# Boolean algebras with operators

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### Definition

A Boolean algebra with operators is a structure A = (A,∨,0, ∧,1,¬,fi (i ∈ I)) such that

(A,∨,0,∧,1,¬) is a Boolean algebra,
fi is join-preserving in each argument:   fi( . . .,xy, . . .) = fi( . . .,x, . . .)∨fi( . . .,y, . . .), and
fi is normal in each argument:   fi( . . .,0, . . .) = 0 for each i ∈ I.

### Morphisms

Let A and B be Boolean algebras with operators of the same signature. A morphism from A to B is a function h : AB that is a Boolean homomorphism and preserves all the operators: h(fi(x0, . . .,xn−1)) = fi(h(x0), . . .,h(xn−1)).

### Properties

 Classtype variety Equational theory decidable Quasiequational theory 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 no Equationally definable principal congruences no Amalgamation property yes Strong amalgamation property yes Epimorphisms are surjective yes

Modal algebras
Boolean monoids

### Superclasses

Boolean algebras

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