Who first asymmetric relations qua relation, viz., the fact that
A relates to B by some relation
does not always/necessarily imply that
B relates to A by the same relation.
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The Categories, chap.7 /On Relatives/ contains a remarkable discussion in just a few pages and its author, supposedly Aristotle, might well be the first to have 'studied' asymmetric relations. So Aristotle distinguishes contrariety and reciprocation, adding further consideration on simultaneity.
All relatives are 'reciprocated', but no all of them have 'contraries'; simultaneity is optional.
7a22 All relatives, then, if properly defined, have a correlative.
this is the conclusion of a short discussion but earlier it is noted:
6b16 virtue has a contrary, vice, these both being relatives; knowledge, too, has a contrary, ignorance. But this is not the mark of all relatives; 'double' and 'triple' have no contrary, nor indeed has any such term.
So it seems that logical inversion generates symmetrical relations while the rest do not have this property. The contrary of ignorance is knowledge and its contrary is ignorace. 'Master of' and 'slave of' are reciprocal.
Ref: Pamela M. Hood, Aristotle on the Category of Relation 2004
William Rowan Hamilton's discovery of quaternions in the 19th century may be the first studied non-commutative relations. Here is Wikipedia:
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quaternions is that multiplication of two quaternions is noncommutative.
Others may have anticipated this discovery:
Important precursors to this work included Euler's four-square identity (1748) and Olinde Rodrigues' parameterization of general rotations by four parameters (1840), but neither of these writers treated the four-parameter rotations as an algebra. Carl Friedrich Gauss had also discovered quaternions in 1819, but this work was not published until 1900.
One may also look at the symmetry of subject and predicate in the syllogism. According to Irving Copi if one can interchange subject and predicate and maintain validity this would be an immediate inference: (page 190)
It is called conversion and is perfectly valid in the case of E and I propositions.
Identifying where conversion does not work could be viewed as a study of assymetry between subject and predicate.
Copi, I. M. Introduction to Logic. 1982. Macmillan.
Wikipedia contributors. (2019, June 16). Quaternion. In Wikipedia, The Free Encyclopedia. Retrieved 20:09, July 5, 2019, from https://en.wikipedia.org/w/index.php?title=Quaternion&oldid=902089865