Skew Metrics Valued in Sugihara Semigroups

We consider skew metrics (equivalently, transitive relations that are tournaments, linear orderings) valued in Sugihara semigroups on autodual chains. prove that, for odd chains and without a unit, classify certain tree-like structures we call perfect augmented plane towers. When the chain is finite has cardinality \(2K+1\), set X give rise to rooted trees of height K whose frontier preorder X.