# Relation algebras

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### Definition

A relation algebra is a structure A = (A,∨,0, ∧,1,¬,o,,e) such that

(A,∨,0, ∧,1,¬) is a Boolean algebra,
(A,o,e) is a monoid,
o is join-preserving:   (xy)oz = (xoz)∨(yoz)
is an involution:   x = x  and  (xoy) = yox
is join-preserving:   (xy) = xy
o is residuated:   xo(¬(xoy)) ≤ ¬y.

### Morphisms

Let A and B be relation algebras. A morphism from A to B is a function h : A → B that is a Boolean homomorphism and preserves o, , e: h(xoy) = h(x)oh(y)  and  h(x) = h(x)  and  h(e) = e.

### Properties

 Classtype variety Equational theory undecidable Quasiequational theory undecidable First-order theory undecidable Locally finite no Residual size unbounded Congruence distributive yes Congruence modular yes Congruence n-permutable yes, n = 2 Congruence regular yes Congruence uniform yes Congruence extension property yes Definable principal congruences yes Equationally definable principal congruences yes [Discriminator variety]? yes Amalgamation property no Strong amalgamation property no Epimorphisms are surjective no

### Finite members

[Small relation algebras]
[Size 1]?:  1
[Size 2]?:  1
[Size 3]?:  0
[Size 4]?:  3
[Size 5]?:  0
[Size 6]?:  0

### Subclasses

[n-dimensional relation algebras]?
[Representable relation algebras]?
[Commutative relation algebras]?
[Square-increasing relation algebras]?

### Superclasses

Sequential algebras
[Semiassociative relation algebras]?

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