# Nonassociative relation algebras

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### Definition

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

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

Remark:

### 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 Equationally definable principal congruences [Discriminator variety]? no Amalgamation property Strong amalgamation property Epimorphisms are surjective

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### Subclasses

[Weakly associative relation algebras]?

### Superclasses

[Nonassociative sequential algebras]?

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Last edited April 19, 2003 9:32 pm (diff)
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