On the Ranges of Algebraic Functions in Lattices - A Preliminary Report

We investigate ranges of ternary algebraic functions in lukasiewicz-Moisil algebras, where we give a characterization of algebraic functions whose ranges are intervals and we retrieve a canonical form of functions over three-element ternary lukasiewicz-Moisil algebras, a result due to Gr. C. Moisil, one of the founders of switching theory [Moi57]. In the second part of this paper we show that in a Noetherian or Artinian lattice distributivity and boundedness are implied by the condition that every algebraic functions has an interval as its range; this is actually a characterization of boundedness and distributivity in the class of lattices that have finite chains.