Archimedean classes in integral commutative residuated chains

The problem of characterization of the structure of MTL-algebras, which form an equivalent algebraic semantics for Monoidal T-norm Based Logic (see [9]) in the sense of Blok and Pigozzi (see [3]), is still far from being solved. Since MTL-algebras are in fact subdirect products of chains, it suffices to investigate only the structure of MTL-chains if we want to characterize the structure of MTL-algebras. Thus a closely related problem already discussed in the literature [11, 12, 13, 24] is the same task for totally ordered monoids since each MTL-chain forms a totally ordered monoid. As was pointed out in [11], the characterization of the structure of totally ordered monoids could be split into two steps: (1) determine the structure of an arbitrary Archimedean totally ordered monoid; (2) determine the ways in which a given chain of Archimedean totally ordered monoids can be assembled to form a totally ordered monoid having the elements of the chain as its Archimedean classes. In order to solve these two steps, it is clear that the notion of an Archimedean class is crucial. Another closely related problem is to understand better the structure of the lattice of subvarieties of MTL-algebras. It is quite natural to ask whether it is possible to express the number of Archimedean classes in an MTL-chain by an identity. Unfortunately, this is not possible in general. Indeed, there are product chains with arbitrary number of Archimedean classes but the only nontrivial subvariety of product algebras is the variety of Boolean algebras. However, in some cases it is possible as we are going to show in this paper. The obtained results also shed some light on the structure of MTL-chains. The original motivation of our results comes from [15] where the author posed a question whether the variety of ΠMTL-algebras (i.e. the class of cancellative MTL-algebras) is generated by Archimedean ΠMTL-chains. The paper [15] offers only a partial answer by showing it is not generated as a quasivariety. More precisely, the author shows that the quasi-identity