# Heyting algebras

HomePage | RecentChanges | Preferences

### Definition

A Heyting algebra is a structure A = (A,∨,0,∧,1,→) such that
(A,∨,0,∧,1) is a bounded distributive lattice, and
gives the residual of :   xy ≤ z  ⇔   y ≤ xz.

### Definition

A Heyting algebra is a FLew-algebra A = (A,∨,0,∧,1,·,→) such that
xy = x·y.

### Morphisms

Let A and B be Heyting 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 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 [Marcel Erné, Jobst Heitzig and Jürgen Reinhold, On the number of distributive lattices, Electron. J. Combin. 9 (2002) Research Paper 24, 23 pp. (electronic) MRreview]

Goedel algebras

### Superclasses

Bounded distributive lattices

HomePage | RecentChanges | Preferences