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Tense algebrasTense algebras|
x∧py = 0 iff fx∧y = 0 |
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x∧◊py = 0 iff ◊fx∧y = 0 |
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Closure algebras |
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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 ◊fx∧y = 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).
| 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 |