rqigroup Forum Vol. 54 (1997) 230-236 997 Springer-Verlag New York Inc. RESEARCH ARTICLE Semigroups of Order Preserving Mappings on a Finite Chain: A New Class of Divisors Vitor H. Fernandes" Communicated by J.-E. Pin Abstract In this paper we aim to prove that every semigroup of the pseudovariety generated by all semigroups of partial, injective and order preserving transformations on a finite chain belongs to the pseudovariety generated by all semigroups of order preserving mappings on a finite chain. Introduction The study of the pseudovariety (of semigroups) O generated by all semigroups of order preserving transformations on a finite chain was proposed by J.-E. Pin in the "Szeged International Semigroup Colloquium" in 1987. Only recently some essential progress on this problem was made. First P. M. Higgins [1] proved that every finite band belongs to O, and later A. S. Vernitskii and M. V. Volkov [4] generalized Higgins's result, by showing that every finite semigroup whose idempotents form an ideal is in O. Like those results, our main result is also a division theorem for the pseudova- riety O : Theorem. Every semigroup of partial, injective and order preserving transforma- tions on a finite chain belongs to O. This paper is organized as follows: in section 1 we present some definitions and in section 2 we prove several properties of the semigroups of partial, injective and order preserving transformations on a finite chain. Finally, our main theorem is proved in section 3. 1. Preliminaries Let X be a set. We denote by T(X) the monoid of all transformations of X (under composition), hy 7PT(X) the monoid of all partial transformations of X and by :Z(X) the submonoid of 7~T(X) of all injective partial transformations of X. Assume that X= is a chain with n elements, say Xn = {1 < 2 < ... < n}. We say that a mapping a in T(Xn) or in T'T(X=) is order preserving if, for all x,y E Dom((~), x _< y implies x(~ _< ya. As usual, we denote by On the submonoid of T(Xn) of all order preserving mappings on Xn. By definition the pseudovariety O is generated by {O, In E N}. *: This research was done within the project SAL (JNICT, PBIC/C/CEN[1021/92), and the activities of the "Centro de .&lgebra da Universidade de Lisboa".