# Boolean monoids

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### Definition

A Boolean monoid is a structure A = (A,∨,0, ∧,1,¬,·,e) such that

(A,∨,0, ∧,1,¬) is a Boolean algebra,
(A,·,e) is a monoid,
· is join-preserving in each argument:   (xyz = (x·z)∨(y·z) and x·(yz) = (x·y)∨(x·z)
· is normal in each argument:   x = 0 and x·0 = 0.

Remark:

### Morphisms

Let A and B be Boolean monoids. A morphism from A to B is a function h : AB that is a Boolean homomorphism and preserves ·, e: h(x·y) = h(xh(y) and h(e) = e.

### 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

[Size 1]?:  1
[Size 2]?:  1
[Size 3]?:  0
[Size 4]?:  9
[Size 5]?:  0
[Size 6]?:  0
[Size 7]?:  0
[Size 8]?:  258

### Subclasses

Sequential algebras

### Superclasses

Boolean algebras with operators

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