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Mathematical structures with an addition operation but not a subtraction operation include commutative semigroups, commutative monoids and semirings.
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In K-theory, the point of departure is to observe that the category of vector bundles on a topological space has a commutative monoid structure under direct sum.
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Most often, the weights used in implementations of Huffman coding represent numeric probabilities, but the algorithm given above does not require this; it requires only that the weights form a totally ordered commutative monoid, meaning a way to order weights and to add them.
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Given this, the structures can be shown to coincide, and the resulting monoid demonstrated to be commutative.
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For any commutative ring R, the category of R-algebras is monoidal with the tensor product of algebras as the product and R as the unit.
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This turns the natural numbers (N, +) into a commutative monoid with identity element 0, the so-called free object with one generator.
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This turns (N*, ×) into a free commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers. Relationship between addition and multiplication Addition and multiplication are compatible, which is expressed in the distribution law: a × (b + c) = (a × b) + (a × c).
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This turns (N*, ×) into a free commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers. Relationship between addition and multiplicationEdit Addition and multiplication are compatible, which is expressed in the distribution law: a × (b + c) = (a × b) + (a × c).
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In contrast to the classical reflexive spaces the class Ste of stereotype spaces is very wide (it contains, in particular, all Fréchet spaces and thus, all Banach spaces), it forms a closed monoidal category, and it admits standard operations (defined inside of Ste) of constructing new spaces, like taking closed subspaces, quotient spaces, projective and injective limits, the space of operators, tensor products, etc. The category Ste have applications in duality theory for non-commutative groups.
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