MV-algebras

Abbreviation: MV

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 : A→B that is a homomorphism: h(x + y) = h(x) + h(y)  and  h(¬x) = ¬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:   (x→y) ∨(y→x)  = 1, and
BL:   x∧y = x·(x→y).

Definition

A Wajsberg algebra is an algebra A = (A, →, ¬, 1) such that
1→x = x,
(x→y)→((y→z) →(x→z)  = 1,
(x→y)→y  = (y→x)→x, and
(¬x→¬y)→(y→x) = 1.

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

Some results

Examples

Properties

Finite members

Subclasses

Superclasses