Boolean modules over a relation algebra

Abbreviation: BRMod

Definition

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

(A,∨,0, ∧,1,¬) is a Boolean algebra,
f_r is join-preserving: f_r(x∨y) = f_r(x)∨f_r(y)
f_r is normal: f_r(0) = 0
f_1' is the identity map: f_1'(x) = x
f_r(f_s(x)) = f_ros(x).

Remark:

Morphisms

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

Some results

Examples

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

Finite members

[Size 1]?: 1
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Subclasses

[One-element algebras]?

Superclasses

Boolean algebras with operators