Ternary closure operations

Given a ternary relation, the set of functional dependencies upon the slots of the relation form a closure operation that takes some set of slots and that produces the remaining slots that can be determined functionally from the values provided. The fixed points of this closure operation produce a Moore family, which is a set of sets that is intersection closed and that contains a greatest member. For ternary relations, this is an order three Moore family as the union of the sets of the moore family has cardinality three. There are 61 Moore families on three elements, and they are listed below. The dimension of the Moore families corresponds to the functional dimension of the classes of relations. In this way, we can produce a full ontology of all the classes of ternary relations determined by functional dependencies.

Dimension 0

trivial
{{0 1 2}}

Dimension 1

reversible & reversible
{{} {0 1 2}}

constant & reversible
{{0 1 2} {2}}
{{0 1 2} {1}}
{{0 1 2} {0}}

constant & constant
{{0 1 2} {1 2}}
{{0 1} {0 1 2}}
{{0 1 2} {0 2}}

constant & function
{{0 1 2} {2} {1 2}}
{{0 1} {0 1 2} {0}}
{{0 1 2} {2} {0 2}}
{{0 1 2} {0 2} {0}}
{{0 1 2} {1} {1 2}}
{{0 1} {0 1 2} {1}}

reversible binary operation which is partially invariant
{{} {0 1 2} {1}}
{{} {0 1 2} {2}}
{{} {0 1 2} {0}}

product to a reversible pair
{{} {0 1 2} {1 2}}
{{} {0 1 2} {0 2}}
{{0 1} {} {0 1 2}}

product to a functional pair
{{} {0 1 2} {2} {1 2}}
{{} {0 1 2} {0 2} {0}}
{{} {0 1 2} {2} {0 2}}
{{} {0 1 2} {1} {1 2}}
{{0 1} {} {0 1 2} {1}}
{{0 1} {} {0 1 2} {0}}

reversible binary operation
{{} {0 1 2} {1} {0}}
{{} {0 1 2} {2} {0}}
{{} {0 1 2} {2} {1}}

any product function
{{} {0 1 2} {2} {0 2} {0}}
{{0 1} {} {0 1 2} {1} {0}}
{{} {0 1 2} {2} {1} {1 2}}

Dimension 2

constant binary operation (disconnected)
{{0 1} {0 1 2} {1} {1 2}}
{{0 1 2} {2} {1 2} {0 2}}
{{0 1} {0 1 2} {0 2} {0}}

1 goes to 2 and 2 goes to 1 (disconnected)
{{} {0 1 2} {1 2} {0}}
{{} {0 1 2} {1} {0 2}}
{{0 1} {} {0 1 2} {2}}

partially invariant operation
{{} {0 1 2} {2} {1} {1 2} {0 2}}
{{} {0 1 2} {2} {1 2} {0 2} {0}}
{{0 1} {} {0 1 2} {1} {1 2} {0}}
{{0 1} {} {0 1 2} {2} {0 2} {0}}
{{0 1} {} {0 1 2} {1} {0 2} {0}}
{{0 1} {} {0 1 2} {2} {1} {1 2}}

partially reversible binary operation
{{0 1} {} {0 1 2} {2} {1}}
{{0 1} {} {0 1 2} {2} {0}}
{{} {0 1 2} {1} {0 2} {0}}
{{} {0 1 2} {2} {1 2} {0}}
{{} {0 1 2} {1} {1 2} {0}}
{{} {0 1 2} {2} {1} {0 2}}

both 1 and 2 go to 0
{{} {0 1 2} {2} {1 2} {0 2}}
{{0 1} {} {0 1 2} {1} {1 2}}
{{0 1} {} {0 1 2} {0 2} {0}}

completely cancellative binary operaiton
{{} {0 1 2} {2} {1} {0}}

partially cancellative binary operation
{{0 1} {} {0 1 2} {2} {1} {0}}
{{} {0 1 2} {2} {1} {1 2} {0}}
{{} {0 1 2} {2} {1} {0 2} {0}}

any binary operations
{{0 1} {} {0 1 2} {2} {1} {1 2} {0}}
{{} {0 1 2} {2} {1} {1 2} {0 2} {0}}
{{0 1} {} {0 1 2} {2} {1} {0 2} {0}}

Dimension 3

any ternary relation (disconnected)
{{0 1} {} {0 1 2} {2} {1} {1 2} {0 2} {0}}