# Goedel algebras

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### 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
xy = x·y.

### Morphisms

Let A and B be Gödel algebras. A morphism from A to B is a function h : AB that is a homomorphism: h(xy) = h(x)∨h(y)  and  h(0) = 0  and  h(xy) = h(x)∧h(y)  and  h(1) = 1  and  h(xy) = h(x)→h(y).

### 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

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Boolean algebras

### Superclasses

Heyting algebras

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