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On a Question of Frege's about Right‐Ordered Groups

On a Question of Frege's about Right‐Ordered Groups ON A QUESTION OF FREGE'S ABOUT RIGHT-ORDERED GROUPS S. A. ADELEKE, M. A. E. DUMMETT AND PETER M. NEUMANN Dedicated with respect and good wishes to Graham Higman to mark his seventieth birthday, January 1987 1. The problem and its background Let G be a group and P a subset of G. We shall be interested in the following assertions: (1) (2) \$P; (3) x x (4) p,qeP=>p~qePy p = qy q~ peP. Frege's question, which is posed in [5], is whether (4) follows from (1), (2) and (3). We shall amplify the question and answer it negatively below. A binary relation < on G may be defined in terms of P by the rule a < b if and only if ba~ e P. It is easy to see that < is right-invariant (that is, a < b => ag < bg for all geG) and P = {peG\ 1 <p). Properties of < may be characterised either directly or in terms of P. For example, as is well known and easy to prove, < is a strict partial order if and only if (1) and (2) hold. A binary relation p on a set A will be http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

On a Question of Frege's about Right‐Ordered Groups

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/19.6.513
Publisher site
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Abstract

ON A QUESTION OF FREGE'S ABOUT RIGHT-ORDERED GROUPS S. A. ADELEKE, M. A. E. DUMMETT AND PETER M. NEUMANN Dedicated with respect and good wishes to Graham Higman to mark his seventieth birthday, January 1987 1. The problem and its background Let G be a group and P a subset of G. We shall be interested in the following assertions: (1) (2) \$P; (3) x x (4) p,qeP=>p~qePy p = qy q~ peP. Frege's question, which is posed in [5], is whether (4) follows from (1), (2) and (3). We shall amplify the question and answer it negatively below. A binary relation < on G may be defined in terms of P by the rule a < b if and only if ba~ e P. It is easy to see that < is right-invariant (that is, a < b => ag < bg for all geG) and P = {peG\ 1 <p). Properties of < may be characterised either directly or in terms of P. For example, as is well known and easy to prove, < is a strict partial order if and only if (1) and (2) hold. A binary relation p on a set A will be

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Nov 1, 1987

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