# Sequential algebras

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### Definition

A sequential algebra is a structure A = (A,∨,0, ∧,1,¬,o,e,▷,◁) such that

(A,∨,0, ∧,1,¬) is a Boolean algebra,
(A,o,e) is a monoid,
is the right-conjugate of o:   (xoy)∧z = 0   ⇔   (xz)∧y = 0
is the left-conjugate of o:   (xoy)∧z = 0   ⇔   (zy)∧x = 0
▷,◁ are balanced:   xe = ex
o is euclidean:   x·(yz) ≤ (x·y)◁z.

Remark:

### Morphisms

Let A and B be sequential 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(xy) = h(x)▷h(y)  and  h(xy) = h(x)◁h(y)  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 yes Equationally definable principal congruences yes [Discriminator variety]? no Amalgamation property no Strong amalgamation property no Epimorphisms are surjective no

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### Subclasses

Relation algebras
[Representable sequential algebras]?

### Superclasses

Distributive residuated lattices
[Semiassociative sequential algebras]?

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