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M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories with Applications to Wreath Products and Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7. Ronald V. Book and Friedrich Otto, String-rewriting Systems, Springer, 1993, ISBN 0-387-97965-4, chapter 7, "Algebraic Properties"
Many definitions and theorems about monoids can be generalised to small categories with more than one object. For example, a quotient of a category with one object is just a quotient monoid. Monoids, just like other algebraic structures, also form their own category, Mon, whose objects are monoids and whose morphisms are monoid homomorphisms. [8]
History monoids are isomorphic to trace monoids (free partially commutative monoids) and to the monoid of dependency graphs. As such, they are free objects and are universal . The history monoid is a type of semi-abelian categorical product in the category of monoids.
Ordinary monoids are precisely the monoid objects in the cartesian monoidal category Set. Further, any (small) strict monoidal category can be seen as a monoid object in the category of categories Cat (equipped with the monoidal structure induced by the cartesian product).
The regular languages over an alphabet A are the closure of the finite subsets of A*, the free monoid over A, under union, product, and generation of submonoid. [6] For the case of concurrent computation, that is, systems with locks, mutexes or thread joins, the computation can be described with history monoids and trace monoids. Roughly ...
Since liquid water flows, ocean waters cycle and flow in currents around the world. Since water easily changes phase, it can be carried into the atmosphere as water vapour or frozen as an iceberg. It can then precipitate or melt to become liquid water again. All marine life is immersed in water, the matrix and womb of life itself. [7]
Example Bjerrum plot: Change in carbonate system of seawater from ocean acidification.. A Bjerrum plot (named after Niels Bjerrum), sometimes also known as a Sillén diagram (after Lars Gunnar Sillén), or a Hägg diagram (after Gunnar Hägg) [1] is a graph of the concentrations of the different species of a polyprotic acid in a solution, as a function of pH, [2] when the solution is at ...
A biogeochemical cycle, or more generally a cycle of matter, [1] is the movement and transformation of chemical elements and compounds between living organisms, the atmosphere, and the Earth's crust. Major biogeochemical cycles include the carbon cycle, the nitrogen cycle and the water cycle. In each cycle, the chemical element or molecule is ...