Rectangular band

A rectangular band is an algebra X with one binary operation \mathop{?} : X \times X \to X satisfying the following equations:

x \mathop{?} x = x
(x \mathop{?} y) \mathop{?} (a \mathop{?} b) = x \mathop{?} b

This is the theory of the algebraic effect of reading a single bit from memory. The expression x \mathop{?} y can be thought of as reading the bit, and continuing as x if it is 0, and y if it is 1. The monad generated by the free rectangular band is the well-known reader monad on bits, defined as T(X) = \{ 0, 1 \} \to X Staton 2013.

References

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},
}