On monadic MV-algebras

Abstract

We deÿne and study monadic MV -algebras as pairs of MV -algebras one of which is a special case of relatively complete subalgebra named m-relatively complete. An m-relatively complete subalgebra determines a unique monadic operator. A necessary and su cient condition is given for a subalgebra to be m-relatively complete. A description of the free cyclic monadic MV -algebra is also given.

Monadic Bounded Commutative Residuated ℓ-monoids

“…Recall that in the case of monadic MV-algebras both universal and existential quantifiers are defined and that the one-to-one correspondences between them are as follows: If M is an MV-algebra and ∃ is an existential quantifier on M then the corresponding universal quantifier ∃ * is such that ∃ * (x) = (∃(x − )) − for all x, y ∈ M, and dually, if ∀ is a universal quantifier on M then the corresponding existential quantifier ∀ * is such that ∀ * (x) = (∀(x − )) − for all x, y ∈ M -see [9]. Now, we describe some properties of analogously defined operator ∀ * for a universal quantifier ∀ on a monadic Rl-monoid M. …”

“…MMV-algebras were also studied as so-called polyadic MV-algebras in [32,33]. Recently, the theory of MMValgberas has been developed in papers [2,9,13]. Recall that monadic, polyadic and cylindric (Boolean) algebras, as algebraic structures corresponding to the classical predicate logic, have been investigated in [15][16][17].…”

Monadic MV-algebras are Equivalent to Monadic ℓ-groups with Strong Unit

“…These algebras were introduced by Rutledge [10], under the name of monadic Chang algebras, and were recently developed by Di Nola and Grigolia [7], Belluce, Grigolia and Lettieri [1] and Lattanzi and Petrovich [8].…”

“…Monadic MV-algebras have been studied by several authors. In [7], Di Nola and Grigolia study monadic MV-algebras as pairs of MV-algebras one of which is a special case of relatively complete subalgebra. In [1], Belluce, Grigolia and Lettieri obtain a representation theorem for certain classes of monadic MV-algebras and give a characterization of the monadic operators over a finite MV-algebra.…”

Representations of monadic MV -algebras

“…The proof immediately follows from E1 − E6 and Lemma 2.1. [8] A subalgebra A 0 of an MV -algebra A is said to be relatively complete (or an rc-subalgebra) in A if, for every a ∈ A, the set {b ∈ A 0 : a ≤ b} has a least element, which is denoted by…”