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Sequential algebrasSequential algebras
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 ⇔ (x▷z)∧y = 0
◁ is the left-conjugate of o:
(xoy)∧z = 0 ⇔ (z◁y)∧x = 0
▷,◁ are balanced:
x▷e = e◁x
o is euclidean:
x·(y◁z) ≤ (x·y)◁z.
Remark:
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(x▷y) = h(x)▷h(y) and h(x◁y) = h(x)◁h(y) and h(e) = e.
| 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 |