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But now we can explain the basis for the multiplicative/additive distinction in the rules for the two different versions of conjunction: for the multiplicative connective (⊗), the context of the conclusion (Γ, Δ) is split up between the premises, whereas for the additive case connective (&) the context of the conclusion (Γ) is carried ...
The distinction between additive and multiplicative identity is used most often for sets that support both binary operations, such as rings, integral domains, and fields. The multiplicative identity is often called unity in the latter context (a ring with unity).
These two basic examples of commutative linear algebraic groups, the multiplicative and additive groups, behave very differently in terms of their linear representations (as algebraic groups). Every representation of the multiplicative group G m {\displaystyle \mathbf {G} _{\mathrm {m} }} is a direct sum of irreducible representations .
Among the examples above the additive, multiplicative groups and the general and special linear groups are affine. Using the action of an affine algebraic group on its coordinate ring it can be shown that every affine algebraic group is a linear (or matrix group), meaning that it is isomorphic to an algebraic subgroup of the general linear group.
If the additive identity and the multiplicative identity are the same, then the ring is trivial (proved below). In the ring M m × n (R) of m-by-n matrices over a ring R, the additive identity is the zero matrix, [1] denoted O or 0, and is the m-by-n matrix whose entries consist entirely of the identity element 0 in R.
The distinction between additive and multiplicative models has significant implications for social and legal understandings of inequality. Additive models can obscure the unique, compounded forms of discrimination faced by those with intersecting marginalized identities, often reducing these experiences to individual components that fail to ...
While approximation algorithms always provide an a priori worst case guarantee (be it additive or multiplicative), in some cases they also provide an a posteriori guarantee that is often much better. This is often the case for algorithms that work by solving a convex relaxation of the optimization problem on the given input.
Totally additive is also used in this sense by analogy with totally multiplicative functions. If f is a completely additive function then f (1) = 0. Every completely additive function is additive, but not vice versa.