[Home]Tense algebras

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Abbreviation: TA


A tense algebra is a structure A = (A,∨,0, ∧,1,¬,◊f, ◊p) such that both
(A,∨,0,∧,1,¬,◊f) and (A,∨,0,∧,1,¬,◊p) are modal algebras,and
p and f are conjugates:   x∧◊py  = 0 iff fxy  = 0

Remark: Tense algebras provide algebraic models for logic of tenses. The two possibility operators p and f are intuitively interpreted as at some past instance and at some future instance.


Let A and B be tense algebras. A morphism from A to B is a function h : A → B that is a Boolean homomorphism and preserves p and f: h(◊x) = ◊h(x).

Some results



Classtype variety
Equational theory decidable
Quasiequational theory decidable
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
[Discriminator variety]? no
Amalgamation property yes
Strong amalgamation property yes
Epimorphisms are surjective yes

Finite members

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




Boolean algebras with operators

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Last edited March 31, 2004 8:55 pm (diff)