Knowledge Representation Technology
A partial order is a binary relation, ≤, on a set, X if and only if it is reflexive, transitive and antisymmetric. For the Datum Universe, the binary relation is “is” and the set X is the set of all datums in the universe. Let us consider each property of the partial order as it pertains to the datum universe.
Reflexivity: a ≤ a, for all a ∈ X. Any datum “is” itself in the datum universe
Transitivity: if a ≤ b and b ≤ c, then a ≤ c, for all a, b, and c ∈ X. In the datum universe, if we have "a is b" and "b is c", then we can conclude that a is c.
Antisymmetry: if a ≤ b and b ≤ a, then a = b, for all a, b ∈ X. In the datum universe, if a is b and b is a, then this means that a, and b are the same datum.
Notice that datums that are related to datum a by the “is” relation are all datums in a’s extended Class or Value sets. This means that all other datums outside these two sets are not “comparable” to datum a. This is why the datum universe is a partial order and not a total order.
Now let us consider the following poset concepts in the datum universe;
The maximal element: if we have no element b ∈ X such that a ≤ b, then a is the maximal element. In the datum universe, this is the root and is referred to as “thing”. Any datum is eventually, through transitivity, a “thing”.
The minimal element: if we have no element b ∈ X such that b ≤ a, then a is the minimal element. In the datum universe, this is the “nothing” datum. The “nothing” datum is implicitly defined as the single value of all terminal datums.
The immediate predecessor: if b ∈ X so that a < b and there is no c ∈ X so that a < c < b, then b is an immediate predecessor of a. In the datum universe, we call datum “b” a direct class of datum “a” and the set of all such datums is called the Class set of datum “a”.
The immediate successor: if b ∈ X so that b < a, and there is no c ∈ X so that b < c < a, then b is an immediate successor of a. In the datum universe we call datum “b” a direct value of datum “a” and the set of all such datums is called the Value set of datum “a”.
The Least Upper Bound (LUB): An element c is called the least upper bound of two elements a and b, if a ≤ c and b ≤ c and there is no other element d such that d < c and a < d and b < d. In the datum universe, this is the lowest datum that is a class of both a and b.
The Greatest Lower Bound (GLB): An element c is called the greatest upper bound of two elements a and b if c ≤ a, c ≤ b and there is no other element d such that c < d, d < a and d < b. In the datum universe, this is the highest datum that is a value of both a and b.Type your paragraph here.