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We show that for a variety V of Heyting algebras the following conditions are equivalent: (1) V is locally finite; (2) the V-coproduct of any two finite V-algebras is finite;(3) either V coincides with the variety of Boolean algebras or finite V-copowers of the three element chain 3 ∈ V are finite. We also show that a variety V of Heyting algebras is generated by its finite members if, and only if, V is generated by a locally finite V-algebra. Finally, to the two existing criteria for varieties of Heyting algebras to be finitely generated we add the following one: V is finitely generated if, and only if, V is residually finite.

No abstract

We deÿne 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 su cient condition is given for a subalgebra to be m-relatively complete. A description of the free cyclic monadic MV -algebra is also given.

Representations of monadic MV -algebra, the characterization of locally finite monadic MV -algebras, with axiomatization of them, definability of non-trivial monadic operators on finitely generated free MV -algebras are given. Moreover, it is shown that finitely generated m-relatively complete subalgebra of finitely generated free MV -algebra is projective.