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Relational depth of transformation semigroups and their ideals

N. Ruškuc, Z. Yayi

Abstract

We introduce the concept of relational depth of a finite semigroup $S$ whose $J$-classes form a chain. It captures how far down in the ideal structure one is obliged to go in order to define the semigroup by generators and defining relations. We determine the exact value for the relational depth of an arbitrary ideal in the full transformation monoid, symmetric inverse monoid and in the partial transformation monoid.

Relational depth of transformation semigroups and their ideals

Abstract

We introduce the concept of relational depth of a finite semigroup whose -classes form a chain. It captures how far down in the ideal structure one is obliged to go in order to define the semigroup by generators and defining relations. We determine the exact value for the relational depth of an arbitrary ideal in the full transformation monoid, symmetric inverse monoid and in the partial transformation monoid.

Paper Structure

This paper contains 9 sections, 54 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cayley Table Presentation
  4. Lower bound
  5. Upper bound: Preliminaries
  6. Upper bound for $\mathcal{I}_n$
  7. Upper bound for $\mathcal{T}_n$
  8. Upper bound for $\mathcal{PT}_n$
  9. Conclusion

Key Result

Proposition 3.1

Let $S$ be a semigroup whose $\mathcal{J}$-classes form a chain $J_\epsilon<\dots <J_k$. The relational depth of $S$ is $k-i+1$ where $i$ is the largest number for which $\mathcal{C}_i$ defines $S$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (111)

  • proof
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of Proposition \ref{['prop: depth of semigroups same as cayley presentation']}
  • Theorem 3.3
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Lemma 4.4
  • ...and 101 more