An algebraic proof is presented for the finite strong standard completeness of the Involutive Uninorm Logic with Fixed Point ($${{\mathbf {IUL}}^{fp}}$$ IUL fp ). It may provide a first step towards settling the standard completeness problem for the Involutive Uninorm Logic ($${\mathbf {IUL}}$$ IUL, posed in G. Metcalfe, F. Montagna. (J Symb Log 72:834–864, 2007)) in an algebraic manner. The result is proved via an embedding theorem which is based on the structural description of the class of odd involutive FL$$_e$$ (...) e -chains which have finitely many positive idempotent elements. (shrink)

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. Dunnabout finite Sugihara monoids.