This paper initiates the study of the algebra of quotients associated with a given BCK-algebra, analogous to the classical rings of quotients and their subsequent generalizations. MIRAMAKE TRIESTE August 198B * To appear in "Hath. Japonlca" (1988). ** Permanent address: Department of Mathematics, Quaid-i-Azam University, Islamabad, Pakistan.

We present and compare two different implementations of quotient types in Intensional Type Theory. We first introduce quotients as particular inductive-like types following Martin Hofmann’s extension of Calculus of Constructions with quotient types [6]. Then we give an impredicative encoding of quotients. This implementation is reminiscent of Church numerals and more generally of encodings of i...

For any given real number α with bounded partial quotients, we construct explicitly continuum many real numbers β with bounded partial quotients for which the pair (α, β) satisfies a strong form of the Littlewood conjecture. Our proof is elementary and rests on the basic theory of continued fractions.