Starting from an arbitrary MTL-chain, two constructions are considered: the MacNeille completion of the underlying order with suitable extensions of the other operations, and a finite embeddability construction. The preservation of properties via these constructions is investigated, that is, if a property that holds in the initial MTL-chain also holds in the constructed MTL-chain. General syntactic descriptions are given of terms s and t for which the inequality s ≤ t is preserved.

We define and study monadic MV-algebras as pairs of MV-algebras one of which is a special case of relatively complete subalgebra named m-relatively complete. An m-relatively complete subalgebra determines a unique monadic operator. A necessary and sufficient condition is given for a subalgebra to be m-relatively complete. A description of the free cyclic monadic MV-algebra is also given.

This paper deals with the varieties of monadic Heyting algebras, algebraic models of intuitionistic modal logic MIPC. We investigate semisimple, locally finite, finitely approximated and splitting varieties of monadic Heyting algebras as well as varieties with the disjunction and the existence properties. The investigation of monadic Heyting algebras clarifies the correspondence between intuitionistic modal logics over MIPC and superintuitionistic predicate logics and provides us with the solutions of several problems raised by Ono [35].

Monadic MV-algebras are an algebraic model of the predicate calculus of the Łukasiewicz infinite valued logic in which only a single individual variable occurs. GMV-algebras are a non-commutative generalization of MV-algebras and are an algebraic counterpart of the non-commutative Łukasiewicz infinite valued logic. We introduce monadic GMV-algebras and describe their connections to certain couples of GMV-algebras and to left adjoint mappings of canonical embeddings of GMV-algebras. Furthermore, functional MGMV-algebras are studied and polyadic GMV-algebras are introduced and discussed.

In the present paper we show that any at most countable linearly-ordered commutative residuated lattice can be embedded into a commutative residuated lattice on the real unit interval [0, 1]. We use this result to show that Esteva and Godo''s logic MTL is complete with respect to interpretations into commutative residuated lattices on [0, 1]. This solves an open problem raised in.

The study of perfect, local and bipartite IMTL-algebras presented in [29] is generalized in this paper to the general non-involutive case, i.e. to MTL-algebras. To this end we describe the radical of MTL-algebras and characterize perfect MTL-algebras as those for which the quotient by the radical is isomorphic to the two-element Boolean algebra, and a special class of bipartite MTL-algebras,.