### Abstract

We use constructions in monoid and group theory to exhibit an adjunction between the category of partially ordered monoids and lazy monoid homomorphisms and the category of partially ordered groups and group homomorphisms such that the unit of the adjunction is injective. We also prove a similar result for sets acted on by monoids and groups. We introduce the new notion of a lazy homomorphism for a function f between partially ordered monoids such that f(m circle m')

Every monoid can be endowed with the discrete partial ordering (m

Informally, but perhaps informatively, we can describe this work as follows: we give an abstract analysis of how we can sensibly add 'undo' (in the sense of 'control-Z').

Original language | English |
---|---|

Pages (from-to) | 1002-1031 |

Number of pages | 30 |

Journal | Mathematical Structures in Computer Science |

Volume | 23 |

Issue number | 5 |

DOIs | |

Publication status | Published - Oct 2013 |

### Cite this

*Mathematical Structures in Computer Science*,

*23*(5), 1002-1031. https://doi.org/10.1017/S0960129512000849

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*Mathematical Structures in Computer Science*, vol. 23, no. 5, pp. 1002-1031. https://doi.org/10.1017/S0960129512000849

**Imaginary groups : lazy monoids and reversible computation.** / Gabbay, Murdoch J; Kropholler, Peter H.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Imaginary groups

T2 - lazy monoids and reversible computation

AU - Gabbay, Murdoch J

AU - Kropholler, Peter H

PY - 2013/10

Y1 - 2013/10

N2 - We use constructions in monoid and group theory to exhibit an adjunction between the category of partially ordered monoids and lazy monoid homomorphisms and the category of partially ordered groups and group homomorphisms such that the unit of the adjunction is injective. We also prove a similar result for sets acted on by monoids and groups. We introduce the new notion of a lazy homomorphism for a function f between partially ordered monoids such that f(m circle m')Every monoid can be endowed with the discrete partial ordering (mInformally, but perhaps informatively, we can describe this work as follows: we give an abstract analysis of how we can sensibly add 'undo' (in the sense of 'control-Z').

AB - We use constructions in monoid and group theory to exhibit an adjunction between the category of partially ordered monoids and lazy monoid homomorphisms and the category of partially ordered groups and group homomorphisms such that the unit of the adjunction is injective. We also prove a similar result for sets acted on by monoids and groups. We introduce the new notion of a lazy homomorphism for a function f between partially ordered monoids such that f(m circle m')Every monoid can be endowed with the discrete partial ordering (mInformally, but perhaps informatively, we can describe this work as follows: we give an abstract analysis of how we can sensibly add 'undo' (in the sense of 'control-Z').

U2 - 10.1017/S0960129512000849

DO - 10.1017/S0960129512000849

M3 - Article

VL - 23

SP - 1002

EP - 1031

JO - Mathematical Structures in Computer Science

JF - Mathematical Structures in Computer Science

SN - 0960-1295

IS - 5

ER -