MV-algebra - Examples of MV-algebras

Examples of MV-algebras

A simple numerical example is with operations and In mathematical fuzzy logic, this MV-algebra is called the standard MV-algebra, as it forms the standard real-valued semantics of Łukasiewicz logic.

The trivial MV-algebra has the only element 0 and the operations defined in the only possible way, and

The two-element MV-algebra is actually the two-element Boolean algebra with coinciding with Boolean disjunction and with Boolean negation. In fact adding the axiom to the axioms defining an MV-algebra results in an axiomantization of Boolean algebras.

If instead the axiom added is, then the axioms define the MV₃ algebra corresponding to the three-valued Łukasiewicz logic Ł₃. Other finite linearly ordered MV-algebras are obtained by restricting the universe and operations of the standard MV-algebra to the set of equidistant real numbers between 0 and 1 (both included), that is, the set which is closed under the operations and of the standard MV-algebra; these algebras are usually denoted MV_n.

Another important example is Chang's MV-algebra, consisting just of infinitesimals (with the order type ω) and their co-infinitesimals.

Chang also constructed an MV-algebra from an arbitrary totally ordered abelian group G by fixing a positive element u and defining the segment as { x ∈ G | 0 ≤ x ≤ u }, which becomes an MV-algebra with x ⊕ y = min(u, x+y) and ¬x = u−x. Furthermore, Chang showed that every linearly ordered MV-algebra is isomorphic to an MV-algebra constructed from a group in this way.

D. Mundici extended the above construction to abelian lattice-ordered groups. If G is such a group with strong (order) unit u, then the "unit interval" { x ∈ G | 0 ≤ x ≤ u } can be equipped with ¬x = u−x, x ⊕ y = u∧_G (x+y), x ⊗ y = 0∨_G(x+y−u). This construction establishes a categorical equivalence between lattice-ordered abelian groups with strong unit and MV-algebras.