A *-autonomous category may be described as a linearly distributive category with ( left and right ) negations; such categories have two monoidal products linked with a sort of distributive law.
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In mathematics, a "'near-field "'is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws.
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This very basic form of algebraic reasoning is extended in elementary school to recognize patterns in functions and arithmetic operations, such as the distributive law, a key principle for doing high school algebra.
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Symbols of grouping can be removed using the associative and distributive laws, also they can be removed if the expression inside the symbol of grouping is sufficiently simplified so no ambiguity results from their removal.
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However, there is a natural monad structure on the functor " ST " if there is a distributive law of the monad " S " over the monad " T ".
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In the following argument we will ignore this difficulty, by pretending that " all " lambda expressions are Church numerals . ) The distributive law should apply if the church numerals provided a satisfactory implementation of numbers.
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For instance, if one were to simply require the new operations be associative and obey the distributive law like the first multiplicative operation, then clearly, the set is some sort of " multi-ring ".
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In category theory, if and are category " C ", a "'distributive law "'is a natural transformation such that is a lax map of monads and is a colax map of monads.
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In this case we can use the arithmetic distributive law to interchange some of the multiplications and additions, to reduce the number of multiplications necessary to implement the filter, but the number of multiplications is still proportional to the filter order.
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III . Every citizen will make his particular contribution to the activities of the community according to his capacity, his talent and his age; it is on this basis that his duties will be determined, in conformity with the distributive laws.
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