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September 5, 2025

Semigroup of Partition-Preserving Transformations Whose Images Intersect Each Block

MSMosarof SarkarCentral University of South Bihar SSShubh N. SinghIndian Institute of Technology Guwahati

Key Points

Green's relations on the semigroup are identical if the partition is finite, highlighting a significant characteristic of transformations.
We characterize ideals and the kernel of these transformations, providing clarity on their structure within partitions.
The research determines conditions under which the semigroup is unit-regular, orthodox, and completely regular, enhancing our understanding of algebraic structures.
This work lays a foundation for exploring how transformations intersect with blocks in various mathematical contexts.

Abstract

For a partition Formula: see text of a set Formula: see text, we say that a transformation Formula: see text of Formula: see text preserves Formula: see text if for every Formula: see text there exists Formula: see text such that Formula: see text. Let Formula: see text denote the semigroup of all transformations Formula: see text of Formula: see text such that Formula: see text preserves Formula: see text and its image Formula: see text intersects each block of Formula: see text. We describe Green's relations on Formula: see text and prove that Green's relations Formula: see text and Formula: see text on Formula: see text are equal if and only if Formula: see text is finite. Moreover, we characterize ideals and the kernel of Formula: see text and determine when the semigroup Formula: see text is unit-regular, orthodox, inverse, and completely regular separately.

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