Historically, the current "standard" set of chess pieces wasn't the only existing alternative or even the standard one. For instance, the famous Al-Suli's Diamond Problem (which remained open for more than one millennium before getting solved by Grandmaster Yuri Averbakh) was formulated in an ancient Persian variant of chess, called Shatranj, using a fairy chess piece, called Wazir (Persian: counsellor), rather than the conventional queen.

There is a long-standing discussion amongst chess players concerning the best possible configuration of chess pieces which makes the game more exciting and complicated. Also, one might be interested in knowing whether, in a fixed position on the infinitary chessboard, the game value could be changed into an arbitrary ordinal merely by replacing the pieces with new (possibly unconventional) ones rather than changing their positions.

In order to address such questions one first needs to have a reasonable mathematical definition of the notion of a "chess piece" in hand.

Maybe a promising approach inspired by Rook, Knight, and King's graphs is to simply consider a chess piece a graph which satisfies certain properties. Though, due to the different nature of all "reasonable" chess pieces, it seems a little bit hard to find principles which unify all of them into one single "neat" definition. For example, some pieces can move only in one direction, some others can jump out of the barriers, some have a/an finite/infinite range, some can only move among positions of a certain color, etc.

Here the following question arises:

Question.What are examples of mathematical papers (or unpublished notes) which present an abstract mathematical definition of a chess piece? Is such a definition unique or there are several variants?

Update 1.In view of Todd and Terry's comments (here and here), it seems a more generalized question could be of some interest. The problem simply is to formulate an abstract mathematical definition of a "game piece" in general. Are there any references addressing such a problem?

Update 2.As a continuation of this line of thought, Joel has asked the following question as well:When is a game tree the game tree of a board game?

every variant of chess, we have rigorous well-defined rules but as a game with a long history, "chess" has many variants with different rules and pieces. For example, there are chess games played on a three-dimensional chessboard! (The theory of infinite three-dimensional chess has been studied in some of Joel Hamkins' works. See page 443 of this slides for more information: jdh.hamkins.org/wp-content/uploads/2014/08/… ). I don't see how and why one can consider one of them the standard version and the rest just "non-chess". $\endgroup$ – Morteza Azad Dec 7 '17 at 12:23