[Home]Sequential algebras

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Abbreviation: SeA

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.

Some results

Examples

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

Finite members

[Size 1]?:  1
[Size 2]?:  
[Size 3]?:  
[Size 4]?:  
[Size 5]?:  
[Size 6]?:  

Subclasses

Relation algebras
[Representable sequential algebras]?

Superclasses

Distributive residuated lattices
[Semiassociative sequential algebras]?


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Last edited April 19, 2003 9:20 pm (diff)
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