ALGEBRAIC STRUCTURE OF PERFECT-PRESERVING FULL TRANSFORMATION SEMIGROUP.
Keywords:
Full transformation semigroup, perfect-preserving transformation, Pseudo-inverse, kernel, Ideals, regular semigroup, idempotent, Green’s relations, singular transformationsAbstract
The full transformation semigroup Tn is a fundamental object in finite semigroup theory, and imposing restrictions on the fibres of its transformations gives rise to important structural subclasses. This paper investigates the algebraic structure of the perfect-preserving full transformation semigroup , a subclass of the full transformation semigroup Tn consisting of all transformations f : Xn Xn where Xn = {1,2,…,n}, satisfying the condition: for every y in the image of f. The study aims to characterize the fundamental algebraic properties of including its rank structure, idempotents, regularity, ideals, and Green's relations. An algebraic approach based on kernel block/partitions, image sets, rank, and function composition is employed to study the elements of . The regularity of the semigroup is established by constructing a perfect-preserving pseudo-inverse for an arbitrary element, while kernel and image classifications are used to investigate its Green classes and rank-based ideal structure. The results show that every element ofis singular and has rank at most , while non-constant perfect-preserving transformations exist if and only if . The idempotents in are characterized as those idempotent transformations for which every fixed point has at least one additional preimage. Furthermore, its ideals are organised according to admissible ranks, with the constant transformations forming the minimal ideal, and its Green-class decomposition is described in terms of kernel and image structures. These results provide a foundation for further combinatorial and algebraic investigations of perfect-preserving transformation semigroups.
DOI: https://doi.org/10.5281/zenodo.23144659
