Gödel spaces and perfect MV-algebras

Abstract

The category of Gödel spaces GS (with strongly isotone maps as morphisms), which are dually equivalent to the category of Gödel algebras, is transferred by a contravariant functor H into the category MV(C)^G of MV-algebras generated by perfect MV-chains via the operators of direct products, subalgebras and direct limits. Conversely, the category MV(C)^G is transferred into the category GS by means of a contravariant functor P. Moreover, it is shown that the functor H is faithful, the functor P is full and the both functors are dense. The description of finite coproduct of algebras, which are isomorphic to Chang algebra, is given. Using duality a characterization of projective algebras in MV(C)^G is given

On the free S^{ω}_{1} -algebras

“…The class of perfect M V -algebras does not form a variety and contains non-simple subdirectly irreducible M V -algebras [15]. It is worth stressing that the variety V(S ω 1 ), denoted by MV(C) in [11], generated by all perfect M V -algebras is also generated by a single M V -chain C( ∼ = S ω 1 ) defined by Chang in [5]. We name by S ω 1 -algebras all the algebras from the variety generated by S ω 1 ( ∼ = C) [11].…”

“…It is worth stressing that the variety V(S ω 1 ), denoted by MV(C) in [11], generated by all perfect M V -algebras is also generated by a single M V -chain C( ∼ = S ω 1 ) defined by Chang in [5]. We name by S ω 1 -algebras all the algebras from the variety generated by S ω 1 ( ∼ = C) [11]. Notice that the Lindenbaum algebra of the logic L P is an S ω 1 -algebra where L P is the logic corresponding to the variety V(S ω 1 ).…”

“…Notice, that the algebra S ω 1 was firstly defined as algebra C by Chang in [5]. Not exact algebraic description of free m-generated S ω 1 -algebra (or M V (C)-algebra) have been given in [11] where the free algebras are represented by subdirect product of infinite family of chains. In this paper we represent free algebras by means of subdirect product of finite family of chains according to Panti's result in [24].…”