A note on the order type of minoring orderings and some algebraic properties of ω2-well quasi-orderings

The minoring ordering between finite sets of (X,≤) is a well quasi-ordering provided (X,≤) is an ω²-well quasi-ordering. We mention some known facts about ω²-well quasi orderings, and we prove some new results about them. We also study some algebraic properties of the minoring well-quasi ordering over finite sets of (X, ≤), such as the behaviour of its order type when the underlying set is a disjoint sum, and give a tight lower bound for its maximal order type in terms of the maximal order type of (X,≤). We also state some observations regarding the upper bound.