"Historically the notion of monoid (or of group in particular) was abstracted from the actions, a pivotally important abstraction since as soon as a particular action is constructed or noticed, the demands of learning, development, and use mutate it into: 1) other actions on the same object, 2) actions on other related objects, and 3) actions of related monoids. "
Our foundry used to be much like other foundries. When we cast the first “Model T” cylinders in 1910, everything in the place was done by hand; shovels and wheelbarrows abounded. The work was then either skilled or unskilled; we had moulders and we had labourers.
Now we have about five per cent. of thoroughly skilled moulders and core setters, but the remaining 95 per cent. are unskilled, or to put it more accurately, must be skilled in exactly one operation which the most stupid man can learn within two days. The moulding is all done by machinery.
--- Henry Ford, My Life and Work.
The average NFL career is short — about 3.3 years.
Quarterbacks fare longer, with an average career span of 4.4 years.
https://fivethirtyeight.com/features/the-backup-qb-who-gets-paid-a-lot-to-never-play/
...the most basic questions about a resource theory: When are two resources equivalent under the free operations, in the sense that one can convert one to the other and vice-versa? What are the necessary and sufficient conditions on two resources such that the first can be converted to the second (possibly irreversibly) under the free operations? How can we find measures of the quality of a resource, that is, functions from the resources to the reals which are nonincreasing under the transformations allowed by the resource theory? ... Can a given resource conversion be made possible by the presence of a catalyst, i.e. a resource which must be present but is not consumed in the process?
We refer to this minimal framework as a theory of resource convertibility. The requisite mathematical formalism turns out to be that of a commutative (equivalently, Abelian or symmetric) preordered monoid.
First, consider the preorder. This is the order over resources that captures whether one resource can be converted to another or not. Recall that the first of the questions above called us to characterize the equivalence classes of resources under the free operations, and the second called us to find the partial order over these equivalence classes that is induced by the free operations. But this is nothing more than a call to find the preorder over the resources induced by the free relation of convertibility. Once the preorder is identified, one can define measures of the resource, or “resource monotones”, as any map from the resources to the reals which respects the preorder.
BFS. https://doi.org/10.1016/j.ic.2016.02.008
For every set M we may construct the monoid of endomorphisms (M → M,◦,id), where ◦ is function composition and id is the identity function.
Theorem 1.1 (Cayley representation for (Set) monoids)
Every monoid (M,⊕,e) is a sub-monoid of the monoid of endomorphisms on M.
Exequiel Rivas, Mauro Jaskelioff. 2014.
https://arxiv.org/abs/1406.4823v1
The general idea was that if the altura, or height above the horizon, of the sun or a star (normally the Pole Star) could be determined and compared with the known altura of the port of destination, then the ship could sail north or south until it reached that latitude, and then sail, as appropriate, east or west in the certainty of finding its point of destination. The measurement of the altura was possible with the use of either the quadrant or the astrolabe.
Both devices were standard university instruments of astronomy and astrology that carried a great deal of information that was both unnecessary to the calculation of latitude and simply incomprehensible to the layman. On the back of the astrolabe there was, however, an alidade, which was a rule on a swivel with two pinhole sights. The observer held the instrument upright by a swivel suspension ring, peered along the alidade, and measured the altura of the star by reading off the position of the alidade on a scale marked on the rim ( figure 3 ). The quadrant was an instrument with similar functions.
But instruments, inscriptions, and stars were not enough. Part of the association of elements to convert stars into latitudes lay in the practices of the mariners, and it was this element that was the most prone to distortion. It was difficult, although not ultimately impossible, to create a new social group necessary for closure: the astronomical navigator.
- John Law, ibid
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