Goedel algebras

Abbreviation: GödA

Definition

A Gödel algebra is a Heyting algebra A = (A,∨,0,∧,1,→) such that (x → y)∨(y → x) = 1.

Remark: Gödel algebras are also called linear Heyting algebras since subdirectly irreducible Gödel algebras are linearly ordered Heyting algebras.

Definition

A Gödel algebra is a [representable FLew-algebra]? A = (A,∨,0,∧,1,·,→) such that
x∧y = x·y.

Morphisms

Let A and B be Gödel 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(0) = 0 and h(x∧y) = h(x)∧h(y) and h(1) = 1 and h(x→y) = h(x)→h(y).

Some results

Examples

Properties

Classtype	variety
Equational theory	decidable
Quasiequational theory	decidable
First-order theory
Locally finite
Residual size	countable
Congruence distributive	yes
Congruence modular	yes
Congruence n-permutable	yes, n = 2
Congruence e-regular	yes, e = 1
Congruence uniform
Congruence extension property	yes
Definable principal congruences	yes
Equationally definable principal congruences	yes
Amalgamation property
Strong amalgamation property
Epimorphisms are surjective

Finite members

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Subclasses

Boolean algebras

Superclasses

Heyting algebras