Abstract
For odd and for even involutive, commutative residuated chains a representation theorem is presented in this paper by means of direct systems of abelian o-groups equipped with further structure. This generalizes the corresponding result of J. M. Dunn about finite Sugihara monoids.
Highlights
In line with the tradition of representing classes of residuated lattices by simpler and better known structures such as groups or Boolean algebras, Mundici’s celebrated categorical equivalence theorem represents MValgebras – a variety which corresponds to Lukasiewicz logic L [12] – by -groups with strong units using a truncation construction [43]
By replacing the integrality axiom of IMTL-algebras by one of its two natural non-integral analogues one obtains the class of odd and the class of even involutive semilinear FLe-algebras, the former of which is known as the class of Involutive Uninorm Logics with fixed point (IULfp)-algebras corresponding
A representation theorem has been presented in [31,32] for those odd involutive FLe-chains where the number of idempotent elements of the algebra is finite by means of partial sublex products of abelian o-groups, which are well understood mathematical objects that are much more regular than what had been expected to need for describing these particular FLe-chains
Summary
A representation theorem has been presented in [31,32] for those odd involutive FLe-chains where the number of idempotent elements of the algebra is finite by means of partial sublex products of abelian o-groups, which are well understood mathematical objects that are much more regular than what had been expected to need for describing these particular FLe-chains. While the construction in the representation of [31,32] is done by starting with an abelian o-group and iteratively enlarging it by other abelian o-groups until the obtained structure becomes isomorphic to the given algebra, here we present a structural description using direct systems, without referring to iteration To this end, a core auxiliary result concerns a one-to-one correspondence between two lattice ordered classes: odd involutive FLe-algebras and even involutive FLe-algebras with an idempotent falsum constant. Applications of the main result of this paper to amalgamation and densification problems are foreshadowed in [30]
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