Subdirect product decompositions of MV-algebras
Ján Jakubík · Czechoslovak Mathematical Journal · 1999
Each M V -algebra A can be represented by means of an appropriate abelian lattice ordered group G with a strong unit u. (Cf.[4], [5], [7].)We denote by Con A and Con G the system of all congruence relations of A or of G, respectively.Both Con A and Con G are partially ordered in the usual way.In the present paper it will be shown that there exists an isomorphism of Con A onto Con G.This result will be applied for characterizing the relations between subdirect product decompositions of A and those of G.To each direct product decomposition of G there corresponds a direct product decomposition of A (cf. [5]).Let us remark that each direct product decomposition of G has only a finite number of nonzero direct factors; on the other hand, A can have direct product decompositions with an infinite number of nonzero direct factors.The mentioned result from [5] concerning direct product decompositions will be sharpened.Some notions making possible to clasify subdirect product decompositions of lattice ordered groups are contained in [9].We show that these notions can be adapted for the case of M V -algebras.In [3], congruence relations on and subdirect product decompositions of M Valgebras have been applied in the context of Priestley duality.In [8], congruence relations on M V -algebras were dealt with by using the results of the theory of DRℓsemigroups.For the terminology and undefined notions concerning M V -algebras cf.[2], [4], [5].