A rectangular band is an algebra with one binary operation satisfying the following equations:
This is the theory of the algebraic effect of reading a single bit from memory. The expression can be thought of as reading the bit, and continuing as if it is 0, and if it is 1. The monad generated by the free rectangular band is the well-known reader monad on bits, defined as [Staton 2013].
Staton, Sam. ‘Instances of Computational Effects: An Algebraic Perspective’. In 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science, 519–519. New Orleans, LA, USA: IEEE, 2013. https://doi.org/10.1109/LICS.2013.58. [pdf]
@inproceedings{staton_instances_2013,
location = {New Orleans, {LA}, {USA}},
title = {Instances of Computational Effects: An Algebraic Perspective},
doi = {10.1109/LICS.2013.58},
pages = {519--519},
booktitle = {2013 28th Annual {ACM}/{IEEE} Symposium on Logic in Computer Science},
publisher = {{IEEE}},
author = {Staton, Sam},
date = {2013},
}