The structure of elements in finite full transformation semigroups

Abstract

The index and period of an element a of a finite semigroup are the smallest values of m ^ 1 and r ^ 1 such that a m+T = a m . An element with index m and period 1 is called an m-potent element. For an element a of a finite full transformation semigroup with index m and period r, a unique factorisation a = of} such that Shift(cr) PI Shift(/?) = 0 is obtained, where a is a permutation of order r and 0 is an m-potent. Some applications of this factorisation are given.

On transformations of index $1$

“…, Ω t (t ≥ 1). The orbits are the connected components of the function graph and provide valuable information about the structure of the map α (for example, see [1], [3]). Typically, an orbit consists of a cycle with some trees attached.…”

Ranks of nilpotent subsemigroups of order-preserving and decreasing transformation semigroups

“…The number of nilpotent elements in C n have been calculated by Laradji and Umar in[11]. The number of m -potent elements and (m, r) -potent elements in ST n , the subsemigroup of all singular transformations of T n , were computed by Ayık, Ayık, Ünlü and Howie in[3]. The combinatorial results relating the cardinalities of the subsets {α ∈C n : |im (α)| = r and nα = k} and {α ∈ C n : im (α) = r} were given by Umar in [12, Propositions 3.4 and 3.6].…”