Some Results of Ideals for Partial Transformation Semigroups

Abstract

Let 〖PT〗_n  be the set of all partial functions from the set N to the set  M, where N, M⊆X_n  and X_n={1,2,…,n}. Then  〖PT〗_n  is a semigroup subject to the composition of partial mapping or a monoid with identity PI_n. Here,  the ideal of  〖PT〗_n  in terms of an element ξ∈ T ̃_(n,0)  was studied, where T ̃_(n,0)  is full transformation monoid from the set X_(n,0)  to itself such that X_(n,0)={0}∪X_n. Finding that there are two kinds of ideals in 〖PT〗_n, two-sided ideals, and a left ideal. As well as the ideals of partial transformation monoid of a free (left) G -act on n-generators, 〖PT〗_(F_n (G))  where G is a finite group and F_n (G)=∪ ̇_(i=1)^n Gx_i  were considered. Finally, the number of elements of a two-sided ideal and left ideal of 〖PT〗_n  and 〖PT〗_(F_n (G))  were also found.