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Presentation of monoids generated by a projection and an involution

Pascal Caron, Jean-Gabriel Luque, Bruno Patrou

TL;DR

The paper investigates finite monoids generated by a projection and an involution (strict 2-PIMs), showing that any such monoid admits a presentation reducible to a single extra equation beyond the defining relations $\boxempty^2=\boxempty$ and $\Diamond^2=\mathrm{Id}$, and providing a parameterized classification of these equations. It develops a reduction framework, proving that homogeneous and heterogeneous equation systems collapse to canonical forms and establishing a lattice structure over equation-classes that organizes the resulting monoids into distinct families $M_i^s$ with computed sizes and Hilbert-series. The Kuratowski monoid arises as the $eq_{00}(2,2)$ case, illustrating the connection to Kuratowski-type theorems, and the results lay the groundwork for a complete classification, including the future idempotent-idempotent case and broader ties to language theory monoids. The work contributes a structured, lattice-based view of monoids generated by a projection and an involution, clarifying how single additional relations constrain finite models and enabling precise counting and isomorphism distinctions.

Abstract

Monoids generated by elements of order two appear in numerous places in the literature. For example, Coxeter reflection groups in geometry, Kuratowski monoids in topology, various monoids generated by regular operations in language theory and so on. In order to initiate a classification of these monoids, we are interested in the subproblem of monoids, called strict 2-PIMs, generated by an involution and an idempotent. In this case we show, when the monoid is finite, that it is generated by a single equation (in addition to the two defining the involution and the idempotent). We then describe the exact possible forms of this equation and classify them. We recover Kuratowski's theorem as a special case of our study.

Presentation of monoids generated by a projection and an involution

TL;DR

The paper investigates finite monoids generated by a projection and an involution (strict 2-PIMs), showing that any such monoid admits a presentation reducible to a single extra equation beyond the defining relations and , and providing a parameterized classification of these equations. It develops a reduction framework, proving that homogeneous and heterogeneous equation systems collapse to canonical forms and establishing a lattice structure over equation-classes that organizes the resulting monoids into distinct families with computed sizes and Hilbert-series. The Kuratowski monoid arises as the case, illustrating the connection to Kuratowski-type theorems, and the results lay the groundwork for a complete classification, including the future idempotent-idempotent case and broader ties to language theory monoids. The work contributes a structured, lattice-based view of monoids generated by a projection and an involution, clarifying how single additional relations constrain finite models and enabling precise counting and isomorphism distinctions.

Abstract

Monoids generated by elements of order two appear in numerous places in the literature. For example, Coxeter reflection groups in geometry, Kuratowski monoids in topology, various monoids generated by regular operations in language theory and so on. In order to initiate a classification of these monoids, we are interested in the subproblem of monoids, called strict 2-PIMs, generated by an involution and an idempotent. In this case we show, when the monoid is finite, that it is generated by a single equation (in addition to the two defining the involution and the idempotent). We then describe the exact possible forms of this equation and classify them. We recover Kuratowski's theorem as a special case of our study.
Paper Structure (17 sections, 28 theorems, 79 equations, 3 figures, 4 tables)

This paper contains 17 sections, 28 theorems, 79 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The presentation of a strict 2-PIM
  4. Generic cases
  5. Degenerated cases
  6. Equivalent sets of equations
  7. A simple parameterization
  8. Reducing equations
  9. Homogeneous systems
  10. Heterogeneous systems
  11. Theorem of reduction
  12. Lattice structure on sets of strict 2-PIMs
  13. An inclusion diagram
  14. Intersections of the sets $\mathbb{M}_i$
  15. Size of strict 2-PIMs
  16. ...and 2 more sections

Key Result

Lemma 1

For every $k>0$, then $\boxempty=\mathrm{Id}$.

Figures (3)

  • Figure 1: Lattice of the $\mathbb{E}^s_i$ classes
  • Figure 2: Inclusion lattice of the $\mathbb{M}^s_i$ sets
  • Figure 3: Summary of the situation

Theorems & Definitions (58)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 1
  • Definition 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 48 more