Brouwerian algebras

Abbreviation: BrA

Definition

A Brouwerian algebra is a structure A = (A, ∨, ∧, 1, →) such that
(A, ∨, ∧, 1) is a distributive lattice with top, and
→ gives the residual of ∧: x∧y ≤ z ⇔ y ≤ x→z.

Morphisms

Let A and B be Brouwerian algebras. A morphism from A to B is a function h : A→B that is a homomorphism: h(x∨y) = h(x)∨h(y) and h(x∧y) = h(x)∧h(y) and h(1) = 1 and h(x→y) = h(x)→h(y).

Definition

A Brouwerian algebra is a BL-algebra A = (A, ∨, ∧, 1, ·, →) such that x∧y = x·y.

Some results

Examples

Properties

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 e-regular	yes, e = 1
Congruence uniform	no
Congruence extension property	yes
Definable principal congruences	yes
Equationally definable principal congruences	yes
Amalgamation property	yes
Strong amalgamation property	yes
Epimorphisms are surjective	yes

Finite members

[Size 1]?: 1
[Size 2]?: 1
[Size 3]?: 1
[Size 4]?: 2
[Size 5]?: 3
[Size 6]?: 5
[Size 7]?: 8
[Size 8]?: 15
[Size 9]?: 26
[Size 10]?: 47
[Size 11]?: 82
[Size 12]?: 151
[Size 13]?: 269
[Size 14]?: 494
[Size 15]?: 891
[Size 16]?: 1639
[Size 17]?: 2978
[Size 18]?: 5483
[Size 19]?: 10006
[Size 20]?: 18428
Values known up to size 49 [Erne, Heitzig, Reinhold (2002)]

Subclasses

Generalized Boolean algebras
Heyting algebras

Superclasses

Distributive lattices
Basic logic algebras