Representations of monadic MV -algebras

Abstract

Representations of monadic MV -algebra, the characterization of locally finite monadic MV -algebras, with axiomatization of them, definability of non-trivial monadic operators on finitely generated free MV -algebras are given. Moreover, it is shown that finitely generated m-relatively complete subalgebra of finitely generated free MV -algebra is projective.

Monadic BE-algebras

“…By Proposition 3.3(5), ∃x, ∃y ≤ ∃(x ∨ y). If there exists d ∈ A such that ∃x, ∃y ≤ d, then x ≤ ∃x ≤ ∀d and similarly y ≤ ∀d, so x ∨ y ≤ ∀d.…”

“…It were also studied as polyadic MV -algebras in [22,23]. Recently, the theory of Monadic MV -algebras has been developed in [5,8,9]. In [24], Rachunek andŠalounová extended the notion of a Monadic MV -algebras to an arbitrary GMV -algebra which need not be commutative and defined the monadic non-commutative Łukasiewicz propositional calculus MPL using the non-commutative Łukasiewicz propositional calculus PL from [18].…”

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].…”

“…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. Finally in [8], Lattanzi and Petrovich give categorical equivalences between the varieties of monadic (n + 1)-valued MV-algebras and the class of monadic Boolean algebras endowed with certain family of their filters.…”

Monadic Bounded Commutative Residuated ℓ-monoids

“…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].…”