Equivalence and order: two ways to compare states

Our original cards use X for dot and Y for stripe. A has only a dot, B both, C only a stripe, and D neither. The two-bit annotations follow X,Y order. Selecting just one question keeps only that answer in the selected description; it leaves the complete card unchanged. Observe only X on our four valuations. A and B share the selected description true; C and D share the selected description false. The complete states still differ, but the chosen description groups them into two equivalence classes. An equivalence class collects points related by this equality of descriptions. Classes do not overlap and cover the carrier. Replacing each class by one label is a later operation; today we keep track of the grouping itself. Now select both X and Y. Instead of grouping equal descriptions, collect the selected formulas true at each point. This collection is its positive support. D has none, A has only X, C has only Y, and B has both. Here each formula occurs once; later we will use position labels when a selected list can repeat a formula. The description containing X is included in the description containing X and Y. Every positive item in the first remains present in the second. Inclusion expresses a one-way comparison, not equality between the two descriptions. Put the empty support at the bottom, the two singleton supports in the middle, and the two-item support at the top. Draw upward links only when one item is added. The diagram records inclusion, not physical or numerical distance. The two middle supports are incomparable. Each has something the other lacks. Different supports need not be comparable, so this diagram is not a single line ranking points from universally worse to universally better. Every support includes itself. A relation with this self-comparison property is called reflexive. We can record comparisons in a table: the row is the first object and the column is the proposed second object. If every item in the first support occurs in the second, and every item in the second occurs in the third, then every item in the first occurs in the third. This chaining property is called transitive. An upper set keeps every description above any description it contains. If it contains the support with X alone, it must also contain the support with both X and Y. We check this by following the inclusion links upward. An antichain is a collection of distinct supports with no containment comparison between different members. Our two middle supports form an antichain. This is different from an upper set: neither term means merely a random subset of the diagram. Under the one-test menu X, compare A with B, and then D with A. Which pair has equal descriptions? For which pair does inclusion work in just one direction? A and B have equal descriptions, so each support includes the other. D has the empty support and A has the support containing X. The empty support is included in A's support, but the reverse inclusion fails. Equality of descriptions groups points into classes. Inclusion orders the descriptions, possibly asymmetrically. Later we will prove that this inclusion direction is exactly the specialization convention used by the observation topology.