Some combinatorial results in full and partial transformation semigroups

Let [𝑛] = {1, 2, . . . , 𝑛}. A function 𝛼 with domain Dom 𝛼 ⊆ [𝑛] and image Im 𝛼 ⊆ [𝑛] is called a partial transformation. It is a full transformation if Dom 𝛼 = [𝑛]. Partial transformations form a semigroup under the composition of maps. The index and period of an element 𝛼 in this semigroup are the smallest values of positive integers 𝑚 and r such that 𝛼^(𝑚+𝑟)= 𝛼^𝑚. 𝛼 is called 𝑚-potent if 𝑟 = 1 and it is a potent element if it is 𝑚-potent for some 𝑚. The number of 𝑚-potent and potent elements are investigated. We give a discussion of the fact that every 𝛼 in full transformation with index 𝑚 and period 𝑟 can be uniquely written as a product of a permutation of order 𝑟 and an 𝑚-potent element where the permutation and potent element have disjoint shifts. If S𝑛 is the group of permutations, it acts on full transformation group by conjugation. We calculate the cardinality of equivalence classes of conjugates of 1-potent, more commonly known as idempotent, elements. In the last two chapters we investigate the cardinality properties of nilpotent semigroups. We utilize graph theoretic methods in this investigation. Interpreting the results obtained for the cardinality of maximal symmetric inverse semigroups we obtain some relationships involving Bell numbers and Stirling numbers.