'On one's half' definitions:

Definition of 'On one's half'

From: GCIDE

Half \Half\ (h[aum]f), n.; pl. Halves (h[aum]vz). [AS. healf. See Half, a.]
1. Part; side; behalf. [Obs.] --Wyclif. [1913 Webster]
The four halves of the house. --Chaucer. [1913 Webster]
2. One of two equal parts into which anything may be divided, or considered as divided; -- sometimes followed by of; as, a half of an apple. [1913 Webster]
Not half his riches known, and yet despised. --Milton. [1913 Webster]
A friendship so complete Portioned in halves between us. --Tennyson. [1913 Webster]
Better half. See under Better.
In half, in two; an expression sometimes used improperly instead of in halves or into halves; as, to cut in half. [Colloq.] --Dickens.
In one's half or On one's half, in one's behalf; on one's part. [Obs.]
To cry halves, to claim an equal share with another.
To go halves, to share equally between two. [1913 Webster]