Conjugacy classes of left ideals of Sweedler's four-dimensional algebra $ H_{4} $

Let $ A $ be a finite-dimensional algebra with identity over the field $ \mathbb{F} $, $ U(A) $ be the group of units of $ A $ and $ L(A) $ be the set of left ideals of $ A $. It is well known that there is an equivalence relation $ \sim $ on $ L(A) $ by defining $ L_1\sim L_2\in L(A) $ if and only if there exists some $ u\in U(A) $ such that $ L_{1} = L_{2}u $. $ C(A) = \{[L]|L\in L(A)\} $ is the set of equivalence classes determined by the relation $ \sim $, which is a semigroup with respect to the natural operation $ [L_1][L_2] = [L_1L_2] $ for any $ L_1, L_2 \in L(A) $. The purpose of this paper is to describe the structures of semigroup of conjugacy classes of left ideals for the Sweedler's four-dimensional Hopf algebra $ H_{4} $.