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Monoids generated by projections

Matthew Fayers

Abstract

We define and explore semireflection monoids on a finite-dimensional vector space. These are monoids generated by semireflections: linear maps fixing a subspace of codimension 1. We mostly focus on the case of projection monoids (where the generating semireflections are non-invertible). After exploring some general theory, we give some important examples, and give classification results for projection monoids on $\mathbb C^2$ and $\mathbb R^3$. We then briefly introduce affine projection monoids.

Monoids generated by projections

Abstract

We define and explore semireflection monoids on a finite-dimensional vector space. These are monoids generated by semireflections: linear maps fixing a subspace of codimension 1. We mostly focus on the case of projection monoids (where the generating semireflections are non-invertible). After exploring some general theory, we give some important examples, and give classification results for projection monoids on and . We then briefly introduce affine projection monoids.

Paper Structure

This paper contains 22 sections, 12 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Semi-reflection mon-oids
  3. Some more definitions
  4. Irreducibility and complete reducibility
  5. Images and kernels
  6. Two examples
  7. Type $A$
  8. Definition, irreducibility and cardinality
  9. Normalising reflections
  10. Minimal generating sets
  11. Type $B$
  12. Definition, irreducibility and cardinality
  13. Normalising reflections
  14. Minimal generating sets
  15. Classification of irreducible projection monoids on $\bbc^2$
  16. ...and 7 more sections

Key Result

lemma 1

Suppose $M$ is a projection monoid on $V$, and $0\ls W\ls V$. Then $W$ is invariant for $M$ for every projection $p\in M$ either $\ker(p)\ls W$ or $W\ls\operatorname{im}(p)$.

Theorems & Definitions (12)

  • lemma 1
  • lemma 2
  • lemma 3
  • lemma 4
  • lemma 5
  • lemma 6
  • lemma 7
  • lemma 8
  • lemma 9
  • lemma 10
  • ...and 2 more