# MV-algebras

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### Definition

An MV-algebra (short for multivalued logic algebra) is a structure A = (A, + , 0, ¬) such that
(A, + , 0) is a commutative monoid,
¬¬x = x,
x + ¬0  = ¬0, and
¬(¬x + y) + y  = ¬(¬y + x) + x.

Remark: This is the definition from [Roberto L. O. Cignoli, Itala M. L. D'Ottaviano, Daniele Mundici, Algebraic foundations of many-valued reasoning, Trends in Logic---Studia Logica Library 7 Kluwer Academic Publishers (2000) x+231 MRreview]

### Morphisms

Let A and B be MV-algebras. A morphism from A to B is a function h : AB that is a homomorphism: h(x + y) = h(x) + h(y)  and  hx) = ¬h(x)  and  h(0) = 0.

### Definition

An MV-algebra is a structure A = (A, + , 0, ·, 1, ¬) such that
(A, ·, 1) is a commutative monoid,
¬ is a DeMorgan involution for + ,·:   ¬¬x = x  and  x + y = ¬( ¬x·¬y) ,
¬0 = 1  and  0·x = 0  and  ¬( ¬x + y) + y = ¬( ¬y + x) + x.

### Definition

An MV-algebra is a basic logic algebra A = (A,∨,0,∧,1,·,→) such that
linearity:   (xy) ∨(yx)  = 1, and
BL:   xy = x·(xy).

### Definition

A Wajsberg algebra is an algebra A = (A, →, ¬, 1) such that
1→x = x,
(xy)→((yz) →(xz)  = 1,
(xy)→y  = (yx)→x, and
x→¬y)→(yx) = 1.

Remark: Wajsberg algebras are term-equivalent to MV-algebras via xy = ¬x + y, 1 = ¬0 and x + y = ¬xy, 0 = ¬1.

### Definition

A bounded hoop is an algebra A = (A, ·, →, 0, 1) such that (A, ·, →, 1) is a hoop, and 0→x = 1.

Remark: Bounded hoops are term-equivalent to Wajsberg algebras via x·y = ¬(x→¬y), 0 = ¬1, and ¬x = x→0. See [W. J. Blok, D. Pigozzi, On the structure of varieties with equationally definable principal congruences. III, Algebra Universalis 32 (1994) 545--608 MRreview] for details.

### Properties

 Classtype variety Equational theory decidable Quasiequational theory First-order theory 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 Congruence extension property yes Definable principal congruences Equationally definable principal congruences no Amalgamation property yes [Daniele Mundici, Bounded commutative BCK-algebras have the amalgamation property, Math. Japon. 32 (1987) 279--282 MRreview] Strong amalgamation property Epimorphisms are surjective

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Boolean algebras

### Superclasses

Generalized MV-algebras
Basic logic algebras
Wajsberg hoops

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