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Partial groups as partial groups

Philip Hackney, Justin Lynd, Edoardo Salati

Abstract

There are many examples of `binary' partial groups in the literature: sets equipped an identity and a partially-defined binary operation, such that each element admits an inverse. We show that many of these may be regarded as partial groups in the sense of Chermak, and single out the largest class of such objects.

Partial groups as partial groups

Abstract

There are many examples of `binary' partial groups in the literature: sets equipped an identity and a partially-defined binary operation, such that each element admits an inverse. We show that many of these may be regarded as partial groups in the sense of Chermak, and single out the largest class of such objects.
Paper Structure (4 sections, 8 theorems, 7 equations)

This paper contains 4 sections, 8 theorems, 7 equations.

Table of Contents

  1. Chermak partial groups
  2. Binary partial groups
  3. The big embedding of binary partial groups
  4. The small embedding of binary partial groups

Key Result

Lemma 3

In a binary partial group, $\dagger$ is an involutive anti-isomorphism, i.e. $(a^\dagger)^\dagger = a$ and $(ab)^\dagger \asymp b^\dagger a^\dagger$.

Theorems & Definitions (21)

  • Definition 1
  • Remark 2
  • Lemma 3
  • proof
  • Example 4
  • Remark 5
  • Lemma 6
  • proof
  • Theorem 7
  • proof
  • ...and 11 more