Math 2.0

Mathematical logic and foundations Definition

Relation algebra

Builds on Boolean algebraMonoidBinary relationComposition of relationsConverse relation

Definition

A relation algebra is a structure (A,∨,∧,−,0,1,;,c,1′)(A, \vee, \wedge, -, 0, 1, ;, c, 1'), where ;; is a binary operation, cc is a unary operation and 1′1' is an element of AA. Write x˘\breve{x} for c(x)c(x). The following hold for all x,y,z∈Ax, y, z \in A:

  1. (A,∨,∧,−,0,1)(A, \vee, \wedge, -, 0, 1) is a Boolean algebra, with order ≤\le.
  2. (A,;,1′)(A, ;, 1') is a monoid.
  3. (x∨y);z=(x;z)∨(y;z)(x \vee y) ; z = (x ; z) \vee (y ; z).
  4. x˘˘=x\breve{\breve{x}} = x and (x∨y)˘=x˘∨y˘\breve{(x \vee y)} = \breve{x} \vee \breve{y}.
  5. (x;y)˘=y˘;x˘\breve{(x ; y)} = \breve{y} ; \breve{x}.
  6. x˘;−(x;y)≤−y\breve{x} ; -(x ; y) \le -y.

Remarks

Example: let UU be a set and let AA be the set of all binary relations on UU, with the set operations. Let ;; be the composition R;S={(a,c):(a,b)∈R and (b,c)∈S for some b}R ; S = \set{(a, c) : (a, b) \in R \text{ and } (b, c) \in S \text{ for some } b}. Let R˘\breve{R} be the converse relation and 1′1' the identity relation. Check of clause 6: if (a,c)∈R˘;−(R;S)(a, c) \in \breve{R} ; -(R ; S), there is bb with (b,a)∈R(b, a) \in R and (b,c)∉R;S(b, c) \notin R ; S. If (a,c)(a, c) were in SS, then (b,c)∈R;S(b, c) \in R ; S; so (a,c)∉S(a, c) \notin S.

Non-example: on the relations of a set UU with at least two elements, keep the Boolean operations, ;; and 1′1', but take R˘=R\breve{R} = R. Clause 5 then says R;S=S;RR ; S = S ; R for all relations, which fails for U={a,b}U = \set{a, b}, R={(a,b)}R = \set{(a, b)} and S={(b,b)}S = \set{(b, b)}, since R;S={(a,b)}R ; S = \set{(a, b)} and S;R=∅S ; R = \emptyset.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
Binary relation, concept, no evidence levelBinary relationconceptBoolean algebra, concept, no evidence levelBoolean algebraconceptComposition of relations, concept, no evidence levelComposition of relati…conceptConverse relation, concept, no evidence levelConverse relationconceptMonoid, concept, no evidence levelMonoidconceptRelation algebra, definition, no evidence levelRelation algebradefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Binary relation concept
  2. Boolean algebra concept
  3. Composition of relations concept
  4. Converse relation concept
  5. Monoid concept
  6. Relation algebra definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Relation_algebra
About this page
TagRDSY11X6N2
URIm2://RDSY11X6N2
Kinddefinition
WritingW0 stub
MSC03G15
DefinesRelation algebra
UsesBoolean algebra, Monoid, Binary relation, Composition of relations, Converse relation
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.