# Boolean modules over a relation algebra

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### Definition

A Boolean module over a relation algebra R is a structure A = (A,∨,0, ∧,1,¬,fr (r ∈ R)) such that

(A,∨,0, ∧,1,¬) is a Boolean algebra,
fr is join-preserving:   fr(xy) = fr(x)∨fr(y)
fr is normal:   fr(0) = 0
f1' is the identity map:   f1'(x) = x
fr(fs(x)) = fros(x).

Remark:

### Morphisms

Let A and B be Boolean modules over a realtion algebra. A morphism from A to B is a function h : AB that is a Boolean homomorphism and preserves all fr: h(fr(x)) = fr(h(x)).

### Properties

 Classtype variety Equational theory Quasiequational theory First-order theory 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 no Equationally definable principal congruences no Amalgamation property Strong amalgamation property Epimorphisms are surjective

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### Subclasses

[One-element algebras]?

### Superclasses

Boolean algebras with operators

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