Relation Algebra - Expressing Properties of Binary Relations in RA

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 AB is shorthand for the Boolean equation AB = 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 RRI
Total or Connected IRR (R is surjective)
Function functional and total.
Injective
RRI (R is functional)
Surjective IRR (R is total)
Bijection RR = RR = I (Injective surjective function)
Reflexive IR
Coreflexive RI
Irreflexive RI = 0
Transitive RRR
Preorder R is reflexive and transitive.
Antisymmetric RRI
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 RR = R. R is a symmetric preorder.
Asymmetric RR
Dense RI– ≤ (RI–)•(RI–).

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