Expressing Properties of Binary Relations in RA
The following table shows how many of the usual properties of binary relations can be expressed as succinct RA equalities or inequalities. Below, an inequality of the form A≤B is shorthand for the Boolean equation A∨B = B.
The most complete set of results of this nature is chpt. C of Carnap (1958), where the notation is rather distant from that of this entry. Chpt. 3.2 of Suppes (1960) contains fewer results, presented as ZFC theorems and using a notation that more resembles that of this entry. Neither Carnap nor Suppes formulated their results using the RA of this entry, or in an equational manner.
| R is | If and only if: |
|---|---|
| Functional | R•R ≤ I |
| Total or Connected | I ≤ R•R (R is surjective) |
| Function | functional and total. |
| Injective |
R•R ≤ I (R is functional) |
| Surjective | I ≤ R•R (R is total) |
| Bijection | R•R = R•R = I (Injective surjective function) |
| Reflexive | I ≤ R |
| Coreflexive | R ≤ I |
| Irreflexive | R ∧ I = 0 |
| Transitive | R•R ≤ R |
| Preorder | R is reflexive and transitive. |
| Antisymmetric | R ∧ R ≤ I |
| Partial order | R is an antisymmetric preorder. |
| Total order | R is a total partial order. |
| Strict partial order | R is transitive and irreflexive. |
| Strict total order | R is a total strict partial order. |
| Symmetric | R = R |
| Equivalence | R•R = R. R is a symmetric preorder. |
| Asymmetric | R ≠ R |
| Dense | R ∧ I– ≤ (R ∧ I–)•(R ∧ I–). |
Read more about this topic: Relation Algebra
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