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Such algebraic structures occur in several branches of mathematics. The functions from a set into itself form a monoid with respect to function composition. More generally, in category theory , the morphisms of an object to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object.
The ocean plays a key role in the water cycle as it is the source of 86% of global evaporation. [2] The water cycle involves the exchange of energy, which leads to temperature changes. When water evaporates, it takes up energy from its surroundings and cools the environment. When it condenses, it releases energy and warms the environment.
A monoid object in the category of monoids (with the direct product of monoids) is just a commutative monoid. This follows easily from the Eckmann–Hilton argument. A monoid object in the category of complete join-semilattices Sup (with the monoidal structure induced by the Cartesian product) is a unital quantale.
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 ...
Congruence classes and factor monoids are the objects of study in string rewriting systems. A nuclear congruence on S is one that is the kernel of an endomorphism of S. [6] A semigroup S satisfies the maximal condition on congruences if any family of congruences on S, ordered by inclusion, has a maximal element.
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In graph theory, a graph product is a binary operation on graphs. Specifically, it is an operation that takes two graphs G 1 and G 2 and produces a graph H with the following properties: The vertex set of H is the Cartesian product V ( G 1 ) × V ( G 2 ) , where V ( G 1 ) and V ( G 2 ) are the vertex sets of G 1 and G 2 , respectively.
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]