[Paper Review] The Ranks of the Additive Semigroup Reduct of Affine Near-Semiring over Brandt Semigroup

Jitender Kumar, K. V. Krishna|arXiv (Cornell University)|Aug 19, 2013

semigroups and automata theory9 references3 citations

TL;DR

This paper investigates the five rank properties—small, lower, intermediate, upper, and large rank—of the additive semigroup reduct $A^{+}(B_n)$, the semigroup of affine maps over the Brandt semigroup $B_n$. Using structural analysis and prime subset techniques, it determines all five ranks for $n \geq 6$, provides a lower bound for the upper rank $r_4$ when $2 \leq n \leq 5$, and conjectures these bounds are tight.

ABSTRACT

This work investigates the rank properties of $A^+(B_n)$, the additive semigroup reduct of affine near-semiring over Brandt semigroup $B_n$. In this connection, this work reports the ranks $r_1$, $r_2$, $r_3$ and $r_5$ of $A^+(B_n)$ and identifies a lower bound for the upper rank $r_4(A^+(B_n))$. While this lower bound is found to be the $r_4(A^+(B_n))$ for $n \ge 6$, in other cases where $2 \le n \le 5$, the upper rank of $A^+(B_n)$ is still open for investigation.

Motivation & Objective

• To determine the five rank properties—small, lower, intermediate, upper, and large rank—of the additive semigroup reduct $A^{+}(B_n)$ of the affine near-semiring over the Brandt semigroup $B_n$.
• To establish exact values for $r_1, r_2, r_3, r_5$ and a lower bound for $r_4$ in $A^{+}(B_n)$ for all $n \geq 1$.
• To identify the upper rank $r_4(A^{+}(B_n))$ for $n \geq 6$, and provide a lower bound for $r_4$ when $2 \leq n \leq 5$.
• To conjecture that the derived lower bounds for $r_4$ in the cases $2 \leq n \leq 5$ are in fact the exact values of the upper rank.
• To extend the understanding of rank properties in semigroups of affine maps over finite semigroups, particularly over Brandt semigroups.

Proposed method

• Utilizes the structural characterization of $A^{+}(B_n)$ from prior work [9], which describes the semigroup via Green’s relations and element decomposition.
• Applies the concept of prime subsets in finite semigroups: a nonempty subset $U$ is prime if $a + b \in U$ implies $a \in U$ or $b \in U$, and uses Lemma 5.4 to relate minimal prime subsets to maximal proper subsemigroups.
• Identifies $V = \{\xi_{(n,k)} \mid 1 \leq k \leq n-1\}$ as a minimal prime subset of $A^{+}(B_n)$, which enables the computation of the large rank $r_5$.
• Employs decomposition techniques to show that for $n \geq 3$, all elements in $A^{+}(B_n)$ are decomposable, implying no indecomposable elements exist.
• Constructs explicit independent sets to derive lower bounds for the upper rank $r_4(A^{+}(B_n))$ for $2 \leq n \leq 5$, using case analysis on element types: zero, singleton support, $n$-support maps, and full support maps.
• Uses the known formula for $r_5(A^{+}(B_n)) = (n!)n^2 + n^2 + n^4 - n + 3$ to derive bounds and conjecture equality for $r_4$ in small $n$ cases.

Experimental results

Research questions

• RQ1 What are the values of the five rank properties—$r_1, r_2, r_3, r_4, r_5$—for the additive semigroup reduct $A^{+}(B_n)$ of the affine near-semiring over the Brandt semigroup $B_n$?
• RQ2 For which values of $n$ is the upper rank $r_4(A^{+}(B_n))$ exactly determined, and what is the nature of the bound for $r_4$ when $n < 6$?
• RQ3 Can the derived lower bounds for $r_4(A^{+}(B_n))$ for $2 \leq n \leq 5$ be proven to be the exact values of the upper rank?
• RQ4 How do the structural properties of $A^{+}(B_n)$, such as element decomposability and support types, influence the rank computation?
• RQ5 What role do minimal prime subsets play in computing the large rank $r_5(A^{+}(B_n))$?

Key findings

• For $n \geq 6$, the upper rank $r_4(A^{+}(B_n))$ is exactly equal to the lower bound derived from the structure of $A^{+}(B_n)$, which is $ (n!)n^2 + n^2 + n^4 - n + 3 $.
• The large rank $r_5(A^{+}(B_n))$ is exactly $ (n!)n^2 + n^2 + n^4 - n + 3 $ for all $n \geq 2$, derived using the minimal prime subset $V = \{\xi_{(n,k)} \mid 1 \leq k \leq n-1\}$.
• For $n = 2$, the large rank $r_5(A^{+}(B_2))$ is exactly 29, which equals the size of the entire semigroup $A^{+}(B_2)$, indicating all elements are independent.
• For $2 \leq n \leq 5$, a lower bound for the upper rank $r_4(A^{+}(B_n))$ is established: 14 for $n=2$, and $ (n!)n^2 + n $ for $n \geq 3$.
• All elements in $A^{+}(B_n)$ are decomposable for $n \geq 3$, meaning no indecomposable elements exist, which implies that the semigroup is not freely generated.
• The authors conjecture that the derived lower bounds for $r_4(A^{+}(B_n))$ when $2 \leq n \leq 5$ are in fact the exact values of the upper rank, based on structural and constructive evidence.