Single-tile semigroups of shift operators

We introduce some new subsemigroups of the finite full transformation semigroups tn. we consider various irregular boards of different shapes and sizes to generate subsemigroups of tn using the four idempotent operators l,r,u,d. these operators shift tiles on a board in four different directions (left,right,up and down) in their respective rows and columns. in this way each operator o € g = {l,r,u,d) defines a member of the base setx = {1,2,…,n} and the semigroups of various properties are given as (formula presented). © 2016,pakistan academy of sciences.