[Paper Review] On certain semigroups of partial contractions of a finite chain

Abdullahi Umar, M. M. Zubairu|arXiv (Cornell University)|Mar 7, 2018

semigroups and automata theory9 references3 citations

TL;DR

This paper investigates the algebraic structure of semigroups of partial contractions on a finite chain $[n]$, focusing on regularity and abundance properties. It proves that $\mathcal{CP}_n$, $\mathcal{ORCP}_n$, and $\mathcal{OCP}_n$ are left abundant for all $n$ but not right abundant for $n \geq 4$, and identifies the set of strongly regular elements in $\mathcal{ORCP}_n$ as a regular subsemigroup, resolving structural questions left open by prior work on non-regularity.

ABSTRACT

Let $[n]=\{1,2,\ldots,n\}$ be a finite chain and let $\mathcal{P}_{n}$ be the semigroup of partial transformations on $[n]$. Let $\mathcal{CP}_{n}=\{α\in \mathcal{P}_{n}: (for ~all~x,y\in Dom~α)~|xα-yα|\leq|x-y|\}$ be the subsemigroup of partial contraction mappings on $[n]$. We have shown that the semigroup $\mathcal{CP}_{n}$ and some of its subsemigroups are nonregular left abundant semigroups for all $n$ but not right abundant for $n\geq 4$.

Motivation & Objective

• To determine the abundance properties (left/right) of the semigroups $\mathcal{CP}_n$, $\mathcal{ORCP}_n$, and $\mathcal{OCP}_n$ of partial contractions on a finite chain $[n]$.
• To characterize the set of strongly regular elements in $\mathcal{ORCP}_n$ and investigate whether they form a subsemigroup.
• To extend prior results on non-regularity of $\mathcal{CP}_n$ and $\mathcal{ORCP}_n$ by identifying their place within the broader class of abundant semigroups.
• To clarify the relationship between convex transversals, admissible transversals, and regularity in the context of partial contractions.

Proposed method

• Characterization of Green’s relations $\mathcal{D}^*$ and $\mathcal{J}^*$ in $\mathcal{CP}_n$, $\mathcal{ORCP}_n$, and $\mathcal{OCP}_n$, showing $\mathcal{D}^* = \mathcal{J}^*$.
• Use of partition-based representation of partial transformations via kernel $\text{Ker}~{}\alpha$ and image $\text{Im}~{}\alpha$ to analyze structure.
• Definition and analysis of convex and admissible transversals of $\text{Ker}~{}\alpha$ to determine regularity conditions.
• Application of Hall’s Proposition 1 to link regularity of products of idempotents with regularity of subsemigroups generated by idempotents.
• Proof that the set of strongly regular elements in $\mathcal{ORCP}_n$ is closed under multiplication, using case analysis on overlapping supports and image intervals.
• Use of the representation $\alpha = \left(\begin{array}{ccc}A_1 & \cdots & A_p \\ x+1 & \cdots & x+p \end{array}\right)$ for idempotents with convex transversals to derive structural constraints.

Experimental results

Research questions

• RQ1 Are the semigroups $\mathcal{CP}_n$, $\mathcal{ORCP}_n$, and $\mathcal{OCP}_n$ left abundant for all $n$?
• RQ2 For which $n$ are these semigroups right abundant, and why do they fail to be right abundant for $n \geq 4$?
• RQ3 Does the set of strongly regular elements in $\mathcal{ORCP}_n$ form a subsemigroup?
• RQ4 What is the role of convex transversals in characterizing strong regularity of partial contractions?
• RQ5 How do the Green’s relations $\mathcal{D}^*$ and $\mathcal{J}^*$ relate in these semigroups?

Key findings

• The semigroups $\mathcal{CP}_n$, $\mathcal{ORCP}_n$, and $\mathcal{OCP}_n$ are left abundant for all $n$, but not right abundant for $n \geq 4$.
• The set of strongly regular elements in $\mathcal{ORCP}_n$, defined as those with a convex transversal, forms a regular subsemigroup.
• The product of two idempotent elements in $SReg(\mathcal{ORCP}_n)$ is strongly regular, as shown through case analysis on overlapping image intervals.
• For $n \leq 3$, the semigroups $\mathcal{CP}_n$, $\mathcal{ORCP}_n$, and $\mathcal{OCP}_n$ are right abundant, but this fails for $n \geq 4$.
• The relation $\mathcal{D}^* = \mathcal{J}^*$ holds in $\mathcal{CP}_n$, $\mathcal{ORCP}_n$, and $\mathcal{OCP}_n$, confirming a structural equivalence in their Green’s relations.
• An idempotent in $SReg(\mathcal{ORCP}_n)$ must have the form $\left(\begin{array}{ccccc}A_1 & a+2 & \cdots & a+p-1 & A_p \\ a+1 & a+2 & \cdots & a+p-1 & a+p \end{array}\right)$ with $\max A_1 = a$ and $\min A_p = a+p$.