# Tense algebras

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### Definition

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.

### Morphisms

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).

### Properties

 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

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### Superclasses

Boolean algebras with operators

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