us mention that every non empty relational structure which is trivial and reflexive is also complete. Let T be a relational structure. A type of T is an element of T . Let T be a relational structure. We say that T is Noetherian if and only if: (Def. 1) The internal relation of T is reversely well founded. Let us observe that every non empty relational structure which is trivial is also Noetherian. Let T be a non empty relational structure. Let us observe that T is Noetherian if and only if the condition (Def. 2) is satisfied. (Def. 2) Let A be a non empty subset of T . Then there exists an element a of T such that a ∈ A and for every element b of T such that b ∈ A holds a 6< b. Let T be a poset. We say that T is Mizar-widening-like if and only if: (Def. 3) T is a sup-semilattice and Noetherian.