Expressing Partial Order-Preserving Transformations as Products of Nilpotents

Abstract

Let S be a semigroup with zero, then an element a∈S is called nilpotent, if there exists a positive integer n such that . In the partial transformation semigroup on , where  is a non-empty set, denoted by , the empty map is the zero of . Further, nilpotency is a structural property of fundamental importance which arises in many situations, for instance in linear algebra, we have nilpotent matrices and much more in other branches of mathematics. In this study, we will investigate, some elements of semigroups of partial one-to-one order-preserving transformations (denoted by ) and partial order-preserving transformations (denoted by ), and some basic properties of nilpotent transformations. We have studied certain properties of nilpotent elements in , symmetric inverse semigroup, where  is finite and infinite, certain properties of nilpotent elements in , descriptions of the elements of 〈N〉, where N is the set of nilpotents of  and we used the same letter N to denote that of , and used them to establish the following results, in this paper (that is, we have used the knowledge that we have acquired to establish the following results): - a partial ono-to-one order-preserving transformation , has from 1 to  fixed,  and there is an upper jump of length two(2) between  and , and lower jumps of length one(1) and one(1), immediately after , can be expressed as a product of three nilpotents. – a partial one-to-one order-preserving transformation , has from 1 to  fixed,  and there is an upper jump of length three (3) between  and  and lower jumps of length two (2) and one (1), immediately after , can be expressed as a product of fewer than three nilpotents. – a partial order-preserving transformation , has from 1 to  fixed, but  belongs to ,  and there is an upper jump of length two(2) between  and , and lower jumps of length two(2)  and one(1), immediately after , cannot be expressed as a product of fewer than three nilpotents. – a partial order-preserving transformation , has from 1 to  fixed, but  belongs to ,  and there is an upper jump of length three (3) between  and , and lower jumps of length three (3) and one (1), immediately after n-5, can be expressed as a product of fewer than three nilpotents. – a transformation , partial transformation semigroup, is not nilpotent if and only if the graph  of the transformation  has a loop, and - a transformation , partial one-to-one transformation semigroup, which has a nilpotent path and a permutation part is not nilpotent